SAVING VERIFICATIONISM (AND THE PROBLEM OF INDUCTION)##

Draft for the book Philosophical Semantics 

Chapter V

Verificationism Redeemed

 

 

 

There is no distinction of meaning so fine as to consist in anything but a possible difference in practice.

—C. S. Peirce

 

Es ist schwer einem Kurzsichtigen einen Weg zu beschreiben. Weil man ihm nicht sagen kann: ‘Schau auf dem Kirchturm dort 10 Meilen von uns und geh’ in dieser Richtung.

[It is difficult to tell a near-sighted man how to get somewhere. Because you cannot say to him: ‘Look at the Church tower ten miles away and go in that direction.’]

—Wittgenstein

 

Verificationism is now commonly viewed as a relic of philosophy as practiced in the first half of the 20^th century. Although initially advocated by members of the Vienna Circle, it soon proved unable to withstand an ever expanding range of opposing arguments, which came from both within and outside of the Circle. My aim in this chapter is to show that we can achieve an understanding of verifiability that is both intuitively acceptable and resistant to the most widespread objections. In my view, the Vienna Circle failed to successfully defend verificationism because it used the wrong approach of beginning by formally clarifying the principle of verification initially proposed by Wittgenstein without paying sufficiently detailed attention to what we really do when we verify statements. When their arguments in favor of the principle were shown to be faulty, most of them, along with their offspring, unwisely concluded that the principle itself should be rejected. In my view, they were exhibiting the same reaction of the proverbial fox in Aesop’s fable: unable to reach the grapes, he consoled himself by imagining they were sour...

     Returning to the methodology and assumptions of the later Wittgenstein, my aim in this chapter is twofold: first to sketch a plausible version of what I call semantic verificationism, which consists in the proposal that the epistemic contents of declarative sentences, that is, the e-thought-contents or propositions expressed by them are constituted by their verifiability rules; second, to confirm and better explain semantic verificationism by answering the main counter-arguments.

1. Origins of semantic verificationism

The first point to be remembered is that, contrary to a mistaken popular belief, the idea that a sentence’s meaning is its method of verification didn’t stem from the logical positivists. The first to propose the principle was actually Wittgenstein himself, as members of the Vienna Circle always acknowledged (Cf. Glock: 354). Indeed, if we review his works, we see that he formulated the principle in 1929 conversations with Waismann and referred to it repeatedly in texts over the course of the following years. Furthermore, there is no solid evidence that he abandoned the principle later, replacing it with a merely performative conception of meaning as use, as some have argued. On the contrary, there is clear evidence that from the beginning his verificationism and his subsequent thesis that meaning is a function of use seemed mutually compatible to him. After all, Wittgenstein did not hesitate to conflate the concept of meaning as verification with meaning as use and even with meaning as calculus. As he said:

If you want to know the meaning of a sentence, ask for its verification. I stress the point that the meaning of a symbol is its place in the calculus, the way it is used.[1] (2001: 29)

It is always advisable to check what the original author of an idea really said. If we compare Wittgenstein’s verificationism with the Vienna Circle’s verificationism, we can see that there are some striking contrasts. A first one is that Wittgenstein’s main objective with the principle always seems to have been to achieve a grammatical overview (grammatische Übersicht), that is, to clarify central principles of our factual language, even if this clarification could be put at the service of therapeutic goals. On the other hand, he was always against the positivistic-scientistic spirit of the Vienna Circle, which in its incipient and precocious desire to develop a purely scientific philosophy had as the strongest motivation to develop the verification principle to use it as a powerful reductionist weapon, able to vanquish once and for all the fantasies of metaphysicians. Wittgenstein, for his part, didn’t reject metaphysics in this way. For him, the metaphysical urge was a kind of unavoidable dialectical condition of philosophical inquiry, and the truly metaphysical mistakes have the character of depth (Wittgenstein 1984c sec. 111, 119). Consequently, metaphysical errors were intrinsically necessary for the practice of philosophy as a whole. As he wrote:

The problems arising through a misinterpretation of our forms of language have the character of depth. They are deep disquietudes; their roots are as deep in us as the forms of our language and their significance is as great as the importance of our language. (1984c, sec. 111)

It was this rejection of positivistic-scientistic reductionism that gradually estranged him from the Logical Positivists.

     In these aspects, Wittgenstein was much closer to that great American philosopher, C. S. Peirce. According to Peirce’s pragmatic maxim, metaphysical deception can be avoided when we have a clearer understanding of our beliefs. This clarity can be reached by understanding how these beliefs are related to our experiences, expectations and their consequences. Moreover, the meaning of a concept-word was for Peirce inherent in the totality of its practical effects, the totality of its inferential relations with other concepts and praxis. So, for instance, a diamond, as the hardest material object, can be partially defined as something that scratches all other material objects, but cannot be scratched by any of them.

     Moreover, in contrast to the positivists, Peirce aimed to extend science to metaphysics, instead of reducing metaphysics to science.[2] So, he was of the opinion that verifiability – far from being a weapon against metaphysics – should be elaborated in order to be applicable to it, since the aim of metaphysics is to say extremely general things about our empirical world. As Peirce wrote:

But metaphysics, even bad metaphysics, really rests on observations, whether consciously or not; and the only reason that this is not universally recognized is that it rests upon kinds of phenomena with which every man’s experience is so saturated that he usually pays no particular attention to them. The data of metaphysics are not less open to observation, but immeasurably more so than the data, say, of the very highly observational science of astronomy… (1931, 6.2)[3]

Although overall Peirce’s views were as close to Wittgenstein’s as those of both were distant from the logical positivists and their theories, there is an important difference between both philosophers concerning the analysis of meaning. Peirce was generally interested in the connection between our concepts and praxis, including their practical effects, as a key to conceptual clarification and a better understanding of their meaning. But by proceeding in this way he risked extending the concept of meaning too far; he took a path that can easily lead us to confuse the cognitive and practical effects of meaning with meaning itself. For as we already saw, the cognitive meaning of a declarative sentence, seen as a combination of semantic-cognitive rules, works as a condition for the production of inferential awareness, which consists in the kind of systemic openness (allowing the ‘propagation of content’) that can produce an indeterminate number of subsequent mental states and actions.[4] Meaning as a verifiability rule is one thing; awareness of meaning and inferences that may result from this awareness, together with the practical effects of such inferences, may be a very different thing. Though they can be partially related, they should be distinguished. Hence, within our narrow form of inferentialism, we first have the inferences that construct meanings (like those of the identification rules of singular terms, the ascription rules of predicates, and the verifiability rules of sentences); then we have something usually beyond cognitive meaning, namely, the multiple inferences that enable us to gain something from our knowledge of meaning, along with the multiplicity of behavioral and practical effects that may result from them. Without this separation, we may even have a method that helps us clarify our ideas, but we will lack a boundary that can prevent us from extending the meanings of our expressions beyond a reasonable limit. For instance, the fact that something cannot be scratched helps to verify the assertion ‘This is a diamond’ (the hardest material), whereas the use of diamonds as abrasives will certainly be of little if any relevance for the explanation of the assertion’s meaning. This is why I think that Wittgenstein, restricting cognitive meaning to a method of verification, that is, to combinations of semantic rules able to make a proposition true, proposed a more adequate view of cognitive meaning and its truth.

     Looking for a better example, consider the statement: (i) ‘In October 1942 Chil Rajchman was arrested, put on a train, and deported to Treblinka.’ This promptly leads us to the inference: (ii) ‘Chil Rajchman died in a death camp.’ However, his probable fate would not be part of the verifiability procedure of (i), but rather of the verification of statement (ii). Thus, although (ii) is easily considered a consequence of (i), its thought-content isn’t a real constituent of the cognitive meaning, the thought-content-rule expressed by (i). Statement (ii) has its own verifiability procedures, even if its meaning is strongly associated with that of statement (i) since it is our main reason for being interested in this last statement. So, we could say that considering a statement S, there is something like a cloud of meanings surrounding its cognitive meaning, this cloud being formed by inferentially associated cognitive meanings of other statements with their own verifiability rules. But it is clear that this cloud of meaning does not properly belong to the cognitive meaning of S and should not be confused with it. In short: only by restricting ourselves to the constitutive verifiability procedures of a chosen statement are we able to restrict ourselves to the proper limits of its cognitive meaning.

     Opposition to a reductionist replacement of metaphysics by science was also one reason why Wittgenstein didn’t bother to make his principle formally precise, unlike positivist philosophers from A. J. Ayer to Rudolph Carnap. In saying this, I am not rejecting formalist approaches. I am only warning that such undertakings, if not well supported by a sufficiently careful pragmatic consideration of how language really works, tend to put the logical cart before the semantic horse. In this chapter, I want to show how unwise neglect of some very natural conceptual intuitions has frustrated most attempts by positivist philosophers to defend their own principle.

     Having considered these differences, I want to start by examining some of Wittgenstein’s remarks regarding the verifiability principle, in order to find a sufficiently adequate and reasonably justified formulation. Afterward, I will answer the main objections against the principle, demonstrating that they are much weaker than they seem at first glance.

2. Wittgensteinian semantic verificationism

Here are some of Wittgenstein’s statements presenting the verifiability principle:

 

Each sentence (Satz) is a signpost for its verification. (1984e: 150)

 

A sentence (Satz) without any way of verification has no sense (Sinn). (1984f: 245)

 

If two sentences are true or false under the same conditions, they have the same sense (even if they look different). (1984f: 244)

 

To understand the sense of a sentence is to know how the issue of its truth or falsity is to be decided. (1984e: 43)

 

Determine under what conditions a sentence can be true or false, then determine thereby the sense of the sentence. (This is the foundation of our truth-functions.) (1984f: 47)

 

To know the meaning of a sentence, we need to find a well-defined procedure to see if the sentence is true. (1984f: 244)

 

The method of verification is not a means, a vehicle, but the sense itself. Determine under what conditions a sentence must be true or false, thus determine the meaning of the sentence. (1984f: 226-7)

 

The meaning of a sentence is its method of verification. (1980: 29)[5]

 

What calls attention to statements like these is their strongly intuitive appeal: they seem to be true. They satisfy our need for a methodological starting point that accords with our common knowledge beliefs. To a great extent, they even seem to corroborate Wittgenstein’s controversial view, according to which philosophical theses should be ultimately trivial because they do no more than make explicit what we already know. They are what he would call ‘grammatical sentences’ expressing the rules grounding the linguistic practices that constitute our factual language. In the end the appeal to meaning verificationism involves what we might call a ‘transcendental argument’: we cannot conceive a different way to analyze the cognitive meaning of a declarative sentence, except by appealing to verifiability; hence, if we assume that cognitive meaning is analyzable, some form of semantic verificationism must be right.

     There are some points we can add. The first is terminological and was already extensively discussed in this book: we should not forget that the verifiability rule must be identified with the cognitive content of a declarative sentence. This cognitive content is what we could call, remembering our reconstruction of Frege’s semantics, the e-thought-content-rule expressed by the declarative sentence (being also called the descriptive, informative or factual content of the sentence, if not its proposition or propositional content). A complementary point, already noted, is that we should never confuse cognitive content with grammatical meaning. If you do not know who Tito and Baby are, you cannot understand the cognitive meaning of the sentence ‘Tito loves Baby,’ even if you are already able to understand its gram­matical meaning.

     Another point to be emphasized is that the verifiability rule correctly understood as e-thought-content or proposition must include both, the verification and the falsification of the statement, since this rule can in itself be either true or false.[6] Wittgenstein was explicit about that: ‘The method of verification is not a means, a vehicle, but the sense itself’ (1984f: 226-7). The reason is easy to see: the verifiability e-thought rule either applies to the verifier as such – the truth-maker, which in the last chapter we usually and unequivocally identified with some cognitively independent fact in the world – which verifies the rule – or it does not apply to any expected verifier or fact in the world – which falsifies the rule. Consider, for example, the statement ‘Frege was bearded.’ Here the verifiability e-thought rule applies to a circumstantial fact that the rule is intended to apply to in a world that makes the rule effectively applicable, which means that the verifiability e-thought rule expressed by the statement is true. Consider, by contrast, the statement ‘Wittgenstein was bearded’: here the verifiability e-thought rule does not apply to the intended contextual fact in the world, since this fact does not exist, and that falsifies the statement. But then it is because the verifiability rule expressed by this statement is false, since it is inapplicable.

     A final point concerns the reading of Wittgenstein’s distinction between the verification of a sentence (Satz) and of a hypothesis (Hypothese), which he made in the obscure last chapter of his Philosophical Remarks. As he wrote:

 

A hypothesis is a law for the building of sentences. One could say: a hypothesis is a law for the building of expectations. A sentence is, so to speak, a cut in our hypothesis in a certain place. (1984e XXII, sec. 228)

 

In my understanding, the hypothesis is distinguished here mainly by being more distant from sensory-perceptual experience than what he calls a sentence. As a consequence, only the verification of a sentence (statement) is able to give us certainty. However, this does not mean that the verification of this sentence is infallible. Hence, when Wittgenstein writes that we can verify the truth of the sentence ‘Here is a chair’ by looking only at one side of the chair (1984e, Ch. XXII sec. 225), it is clear that we can increase our degree of certainty by adding new facets, aspects, modes of presentation, sub-facts. We could, e.g., look at the chair from other angles, or make tests to show what the chair consists of, whether it is solid enough to support a person, etc.

     Thus, my take is that what he calls the certainty of a sentence is only postulated as such after we consider it sufficiently verified in the context of some linguistic practice. This is why things can be seen as certain and yet remain fallible, as practical certainties. By contrast, the verification of hypotheses, like sentences stating scientific laws, as this is realized only derivatively, gives us comparatively lower degrees of probability, though they can also be assumed as true.

3. Verifiability rule as a criterial rule

A more important point emphasized by Wittgenstein and ignored by others is that we usually have a choice of ways to verify a statement, each way constituting some different, more or less central aspect of its meaning. As he noted:

Consideration of how the meaning of a sentence is explained makes clear the connection between meaning and verification. Reading that Cambridge won the boat race, which confirms that ‘Cambridge won,’ is obviously not the meaning, but is connected with it. ‘Cambridge won’ isn’t the disjunction ‘I saw the race or I read the result or...’ It’s more complicated. But if we exclude any of the means to check the sentence, we change its meaning. It would be a violation of grammatical rules if we disregarded something that always accompanied a meaning. And if you dropped all the means of verification, it would destroy the meaning. Of course, not every kind of check is actually used to verify ‘Cambridge won,’ nor does any verification give the meaning. The different checks of winning the boat race have different places in the grammar of ‘winning the boat race.’ (2001: 29)

Moreover:

All that is necessary for our sentences to have meaning is that in some sense our experience would agree with them or not. That is: the immediate experience should verify only something of them, a facet. This picture is taken immediately from reality because we say ‘This is a chair’ when we see only a side of it. (1984f: 282, my italics)

In other words: one can verify through the direct observation of facts, that is, by seeing a Cambridge racing boat winning a race or by hearing the judge’s confirmation, or both. These forms of verification are central to the meaning of ‘Cambridge won the boat race.’ It is worth remembering that even this direct observation of the fact is aspectual: each person at the boat race saw the fact from a different perspective, i.e., they saw and heard different sub-facts: different aspects (facets) of the same event. However, we also say that they all did see the grounding fact in the sense that they inferred its totality in the most direct way possible; this is why we can say that the fact-event of Cambridge winning, as a grounding fact, was also directly (that is, in the most direct possible way) experienced. In the same way, we are allowed to say that we see a ship on the sea (the inferred grounding fact), while what we phenomenally see is only one side of a ship (a given aspectual sub-fact).

     However, often enough the way we can know the truth-value of a thought-content like that expressed by the sentence ‘Cambridge won the boat race’ is more indirect: someone can tell us, we can read this in the internet or in a magazine or we can see a trophy in the clubhouse… These ways are secondary, and for Wittgenstein they participate only secondarily in the sentence’s meaning. Finally, they are causally dependent on more direct ways of knowing the truth-value, which are primary verifying criteria. If these first ways of verification did not exist, these dependent forms, being secondary criteria or mere symptoms, would lose their reliability and validity.

     We can say that the verifiability rule applies when we achieve awareness of a fact, which means that we are in a position that allows us to make the relevant inferences from our factual knowledge. This awareness is the most direct when the criterial configuration (a configuration of p-properties or tropes) that satisfies the verifiability rule is at last partially constitutive of the grounding fact, for instance, when we observe a competition being won. But more often verification is indirect, namely, by means of secondary criteria or symptoms, often making the verifiability e-thought-content rule probably or even very probably true.

     Criteria tend to be displayed in the form of criterial configurations, and such conditions can vary indeterminately. Thus, the verifiability rule is said to apply when a criterial configuration demanded by the semantic-cognitive criterial rule is objectively given as belonging to objective facts as their constitutive tropical combinations and arrangements. Furthermore, concerning a basal e-thought-content, also a criterial rule seems to have as a minimum condition for satisfaction of some kind of structural isomorphism between, on the one hand, the interrelated internal elements originating as constituents of the thought-content-criterial-rule, and, on the other hand, the interrelated objective elements (objective tropical combinations) that make up the grounding fact in the world. This is what would constitute the isomorphism with the grounding fact. Since experience is always aspectual and often indirect, this also means that the dependent criterial configurations belonging to the rule must also show a structural isomorphism with aspectual configurations of independent or external criterial arrangements of tropes (given in the world and experienced by the epistemic subject). This generates what we could call isomorphic relations with a sub-fact (say, a ship on the sea seen from one side), and enables us to infer the whole grounding fact (say, a whole ship on the sea). I expect to say more about this complicated issue in the last chapter.[7]

     As this reconstruction of Wittgenstein’s views shows, a sentence’s meaning should be constituted by a verifiability rule that usually ramifies itself, requiring the actual or possible fulfillment of a multiplicity of criterial configurations, allowing us to infer facts in more or less direct ways. Hence, there are definitional criterial configurations (primary criteria) such as, in Wittgenstein’s example, those based on direct observation by a spectator at a boat race. But there are also an indefinite number of secondary criterial configurations depending on the first ones. They are secondary criteria or even symptoms, allowing us to infer that Cambridge (more or less probably) won the boat race, etc. Here too, we can say that the primary criteria have a definitional character: once we accept them as really given and we can agree on this, our verifiability rule should apply with practical certainty by defining the arrangement of tropes (fact) accepted as given. Moreover, we can treat secondary criteria (like reading about an event in a magazine) as less certain, though still very probable, while symptoms (like having heard about the event) make the application of a verifiability rule only more or less probable. Thus, if an unreliable witness tells us that Cambridge won, we can conclude that it is probable that Cambridge won. However, what makes this probability acceptable is, as we noted, that we are assuming it is backed by some observation of the fact by competent judges and eye-witnesses, that is, by primary criterial configurations.

     Investigating the structure of verifiability rules has some consequences for the much-discussed traditional concept of truth-conditions. The truth-condition of a statement S can be defined as the condition sufficient for its e-thought-content-rule actually be the case. The truth condition for the statement ‘Frege had a beard’ is the condition that he actually did have a beard. This means that the truth-condition of S is the condition that a certain fact can be given as S’s truth-maker, that is, as satisfying the verifiability rule for S. The given truth-maker, the fact, is an objective actualization of the truth-condition. Thus, the so-called ‘realist’ view (in Michael Dummett’s sense) is mistaken, since according to it a truth-condition of a statement could possibly be given without at least some conception of criterial configurations (tropical configurations that would possibly warrant its existence), and its related verifiability e-thought rules could to some extent be at least conceivable.

     Now, considering our analysis of the identification rules of proper names (Appendix of Chapter I) and of the ascription rules of predicative expressions (Ch. II, sec. 6), we can consider the verifiability rule of a singular predicative statement to be a combination of both in a more explicit way. We can get an idea of this by examining a very simple predicative statement: ‘Aristotle was bearded.’ For this we have first as the definitional identification rule for Aristotle the same rule already presented at the beginning[8]:

 

IR-Aristotle: The name ‘Aristotle’ is applicable iff its bearer is the human being who sufficiently and more than any other person satisfies the condition(s) of having been born in Stagira in 384 BC, son of Phillip’s court physician, lived the main part of his life in Athens and died in Chalcis in 322 BC and/or was the philosopher who developed the main ideas of the Aristotelian opus. (Auxiliary descriptions may be helpful, though they do not belong properly to the definition…)

 

And for the predicative expression ‘…was bearded’ we may formulate the following definitional ascription rule:

 

AR-bearded: The predicate ‘…is bearded’ is ascribable iff its bearer is a human being who has the tropes (properties) of facial hair growth on the chin and/or cheeks and/or neck.

 

Now, as we already know, we first apply the identification rule of the singular term in order to identify the object, subsequently applying the ascription rule of the general term by means of which we select the tropical cluster of the object identified by the first rule. Not only are there many possible ways in which the identification rule and the ascription rule can be satisfied, there are still more ways of verification for the whole e-thought-content expressed by ‘Aristotle was bearded.’ One of them is by examining the well-known marble bust of Aristotle preserved in Athens, another is by accepting the recorded testimony of his contemporaries that has come down to us, and still another is by learning that most ancient Greeks (particularly among the peripatetics) customarily wore beards as a badge of manhood. All this makes possible the satisfaction of AR-bearded for that human being (the criterial configurations on the chin and cheeks are satisfied), in addition to the satisfaction of IR-Aristotle. As we noted, we assume this criterially-based verification as practically certain, which allows us to say we know that Aristotle was bearded, even if we are aware that this is only indirectly established as highly probable. We can summarize the applicability (or judgment or truth-attribution) of the basal e-thought-content verifiability rule to the grounding fact that Aristotle was bearded by means of the following schema:

 

├ [[IR-Aristotle is applicable to its bearer]AR-bearded is applicable to this same bearer].

 

These brief comments on verificationism à la Wittgenstein suggest the need for more intensive pragmatic research on ways of verification. As we noted, the structure of a verifiability rule is normally ramified, and its details should vary in accordance with the kind of statement that expresses it. A detailed pragmatic investigation of diversified forms of verifiability rules seems to me an important task that as far as I know has not been attempted until now. In what follows, I will not try to correct this limitation. I will restrict myself to answering the main objections to the verifiability principle, showing that they are products of misunderstanding.

4. Objection 1: The principle is self-refuting

The first and most notorious objection to the principle of verifiability is that it is self-defeating. The argument runs as follows. The principle of verifiability must be either analytic or synthetic.[9] If it is analytic it must be tautological, that is, non-informative. However, it seems clearly informative in its task of elucidating cognitive meaning. Furthermore, analytic statements are self-evident, and denying them is contradictory or inconsistent, which does not seem to be the case with the principle of verifiability. Therefore, the principle is synthetic. But if it is synthetic, it needs to be verifiable in order to have meaning. Yet, when we try to apply the principle of verifiability to itself we find that it is unverifiable. Hence, the principle is metaphysical, which implies that it is devoid of meaning. The principle is meaningless by its own standards; and one cannot evaluate meaningfulness based on something that is itself meaningless.

     Logical positivists tried to circumvent that objection by responding that the principle of verifiability has no truth-value, for it is nothing more than a proposal, a recommendation, or a methodological requirement.[10] A. J. Ayer advocated this view by challenging his readers to suggest a more persuasive option (1992: 148). However, a reader with the opposite convictions could respond that he simply doesn’t feel the need to accept or opt for anything of the kind... Moreover, the thesis that the principle is only a proposal appears to be clearly ad hoc. It goes against Wittgenstein’s assumption that all we are doing is exposing the already given intuitions underlying our natural language, the general principles embedded in it. Consequently, to impose on our language a methodological rule that does not belong to it would be arbitrary and misleading as a means of clarifying meaning.[11]

     My suggestion is simply to keep Wittgenstein’s original insight, according to which a principle of verifiability is nothing but a very general grammatical sentence stating the way all our factual language must work to have cognitive content to which a truth-value can be assigned. Once we understand that the principle should make our pre-existing linguistic dispositions explicit, we are entitled to think that it must be seen as an analytic-conceptual principle. More precisely, this principle would consist in the affirmation of a hidden synonymy between the phrases ‘meaning as the cognitive content (e-thought-content-rule or proposition) expressed by a declarative sentence’ and ‘the procedures (combinations of rules) by which we may establish the truth-value of this same cognitive content.’ Thus, taking X to be any declarative sentence, we can define the cognitive value of X by means of the following analytic-conceptual sentence stating the verifiability principle:

 

VP (Df.): Cognitive meaning (e-thought-content) of a declarative sentence X = the verifiability rule for X.

 

Against this, a critic can react by saying that this claim to analytic identity lacks intuitive evidence. Moreover, if the principle of verifiability were analytic, it would be non-informative, its denial being contradictory or incoherent. However, it appears that VP says something to the effect that in principle it can be denied. It seems at least conceivable that the cognitive meaning of statement X, the thought-content expressed by it, isn’t a verifiability rule.

     My reaction to this objection is to recall that an analytic sentence does not need to be transparent; it does not need to be immediately seen as necessarily true, and its negation does not need to be clearly seen as contradictory or incoherent. Assuming that mathematics is analytic, consider the case of the following sentence: ‘3,250 + (3 . 896) = 11,876 ÷ 2.’ At first glance, this identity neither seems to be necessarily true nor does its negation seem incoherent; but a detailed presentation of the calculation shows that this must be the case. We can regard it as a hidden analytic truth, at first view not graspable because of its derivative character and our inability to see its truth on the spot.

     This can be suggested by means of a thought-experiment. We can imagine a person with a better grasp of arithmetic than ours. For a child, 2 + 3 = 5 can be analytically transparent, as it is for me. For me, 12 . 12 = 144 is also transparently analytic (or intuitively true), though not to a child who has just started to learn arithmetic. But 144 . 144 = 20,736 isn’t transparently analytic for me, although it may be so for a person with greater arithmetical skill. Indeed, I would guess that some persons with great arithmetical skill (as in the case of some savants) can recognize at a glance the truth of the identity ‘3,250 + (3 . 896) = 11,876 ÷ 2.’ This means that the boundary line between transparent and derived or non-transparent (but deductively achievable) analytic truths is movable, depending on our cognitive capacities and to some degree affected by training. Thus, from an epistemically neutral point of view, the two types are on the same level since for God (the only epistemic subject able to see all truths at a glance) analytic truths must all be transparent.

     In searching for a better-supported answer, we can now distinguish between transparent and non-transparent analytic-conceptual knowledge.[12] The sentences ‘A triangle has three sides,’ ‘Red is not green’ and ‘Three is greater than two’ express transparent analytic knowledge, since these relations are self-evident and their negation clearly contradictory. But not all analytic sentences are so. Sentences about geometry such as the one stating the Pythagorean Theorem express (I assume) analytic truths in non-applied Euclidean geometry, although this isn’t transparent for me. Non-transparent analytic knowledge is based on demonstrations whose premises are made up of transparent analytic knowledge, namely, analytic truths we can intuitively grasp.

      The arithmetical and geometrical examples of analytic statements presented above are only elucidative, which can mislead us to think that they are informative in the proper sense of the word. This leads us to the suggestion that the principle of verifiability is nothing but a non-transparent, hidden analytic statement.

     Against this last suggestion, one could still object that the principle of verifiability cannot be stated along the same lines as a mathematical or geometrical demonstration. After all, in the case of a proved theorem it is easy to retrace the path that leads to its demonstration; but there is no analogous way to demonstrate the principle of verifiability.

     However, the key to an answer may be found if we compare the principle of verifiability with statements that at first glance do not seem to be either analytic or demonstrable. Close examination reveals that they are in fact only non-transparent analytic truths. A well-known statement of this kind is the following:

 

The same surface cannot be simultaneously red all over and green all over (under the same conditions of observation).

 

This statement isn’t analytically transparent. In fact, it has been regarded by logical positivists and even contemporary philosophers as a serious candidate for what might be called a synthetic a priori judgment (Cf. Bonjour 1998: 100 f.). Nevertheless, we can show that it is actually a hidden analytic statement. We begin to see this when we consider that it seems transparently analytic that (i) visible colors can occupy surfaces, (ii) different colors are things that cannot simultaneously occupy the same surface all over, and (iii) red and green are different colors. From this, it seems to follow that the statement (iv) ‘The same surface cannot be both red and green all over’ must be true. Now, since (i), (ii) and (iii) seem to be intuitively analytic, (iv) should be analytic too, even if not so intuitively clear.[13] Here’s how this argument can be formulated in a standard form:

 

(1) Two different things cannot occupy the same place all over at the same time.

(2) A surface constitutes a place.

(3) (1, 2) Two different things cannot occupy the same surface all over at the same time.

(4) Colors are things that can occupy surfaces.

(5) (3, 4) Two different colors cannot occupy the same surface all over at the same time.

(6) Red and green are different colors.

(7) (5, 6) Red and green cannot occupy the same surface all over at the same time.

 

To most people, premises (1), (2), (4) and (6) can be understood (preserving the intended context) as definitely analytic. Therefore, conclusion (7) must also be analytic, even if it does not appear to be so.

     The suggestion that I want to make is that the principle of verifiability is also a true, non-trivial and non-transparent analytic sentence, and its self-evident character may be demonstrated through an elucidation of its more transparent assumptions in a way similar to that of the above argument. Here is how it can be made plausible by the following ‘cumulative’ argument:

 

(1) Semantic-cognitive rules are criterial rules applicable to (or satisfied by) independent criteria that are tropical properties.

(2) Cognitive (descriptive, representational, factual…) meanings (e-thought-contents) of statements are constituted by proper combinations of (referential) semantic-cognitive rules applicable to real or only conceivable arrangements of tropical properties and their combinations called facts.

(3) The truth-determination of cognitive meanings or e-thought-content-rules of statements lies in the effective applicability of the proper combinations of semantic-cognitive criterial rules constitutive of them by means of their agreement (correspondence) with the arrangements and combinations of those tropical properties called real facts able to satisfy their criteria.

(4) Combinations of semantic-cognitive criterial rules expressible by statements are able to be true or false respectively by their effective applicability or non-applicability to their corresponding real or only conceivable facts, building in this way what we may call their e-thought-content verifiability rules.

(5) (1-4) The cognitive meanings of statements consist in their verifiability rules.

 

To my ears, at least, premises (1), (2), (3), and (4) sound clearly analytic, though conclusion (5) does not seem as clearly analytic. I admit that my view of these premises as analytic derives from the whole background of assumptions gradually reached in the earlier chapters of this book: it is analytically obvious to me that contents, meanings or senses are constituted by the application of rules and their combinations. It is also analytically obvious to me that the relevant rules are semantic-cognitive rules that can be applied in combination to form cognitive meanings or thought-contents expressible by declarative sentences. Moreover, once these combinations of rules are satisfied by the adequate criterial configurations formed by real facts understood as tropical arrangements, they allow us to see them as effectively applicable, that is, as having a verifying fact as their referent and truth-maker. Such semantic-criterial combinations of (normally implicit) cognitive rules, when judged as effectively applicable to their verifying facts, are called true, otherwise they are called false. And these semantic-criterial combinations of cognitive rules can also be called e-thoughts (e-thought-content-rules), propositional contents or simply verifiability rules.

     I am aware that a few stubborn philosophers would still vehemently disagree with my reasoning, insisting that they have different intuitions originating from different starting points. After all I have said until now, I confess to be unable to help. To make things easier, I prefer to avoid discussion, invoking the words of an imaginary character from J. L. Borges: ‘Their impurities forbid them to recognize the splendor of truth.’[14]

5. Objection 2: A formalist illusion

Logic can be illuminating but also deceptive. An example is offered by A. J. Ayer’s attempt to formulate a precise version of the principle of verifiability in the form of a criterion of factual meaningfulness. In his first attempt to develop this kind of verifiability principle, he suggested that:

…it is the mark of a genuine factual proposition… that some experiential propositions can be deduced from it in conjunction with certain other premises without being deducible from these other premises alone. (1952: 38-39)

That is, it is conceivable that a proposition S is verifiable if together with the auxiliary premise P1 it implies an observational result O, as follows:

 

1.      S

2.      P1

3.      O

 

Unfortunately, it was soon noted that Ayer’s criterion of verifiability was faulty. As Ayer himself recognized, his formulation was ‘too liberal, allowing meaning to any statement whatsoever.’ (1952: 11) Why? Suppose that we have as S the meaningless sentence ‘The absolute is lazy.’ Conjoining it with an auxiliary premise P1, ‘If the absolute is lazy, then snow is white,’ we can – considering that the observation that snow is white is true and that the truth of ‘The absolute is lazy’ cannot be derived from the auxiliary premise alone – verify the sentence ‘The absolute is lazy.’

     Now, the core problem with Ayer’s suggestion (which was not solved by his later attempt to remedy it[15]) is this: In order to derive the observation that snow is white, he assumes that a declarative sentence (which he somewhat confusingly called a ‘proposition’) whose meaningfulness is questioned is already able to attain a truth-value. But meaningless statements cannot attain any truth-value: if a sentence has a truth-value, then it must also have a meaning, or, as I prefer to say, it must also express a propositional content as an e-thought verifiability rule that is true only as effectively applicable. By assuming in advance a truth-value for the sentence under evaluation, Ayer’s principle implicitly begs the question, because if a statement must already have a sense in order to have a truth-value, it cannot be proven to be senseless. Moreover, he does not allow the empirical statement in question to reveal its proper method of verification or even if it has one.[16]

     In fact, we cannot imagine any way to give a truth-value to the sentence ‘The absolute is lazy,’ even a false one, simply because it is a grammatically correct but cognitively meaningless word combination. As a consequence, the sentence ‘If the absolute is lazy, then snow is white’ cannot imply that the conclusion ‘Snow is white’ is true in conjunction with the sentence ‘The absolute is lazy.’ To make this obviously clear, suppose we replace ‘The absolute is lazy’ with the equally meaningless symbols @#$, producing the conjunction ‘@#$ & (@#$ → Snow is white).’ We cannot apply a truth-table to show the result of this because @#$, just as much as ‘the absolute is lazy,’ expresses no proposition at all. Even if the statement ‘Snow is white’ is meaningful, we cannot say that this formula allows us to derive the truth of ‘Snow is white’ from ‘The absolute is lazy,’ because @#$, as a meaningless combination of symbols, cannot even be considered false in order to materially imply the truth of ‘Snow is white.’

     A. G. Hempel committed a similar mistake when he pointed out that a sentence of the form ‘S v N,’ in which S is meaningful, but not N, must be verifiable, in this way making the whole disjunction meaningful (1959: 112). Now, as we have seen, the real form of this statement is ‘S v @#$.’ Obviously, we cannot apply any truth-table to this. In this case, only the verifiable S has meaning and allows verification, not the whole disjunction, because this whole cannot be called a disjunction. The true form of this statement, if we wish to preserve this title, is simply S.

     I can develop the point further by giving a contrasting suggestion as a criterion of cognitive meaningfulness, more akin to Wittgenstein’s views. Consider the sentence ‘This piece of metal is magnetized.’ The question of its cognitive meaningfulness suggests verifiability procedures. An affirmative answer results from the application of the following verification procedure that naturally flows from the statement ‘This piece of metal is magnetized’ conjoined with some additional information:

 

(1) This is a piece of metal (observational sentence).

(2) If a piece of metal is magnetized, it will attract other objects made of iron (a criterion for the ascription rule of ‘…is magnetized’),

(3) This piece of metal has attracted iron coins, which remained stuck to it (observational application of the ascription rule’s criterion to the object already criterially identified by the identification rule).

(4) (From 1 to 3): It is certainly true that this piece of metal is magnetized.

(5) If the application of the combination of semantic-cognitive rules demanded by a statement is able to make it true, then this combination must be its cognitive meaning (a formulation of the verifiability principle).

(6) (4 to 6): The statement ‘[It is certainly true that] this piece of metal is magnetized’ is cognitively meaningful (it expresses an e-thought-content verifiability rule).

 

We can see that in cases like this the different possible verifying procedures flow naturally from our understanding of the declarative sentence that we intend to verify, once the conditions for its verification are given. However, in the case of meaningless sentences like ‘The absolute is lazy’ or ‘The nothing nothings,’ we can find no verification procedure following naturally from them, and this is the real sign of their lack of cognitive meaning. Ayer’s statement ‘If the absolute is lazy, then snow is white’ does not follow naturally from the sentence ‘The absolute is lazy.’ In other words: the multiple ways of verifying a statement – themselves expressible by other statements – must contribute, in different measures, to make it fully meaningful; but they do this by building its cognitive meaning and not by being arbitrarily attached to the sentence, as Ayer’s proposal suggests. They must be given to us intuitively as the declarative sentence’s proper ways of verification. The neglect of real ways of verification naturally built into any genuine declarative sentence is the fatal flaw in Ayer’s criterion.

6. Objection 3: Verificational holism

A sophisticated objection to semantic verificationism is found in W. V-O. Quine’s generalization of Duhem’s thesis, according to which it is impossible to confirm a scientific hypothesis in isolation, that is, apart from the assumptions constitutive of the theory to which it belongs. In Quine’s concise sentence: ‘...our statements about the external world face the tribunal of sense experience not individually but only as a corporate body.’ (1951: 9)[17]

     The result of this is Quine’s semantic holism: our language forms a so interdependent network of meanings that it cannot be divided up into verifiability procedures explicative of the meaning of any isolated statement. The implication for semantic verificationism is clear: since what is verified must be our whole system of statements and not any statement alone, it makes no sense to think that each statement has an intrinsic verifiability rule that can be identified with a particular cognitive meaning. If two statements S1 and S2 can only be verified together with the system composed by {S1, S2, S3… Sn}, their verification must always be the same, and if the verifiability rule is the meaning, then all the statements should have the same meaning. This result is so absurd that it leaves room for skepticism, if not about meaning, as Quine would like, at least about his own argument.

     In my view, if taken on a sufficiently abstract level, on which the concrete spatiotemporal confrontations with reality to be made by each statement are left out of consideration, the idea that the verification of any statement in some way depends on the verification of a whole system of statements – or, more plausibly, of a whole molecular sub-system – is very plausible. This is what I prefer to call abstract or structural confirmational holism, and this is what can be seriously meant in Quine’s statement. However, his conclusion that the admission of structural holism destroys semantic verificationism, does not follow. It requires admitting that structural holism implies what I prefer to call a performative, concrete or procedural verificational holism, i.e., a holism regarding the concrete spatiotemporal verification procedures of individual statements, which are the only things really constitutive of their cognitive meanings. But this just never happens.

     Putting things in a somewhat different way: Quine’s holism has its seeds in the fact, well known by philosophers of science, that in order to be true the verification of an observational statement always depends on the truth of an undetermined multiplicity of assumed auxiliary hypotheses and background knowledge. Considered in abstraction from what we really do when we verify a statement, at least structural molecularism is true: verifications are interdependent. After all, our beliefs regarding any domain of knowledge are more or less interdependent, building a complex network. But it is a wholly different matter if we claim that from formal or abstract confirmational holism, a performative procedural or verificational holism follows on a more concrete level. Quine’s thesis is fallacious because, although at the end of the day a system of statements really needs to confront reality as a whole, in their concrete verification, its individual statements do not confront reality either conjunctively or simultaneously.

     I can clarify what I mean with the help of a well-known example. We all know that by telescopic observation Galileo discovered the truth of the statement: (i) ‘The planet Jupiter has four moons.’ He verified this by observing and drawing, night after night, four luminous points near Jupiter, and concluding that these points were constantly changing their locations in a way that seemed to keep them close to the planet, crossing it, moving away and then approaching it again, repeating these same movements in a regular way. His conclusion was that these luminous points could be nothing other than moons orbiting the planet... Contemporaries, however, were suspicious of the results of his telescopic observation. How could two lenses magnify images without deforming them? Some even refused to look through the telescope, fearing it could be bewitched… Historians of science today have realized that Galileo’s contemporaries were not as scientifically naive as they often seem to us.[18] As has been noted (Salmon 2002: 260), one reason for accepting the truth of the statement ‘Jupiter has four moons’ is the assumption that the telescope is a reliable instrument. But the reliability of telescopes was not sufficiently confirmed at that time. To improve the telescope as he did, Galileo certainly knew the law of telescopic magnification, whereby its power of magnification results from the focal length of the telescope divided by the focal length of the eyepiece. But in order to guarantee this auxiliary assumption, one would need to prove it using the laws of optics, still unknown when Galileo constructed his telescope. Consider, for instance, the fundamental law of refraction. This law was established by Snell in 1626, while Galileo’s telescopic observations were made in 1610. With this addition, we can state in an abbreviated way the structural procedure of confirmation as it is known today and which I claim would be unwittingly confused by a Quinean philosopher with the concrete verification procedure. Here it is:

 

(I)

1. Repeated telescopic observation of four points of light orbiting Jupiter.

2. Law of magnification of telescopes.

3. Snell’s law of refraction: sinθ₁/sinθ₂ = v1/v2 = l1/l2 =n2/n1.

4. A telescope cannot be bewitched.

5. Jupiter is a planet.

6. The Earth is a planet.

7. The Earth is orbited by a moon.

8. (All other related assumptions.)

9. Conclusion: the planet Jupiter has at least four moons.

 

If Galileo did not have knowledge of premise 3, this only weakens the inductive argument, which was still strong enough to his lucid mind. From a Quinean verificationist holism, the conclusion, considering all the other constitutive assumptions, would be that the concluding statement 9 does not have a proper verification method, since it depends not only on observation 1, but also on the laws expressed in premises 2 and 3, the well-known premises from 4 to 7, and an undetermined number of other premises constitutive of our system of beliefs, all of them also having their verifiability procedures... As he wrote: ‘our statements should face the tribunal of experience as a corporate body.’ Indeed.

     In this example, the problem with Quine’s reasoning becomes clear. First, we need to remember that the premises belonging to confirmation procedures are not simultaneously checked. The conclusion expressed by statement 9 was actually verified only as a direct consequence of statement 1, resulting from the daily drawings made by Galileo based on his observations of variations in the positions of the four ‘points of light’ aligned near to Jupiter. However, Galileo did not simultaneously verify statement 2 when he made these observations, nor the remaining ones. In fact, as he inferred conclusion 9 from premise 1, he only assumed a previous verification of the other premises, as was the case with premise 2, which he verified as he learned how to build his telescope. Although he didn’t have premise 3 as a presupposition, he had already verified or assumed as verified premises 2, 4, 5, 6, 7 and 8. Now, because in general the verifications of 2 to 8 are already made and presupposed during the verification of 9, it becomes clear that these verifications are totally independent of the actually performed verification of 9 by means of 1. The true form of Galileo’s concrete verification procedure was much simpler than the abstract (holistic or molecularist) procedure of confirmation presented above. In a summarized form, it was:

 

1. Repeated telescopic observation of four points of light orbiting Jupiter.

2. Conclusion: the planet Jupiter has at least four moons.

 

Generalizing: If we call the statement to be verified S, and the statements of the observational and auxiliary hypotheses O and A respectively, the structure of the concrete verifiability procedure of S is not

 

           O

           A₁ & A₂… & A_n

           S

 

But simply:

                                                           

           O

           (Assuming the prior verification of A₁ & A₂... & A_n)

           S

 

This assumption of an anterior verification of auxiliary hypotheses in a way that might hierarchically presuppose sufficient background knowledge is what in practice makes all the difference, as it allows us to separate the verifiability procedure of S from the verifiability procedures of the involved auxiliary hypotheses and the many background beliefs which have been already successfully verified.

     The conclusion is that we can clearly distinguish what verifies each auxiliary hypothesis. For example: the law of telescopic magnification was verified by very simple empirical measurements; and the law of refraction was established and verified later, based on empirical measurements of the relationship between variations in the angle of incidence of light and the density of the transmitting medium. Thus, while it is true that on an abstract level a statement’s verification depends on the verification of other statements of a system, on the level of its proper cognitive and practical procedures, the successful verification of auxiliary and background statements is already assumed. This is what allows us to individuate the concrete verifiability procedure appropriate for a statement as what is actually being verified, identifying it with what we actually mean by the statement, thus with its proper cognitive meaning.

     In the same way, we are able to distinguish the specific concrete modes of verification of each distinctive auxiliary or background statement, whose truth is assumed as verified before employing the verification procedure that leads us to accept S as true. This allows us to distinguish and identify the concrete procedure or procedures whereby each statement of our system is cognitively verified, making the truth of abstract-structural holism irrelevant to the performative structure of semantic verificationism.

     By considering all that is formally involved in confirmation, and by simultaneously disregarding the difference between what is presupposed and what is performed in the concrete spatiotemporal verification procedures, Quine’s argument gives us the illusory impression that verification as such should be a holistic procedure. This seems to imply that the meaning of the statement cannot be identified with a verifiability procedure, since the meanings of the different statements are multiple and diversified, while the holistic confrontation of a system of beliefs with reality is unique and as such undifferentiated.

     However, if we remember that each different statement must have a meaning of its own, it again becomes perfectly reasonable to identify the cognitive meaning of a statement with its verifiability rule! For both the verifiability rule and the meaning are once more individuated together as belonging univocally to each statement, and not to the system of statements or beliefs assumed in the verification. Molecular holism is true regarding the ultimate structure of confirmation. But it would be disastrous regarding meaning, since it would dissolve all meanings into one big, meaningless mush.

     The inescapable conclusion is that Quine’s verificational holism is false. It is false because the mere admission of formal holism, that is, of the fact that statements are in some measure inferentially intertwined with each other is insufficient to lead us to conclude that the verifiability rules belonging to these statements cannot be identified with their meanings because these rules cannot be isolated, as Quine suggested. Finally, one should not forget that in my example I gave only one way of verification for the statement ‘The planet Jupiter has at least four moons.’ Other ways of verification can be added, also constitutive of the meaning and enriching it and univocally related with the same statement.

     Summarizing my argument: an examination of what happens when a particular statement is verified shows us that even assuming formal holism (which I think is generally correct, particularly in the form of a molecularism of linguistic practices), the rules of verifiability are distinguishable from each other in the same measure as the meanings of the corresponding statements – a conclusion that only reaffirms the expected correlation between the cognitive meaning of a statement and its method of verification.

7. Objection 4: Existential-universal asymmetry

The next well-known objection is that the principle of verifiability only applies conclusively to existential sentences, but not to universal ones. To verify an existential sentence such as ‘At least one piece of copper expands when heated,’ we need only observe a piece of copper that expands when heated. To conclusively verify a universal claim like ‘All pieces of copper expand when heated’ we would need to observe all the pieces of copper in the entire universe, including everything future and past, which is impossible. It is true that absolute universality is a fiction and that, when we talk about universal statements, we are always considering some limited domain of entities – some universe of discourse. But even in this case, the problem remains. In the case of metal expanding when heated, for instance, the domain of application remains much broader than anything we can effectively observe, making conclusive verification equally impossible.

     A common reaction to this finding – mainly because scientific laws usually take the form of universal statements – is to ask whether it wouldn’t be better to admit that the epistemic meaning of universal statements consists of falsifiability rules instead of verifiability rules… However, in this case existential sentences like ‘There is at least one flying horse’ would not be falsifiable, since we would need to search through an enormously vast domain of entities in the present, past and future in order to falsify it. Nonetheless, one could suggest that the meanings of universal statements were given by falsifiability rules, while the meanings of existential and singular statements would be given by verifiability rules. Wouldn’t this be a more reasonable answer? (Cf. Hempel 1959)

     Actually, though, I am inclined to think it would and could not do. We can, for example, falsify the statement ‘All ravens are black’ simply by finding a single white raven. In this case, we must simply verify the statement ‘This raven is white.’ In this way, the verifiability rule of this last statement is such that, if applied, it falsifies the statement ‘All ravens are black.’ But if the meaning of the universal statement may be a falsification rule, a rule able to falsify it, and the verifiability rule of the statement ‘That raven is white’ is the same rule that when applied falsifies the statement ‘All ravens are black,’ then – admitting that verifiability is the cognitive meaning of singular statements and falsifiability the meaning of the universal ones – it seems that we should agree that the statement ‘All ravens are black’ must be synonymous with ‘That raven is white.’ However, this would be absurd: the meaning of ‘This raven is white’ has almost nothing to do with the meaning of ‘All ravens are black.’

     The best argument I can think against falsifiability rules, however, is that they do not exist. As already noted, there seems to be no proper falsifiability rule for a statement, as there certainly is no counter-assertoric force (or a force proper to negative judgments, as once believed), no rule of dis-identification of a name, and no rule for the dis-ascription or dis-application of a predicate. This is because what satisfies a rule is a criterion and not its absence. – This is so even in those cases in which, by common agreement, the criterion is the absence of something normally expected, as in the case of a hole, e.g., if someone says: ‘Your shirt has a hole in it,’ or in the case of a shadow, in the statement ‘This shadow is moving.’ In such cases the ascription rule for ‘…has a hole’ and the identification rule for ‘This shadow’ have what could be called ‘negative criteria.’ However, what needs to be satisfied or applied is the verifiability rule for the existence of a hole in the shirt, and not the falsifiability rule for the socially presentable shirt without a hole, since this would be the verifiability rule of a shirt that has no hole. And we use the verifiability rule for a moving shadow and not the falsifiability rule for the absence of a shadow. If I notice a curious moving shadow on a wall, I am verifying it; I am not falsifying the absence of moving shadows on the wall, even if the first observation implies the second.[19]

     It seems, therefore, that we should admit that the cognitive meaning of a statement can only be its verifiability rule, applicable or not. But in this case, it seems at first view inevitable to return to the problem of the inconclusive character of the verification of universal propositions, leading us to the admission of a ‘weak’ together with a ‘strong’ form of verificationism as Ayer attempted to argue (1952: 37).

     However, I doubt if this is the best approach to reach the right answer. My suggestion is that the inconclusiveness objection is simply faulty, since it emerges from a wrong understanding of the true logical form of universal statements; a brief examination shows that these statements are in fact both probabilistic and conclusive. Consider again the universal statement:

 

1.    Copper expands when heated.

 

It is clear that its true logical form is not, as it seems:

 

2.    [I affirm that] it is absolutely certain that all pieces of copper expand when heated,

 

whereby ‘absolutely certain’ means ‘without possibility of error.’ This logical pattern would be suitable for formal truths such as

 

3.    [I affirm that] it is absolutely certain that 7 + 5 = 12,

 

because here there can be no error (except procedural error, which we are leaving out of consideration). However, this same form is not suitable for empirical truths, since we cannot be absolutely certain about their truth. The logical form of what we mean with statement (1) is a different one. This form is that of practical certainty, which can be expressed by

 

4.    [I affirm that] it is practically certain that every piece of copper expands when heated,

 

where ‘practically certain’ means ‘with a probability that is sufficiently high to make us disregard the possibility of error.’ In fact, we couldn’t rationally mean anything different from this. Now, if we accept this paraphrase, a statement such as ‘Copper expands when heated’ becomes conclusively verifiable, because we can clearly find inductive evidence protected by theoretical reasons that become so conclusive that we can be practically certain, namely, that we can assign the statement ‘All pieces of copper expand when heated’ a probability that is sufficiently high to make us very sure about it: we can affirm that we know its truth. In short: the logical form of an empirical universal statement – assuming there is some domain of application – is not that of a universal statement like ‘├ All S are P,’ but usually:

 

5.    [I affirm that] it is practically certain that all S are P.

 

Or (using a sign of assertion-judgment):

 

6.    ├ It is practically certain that all S are P.

 

The objection of asymmetry has its origins in an internal transgression of the limits of language, in the case, the equivocal assimilation of the logical form of empirical universal statements in the logical form of formal universal statements (Chap. III, sec. 11). If the claims of empirical universal statement is nothing beyond a sufficiently high probability, this is enough to make them conclusively verifiable. Hence, the cognitive meaning of an empirical universal statement can still be seen as its verifiability rule. Verification allows judgment; judgment must be treated as conclusive, and verification likewise.

8. Objection 5: Arbitrary indirectness

Another common objection is that the rule of verifiability of empirical statements requires taking as a starting point at least the direct observation of facts that are objects of a virtually interpersonal experience. However, many statements do not depend on direct observation to be true, as is the case with ‘The mass of an electron is 9.109 x 10 kg raised to the thirty-first negative power.’ Cases like this force us to admit that many verifiability rules cannot be based on more than indirect observation of the considered fact. As W. G. Lycan has noted, if we don’t accept this, we will be left with a grotesque form of instrumentalism in which what is real must be reduced to what can be inter-subjectively observed and in which things like electrons and their masses do not exist anymore. But if we accept this, he thinks, admitting that many verifiability rules are indirect, how do we distinguish between direct and indirect observations? ‘Is this not one of those desperately confusing distinctions?’ (2000: 121 f.)

     Here again, problems only emerge if we embark on the narrow formalist canoe of logical positivism, paddling straight ahead, only to tramp against the barriers of natural language with unsuitable requirements. Our assertoric sentences are inevitably uttered or thought in the contexts of language-games, practices, linguistic regions... The verification procedure must be adapted to the linguistic practice in which the statement is uttered. Consequently, the criterion to distinguish direct observation from indirect observation should always be relative to the linguistic practice that we take as a model. We can be misled by the fact that the most common linguistic practice is (A): our wide linguistic practice of everyday direct observational verification. The standard conditions for singling out this practice are:

 

A possible interpersonal observation made by epistemic subjects under normal internal and external conditions and with unbiased senses of solid, opaque and medium-sized objects, which are close enough and under adequate lighting, all other things remaining the same.

 

This is how the presence of my laptop, my table and my chair are typically checked. Because it is the most usual form of observation, this practice is seen as the archetypal candidate for the title of direct observation, to be contrasted with, say, indirect observation through perceptually accessible secondary criteria, as might be the case if we used mirrors, optical instruments, etc. However, it is an unfortunate mistake that some insist on using the widespread model (A) to evaluate what happens in other, sometimes very different, linguistic practices. Let us consider some of them.

     I begin with (B): the bacteriologist’s linguistic practice. Usually, the bacteriologist is concerned with the description of micro-organisms visible under his microscope. In his practice, when he sees a bacterium under a microscope, he says he has made a direct observation; this is his model for verification. But the bacteriologist can also say, for example, that he has verified the presence of a virus indirectly, due to changes he found in the form of the cells he saw under a microscope, even though for him viruses are not directly observable except under an electron microscope. If he does not possess one, he cannot make a direct observation of a virus. Almost nobody would say that the bacteriologist’s procedures are all indirect unless they have in mind a comparison with our everyday linguistic practices (A). Anyway, although unusual, this would be possible. In any case, the right context and utterances clearly show what the speaker has in mind.

     Let us consider now (C) the linguistic practices of archaeology and paleontology. The discovery of fossils is seen here as a direct way to verify the real existence of extinct creatures that died out millions of years ago, such as dinosaurs, since live observation is impossible, at least under any known conditions. But the archaeologist can also speak of indirect verification by comparison and contrast within his practice. So, consider the conclusion that hominids once lived in a certain place based only on damage caused by stone tools to fossil bones of animals that these early hominids once hunted and used for food or clothing. This finding may be regarded as resulting from an indirect verification in archaeological practice, in contrast to finding fossilized remains of early hominids, which would be considered a direct form of verification. Of course, here again, any of these verifications will be considered indirect when compared with verification by the most common linguistic observational practice of everyday life, that is (A). However, the context can easily show what sort of comparison we have in mind. A problem would arise only if the language used were vague enough to create doubts about the model of comparison employed.

     If the practice is (D) one of pointing to linguistically describable feelings, the verification of a sentence will be called direct, albeit subjective, if made by the speaker himself, while the determination of feelings by a third person, based on behavior or verbal testimony, will generally be taken as indirect (e.g., by non-behaviorists and many who accept my objections to the private-language argument). There isn’t any easy way to compare practice (D) with the everyday practice (A) of observing medium-sized physical objects in order to say what is more direct, since they belong to two categorically different dimensions of verification.

     My conclusion is that there is no real difficulty in distinguishing between direct and indirect verification, insofar as we have clarity about the linguistic practice in which the verification is being made, that is, about the model of comparison we have chosen (See Ch. III, sec. 7). Contrasted with philosophers, speakers normally share the contextually bounded linguistic assumptions needed for the applicability and truth-making of verifiability rules. To become capable of reaching agreement on whether a verificational observation or experience is direct or indirect, they merely need to be aware of the contextually established model of comparison that is being considered.

9. Objection 6: Empirical counterexamples

Another kind of objection concerns insidious statements that only seem to have meaning, but lack any effective verifiability rule. In my view, this kind of objection demands consideration on a case-by-case basis.

     Consider, to begin with, the statement ‘John was courageous,’ spoken under circumstances in which John died without having had any opportunity to demonstrate courage, say, shortly after birth. (Dummett 1978: 148 f.) If we add the stipulation that the only way to verify that John was courageous would be by observing his behavior, the verification of this statement becomes practically (and very likely physically) impossible. Therefore, in accordance with the verifiability principle, this statement has no cognitive meaning, however, it still seems more than just grammatically meaningful.

     The explanation is that under the described circumstances the statement ‘John was courageous’ only appears to have a meaning. It belongs to the sizable set of statements whose cognitive meaning is only apparent. Although the sentence has an obvious grammatical sense, given by the combination of a non-empty name with a predicate, we are left without any criterion for the application or non-application of the predicate. Thus, such a statement has no function in language, since it is unable to tell us anything. It is part of a set of statements such as ‘The universe doubled in size last night’ and ‘My brother died the day after tomorrow.’ Although these statements may at first glance appear to have a sense, what they possess is no more than the expressive force of suggesting images or feelings in our minds. But in themselves, they are devoid of cognitive meaning since we cannot test or verify them.

     Wittgenstein discussed an instructive case in his work On Certainty. Consider the statement ‘You are in front of me right now,’ said under normal circumstances for no reason by someone to a person standing before him. He notes that this statement only seems to make sense, given that we are able to imagine situations in which it would have some real linguistic function, for example, when a room is completely dark, so that it is hard for a person to identify another person in the room (1984a, sec. 10).     According to him, we are inclined to imagine counterfactual situations in which the statement would or would not be true, and this invites us to project a truth-value into these possible situations and thus we will get the mistaken impression that the statement has some workable epistemic sense. Against this one could in a Gricean way still argue that even without any practical use the sentence has a literal assertoric sense, since it states something obviously true. However, this would be nothing but a further illusion: it seems to be obviously true only insofar as we are able to imagine situations in which it would make sense (e.g., exemplifying the evidential character of a perceptual assertion).

     Finally, many statements are mediated and are only indirectly verifiable. Because of this, it is easy to make statements like ‘The core of Jupiter is made of marschmallow,’ and say that it is meaningful although unverifiable. However, we know that this statement is obviously false, and the method by which we falsify it is indirect since we cannot make a voyage to the center of Jupiter. We refute ramifications of the verification rule, which would deny our scientific conclusion that the core of this planet is made of water and helium and our awareness that marschmallow is made of milk and that there is no cow in Jupiter… These things show that the verifiability rule is inapplicable.[20]

     What can we say of statements about the past or the future? Here too, it is necessary to examine them on a case-by-case basis. Suppose an expert says: ‘Early Java man lived about 1 million years ago,’ and this statement was fully verified by a reliable carbon dating applied to the fossilized skull. The direct verification of past events in the same way that we observe present events is practically (and it would seem physically) impossible. However, there is no reason to worry, since we are not dealing with the kind of verifiability rule adopted in standard practice (A). Here the linguistic practice assumed is (C), the archaeological, in which direct verification is made on the basis of verifiable empirical traces left by past events.

     There are other, more indirect ways to verify past events. The sentence ‘The planet Neptune existed before it was discovered’ can be accepted as certainly true. Why? Because our knowledge of physical laws (which we trust as sufficiently verified), combined with information about the origins of our solar system, enables us to conclude that Neptune certainly existed a long time before it was discovered, and this inferential procedure is suitable as a form of verification. Finally, it is simply fallacious to say that since we can know about the past only by means of presently available evidence, we cannot say anything about the past, but only about our present, since the resource of present evidence can be the only natural and reliable way to speak about the past.

     Very different is the case of statements about the past such as:

 

1.   On that rock, an eagle landed exactly ten-thousand years ago.

2.   Napoleon sneezed more than 30 times while he was invading Russia.

3.   The number of human beings alive exactly 2,000 years ago was an odd number.

 

For such supposed thought-contents there are no empirical means of verification. Here we must turn to the old distinction between practical, physical and logical verifiability. Such verifications are not practically or technically achiev­able, and as far as I know, they are not even physically realizable (we will probably never be able to visit the past in a time-machine or travel through a worm-hole into the past in a spaceship). The possibility of verification of such statements seems to be only logical. But it is hard to believe that an empirical statement whose verifiability is only logical can be considered as having a non-logical cognitive sense (Cf. Reichenbach 1953: sec. 6).

     To explain this point better: it seems that the well-known distinction between logical, physical and practical forms of verifiability exerts influence on meaningfulness depending on the respective fields of verifiability to which the related statements belong. Statements belonging to a formal field need only be formally verifiable to be fully meaningful: the tautology (P → Q ) ↔ (~P ˅ Q), for instance, is easily verified by the truth-table applying the corresponding logical operators. But statements belonging to the empirical domain (physical and practical) must be not only logically, but also at least in principle empirically verifiable in order to have real cognitive meaning. As a consequence, an empirical statement that is only logically verifiable must be devoid of cognitive significance. This seems to be the case with a statement such as ‘There is a nebula that is moving away from the earth at a speed greater than the speed of light.’ Although logically conceivable, this statement is empirically devoid of sense, insofar as it is impossible according to relativity theory. Similarly, in examples (1), (2) and (3), what we have are empirical statements whose verification is empirically inconceivable. Consequently, although having grammatical and logical meaning and eliciting images in our minds, these statements lack any distinctive cognitive value, for we don’t know what to make of them. Such statements aren’t able to perform the specific function of an empirical statement, which is to be able to truly represent an actual state of affairs. We do not even know how to begin the construction of their proper verifiability rules. All that we can do is to imagine or conceive the situations described by them, but we know of no rule or procedure to link the conceived situation to something that possibly exists in the real world. Although endowed with grammatical and some expressive meaning, they are devoid of genuine cognitive meaning. Finally, we must remember that we are free to reformulate statements (1), (2) and (3) as meaningful empirical possibilities. For instance: (2’) ‘Maybe (it is possible that) Napoleon sneezed more than 30 times when he was invading Russia.’ Although not very dissimilar to (2), this modal statement is verifiable as true by means of its coherence with our belief-system.

     Also unproblematic is the verificational analysis of statements about the future. The great difference here is that in many cases direct verification is practically possible. Consider the sentence (i) ‘It will rain in Caicó seven days from now.’ When a person seriously says something of this sort, what he usually means is (ii) ‘Probably it will rain in Caicó seven days from now.’ And this probability sentence can be conclusively verifiable, albeit indirectly, by a weather forecast. Thus, we have a verifiability rule, a cognitive meaning, and the application of this rule gives the statement a real degree of probability. However, one could not in anticipation affirm (iii) ‘It certainly will rain within seven days.’ Although there is a direct verifiability rule – watch the sky for seven days to determine if the thought-content is true or false – it has the disadvantage that we will only be able to apply it if we wait for a period of time, and we will only be able to affirm its truth (or deny it) within the maximal period of seven days. It is true that we could also use this sentence in certain situations, for example, when making a bet about the future. But in this case, we would not affirm (iii) from the start since we cannot apply the rule in anticipation. In this case, what we mean with sentence (i) can in fact only be (iv) ‘I bet that it will rain in Caicó seven days from now.’ Lacking any empirical justification, the bet has again only an expressive-emotive meaning and no truth-value.

     A similar statement is (v) ‘The first baby to be born on Madeira Island in 2050 will be female,’ which has a verifiability rule that can only be applied at a future point in time. This sentence lacks a practical meaning insofar as we are unable to verify and affirm it at the present moment; right now this sentence, though expressing a thought-content – since it has a verifiability rule whose application can be tested in the future – is able to have a truth-value, but cannot receive it until later. Nonetheless, in a proper context this sentence may also have the sense of a guess: (vi) ‘I guess that the first baby to be born…’ or (vii) a statement of possibility regarding the future ‘It is possible that the first baby to be born…’ In these cases, we are admitting that the sentence has a cognitive meaning since all we are saying is that it has an observational verifiability rule that can be applied (or not), although only in the future. Sentence (v) will only be meaningless if understood as an affirmation of something that is not now the case but will be the case in the year 2050, for in order to be judged to be true this affirmation requires awareness of the effective applicability of the verifiability rule generally based on its real application. (Cf. Ch. IV, sec. 36) When we consider what is really meant by statements regarding future occurrences, we see that even in these cases verifiability and meaning go together.

     Now consider the statement (viii): ‘In about eleven billion years the Sun will expand and engulf Mercury.’ This statement in fact only means ‘Very probably in about eleven billion years the Sun will expand and engulf Mercury,’ This probabilistic prediction can be inferentially verified today, based on what we know of the fate of other stars in the universe that resemble our Sun but are much older, and this inferential verification constitutes its cognitive meaning.

     Jeopardizing positivist hopes, I conclude that there is no general formula specifying the general form of verifiability procedures. Statements about the future can be physically and to some extent also practically verifiable. They cannot make sense as warranted assertions about actual states of affairs since such affirmations require the possibility of present verification. Most of them are concealed probability statements. The kind of verifiability rule required depends on the utterance and its insertion in the linguistic practice in which it is made, only then showing clearly what it really means. Such things are what may lead us to the mistaken conclusion that there are unverifiable statements with cognitive meaning.

     Finally, a word about ethical statements. Positivist philosophers have maintained that they are unverifiable, which has led some to adopt implausible emotivist moral theories. Once again, we find the wrong attitude. I would rather suggest that ethical principles can be only more or less plausible, like metaphysical statements and indeed like any philosophical statement. They have the form: ‘It is plausible that p,’ and as such they are fully verifiable. They cannot be decisively affirmed because we are still unable to state them in adequate ways or make them sufficiently precise, since we lack consensual agreement regarding their most adequate formulation and verifiability rules.

10. Objection 7: Formal counterexamples

The verificationist thesis is naturally understood as extendable to the statements of formal sciences. In this case, the verifiability rules or procedures that demonstrate their formal truth constitute a form of cognitive content deductively, within the assumed formal system in which they are considered. A fundamental difference with respect to empirical verification is that in the case of formal verification, to have a verifiability rule is the same thing as being definitely able to apply it, since the criteria ultimately to be satisfied are the own axioms already assumed as such by the chosen system.

     A much discussed counterexample is Goldbach’s conjecture. This conjecture (G) is usually formulated as:

 

G: Every even number greater than 2 can be expressed as the sum of two prime numbers.

 

The usual objection is that this mere conjecture has cognitive meaning. It expresses a thought-content even if we never manage to prove it, even if a procedure for formal verification of G has not yet been developed. Therefore, its significance cannot be equated with a verifiability procedure.

     The answer to this objection is quite simple and stems from the perception that Goldbach’s conjecture is what its name says: a mere conjecture. Well, what is a conjecture? It’s not an affirmation, a proven theorem, but rather the recognition that an e-thought-content-rule has enough plausibility to be taken seriously as possibly true. One would not make a conjecture if it seemed fundamentally improbable. Thus, the true form of Goldbach’s conjecture is:

 

It is plausible that G.

 

But ‘It is plausible that G,’ that is, ‘[I state that] it is plausible that G,’ or (using a sign of assertion) ‘├It is plausible that G,’ is something other than

 

 I state that G (or ├G),

 

which is what we would be allowed to say if we wanted to state Goldbach’s proved theorem. If our aim were to support the statement ‘I state that G,’ namely, an affirmation of the truth of Goldbach’s theorem as something cognitively meaningful, the required verifiability rule would be the whole procedure for proving the theorem, and this we simply do not have. In this sense, G is cognitively devoid of meaning. However, the verifiability rule for ascribing mere plausibility is far less demanding than the verifiability rule able to demonstrate or prove G, and we have indeed applied this rule many times.

     The plausibility ascription is ‘[I state that] it is plausible that G,’ whereby the verifiability rule consists in something much weaker, namely, a verification procedure able to suggest that G could be proved. This verification procedure does in fact exist. It consists simply in considering random examples, such as the numbers 4, 8, 12, 124, etc., and showing that they are always the sum of two prime numbers. This verifiability rule not only exists, up until now it has been confirmed without exception for every even natural number ever considered! This is the reason why we really do have enough support for Goldbach’s conjecture: it has been fully verified as a conjecture. If an exception had been found, the conjecture would have been proved false, for this would be incompatible with the truth of ‘[I state that] it is plausible that G’ and would from the start be a reason to deny the possibility of Goldbach’s conjecture being a theorem.

     Summarizing: in itself the conjecture is verifiable and – as a conjecture – has been definitely verified: It is simply true that G is highly plausible. And this justifies its cognitive meaningfulness. What remains beyond verification is the statement affirming the necessary truth of G. And indeed, this statement doesn’t really make sense; it has no cognitive content since it consists in a proof, a mathematical procedure to verify it, which we do not have. The mistake consists in the confusion of the statement of a mere conjecture that is true with the ‘statement’ of a theorem that does not exist.

     A contrasting case is Fermat’s last theorem. Here is how this theorem (F) is usually formulated:

 

F: There are no three positive integers x, y and z that satisfy the equation xⁿ + yⁿ = zⁿ, if n is greater than 2.

 

This theorem had been only partially demonstrated up until 1995 when Andrew Wiles finally succeeded in working out a full formal proof. Now, someone could object here that even before Wiles’ demonstration, F was already called ‘Fermat’s theorem.’ Hence, it is clear that a theorem can make sense even without being proved!

     There are, however, two unfortunate confusions in this objection. The first is all too easy to spot. Of course, Fermat’s last theorem has a grammatical sense: it is syntactically correct. But it would be an obvious mistake to confuse the grammatical meaning of F with its cognitive meaning as a theorem. Also an absurd identity, for instance, ‘Napoleon is the number 7,’ has a grammatical sense.

     The second confusion concerns the fact that the phrase ‘Fermat’s theorem’ isn’t appropriate at all. We equivocally used to call F a ‘theorem’ because before his death Fermat wrote that he had proved it, but couldn’t put this proof on paper since the margins of his notebook were too narrow…[21] For these reasons, we have here a misnamed opposite of ‘Goldbach’s theorem.’ Although F was called a theorem, it was in fact only a conjecture of the form:

 

[I state that] it is plausible that F.

 

It was a mere conjecture until Wiles demonstrated F, only then effectively making it a true theorem. Hence, before 1995 the cognitive content that could be given to F was actually ‘[I state that] it is plausible that F,’ a conjecture that was initially demonstrated by the fact that no one had ever found numbers x, y and z that could satisfy the equation. Indeed, the cognitive meaning of the real theorem F, better expressed as ‘I state that F’ or ‘├ F’ (a meaning that very few really know in its entirety), should include the demonstration or verification found by Wiles, which is no more than the application of an exceptionally complicated combination of mathe­matical rules.

     Some would complain that if this is the case, then only very few people really know the cognitive meaning of Fermat’s last theorem. I agree with this, though seeing no reason to complain. The cognitive content of this theorem, its full thought-content, like that of many scientific statements, is really known by very few people indeed. What most of us know is only the weak conjecture falsely called ‘Fermat’s last theorem’. We have applied F to some numbers without finding any exception.

     Finally, there are phrases like (i) ‘the less rapidly convergent series.’ For Frege, this definite description has sense but not reference (1892: 28). We can add that there is a rule that allows us to always find series that are less convergent than any given one, making them potentially infinite. We can state this rule as L: ‘For any given convergent series, we can always find a less rapidly convergent one.’ Since L implies the truth of statement (ii) ‘There is no less rapidly convergent series,’ we conclude that (i) has no referent. Now, what is the identification rule of (i)? What is the sense, the meaning of (i)? One answer would be to say that it is given by failed attempts to create a less rapidly convergent series ignoring L. It would be like the meaning of any mathematical falsity. For instance, the identity (iii) 321 + 427 = 738 is false. Now, what is its meaning? A temptation is to classify it as senseless. But if it were senseless, it would not be false. Consequently, I suggest that its sense resides in the failed usual ways to verify it, which leads to the conclusion that this is a false identity. It seems reasonable to conclude that it is such an external operation that gives a kind of cognitive sense to a false identity. The same holds regarding false statements like 3 > 5. They express misrepresentations, incongruities demonstrating failed attempts to apply rules in the required ways.

11. Objection 8: Skepticism about rules

In his Philosophical Investigations, Wittgenstein formulated a skeptical paradox (1984c, sec. 201) that endangers the possibility of an ongoing common interpretation of rules and, consequently, the idea that our language may work as a system of rules responsible for meaning. Solving this riddle interests us here because if the argument is correct, it seems to imply that it is a mistake to accept that there are verifiability rules consisting in the cognitive meanings of sentences.

     Wittgenstein’s paradox results from the following example of rule-following. Let’s say that a person learns a rule to add 2 to natural numbers. If you give him the number 6, he adds 2 and writes the number 8. If you give him the number 173, he adds 2, writing the number 175... But imagine that for the first time he is presented with a larger number, say the number 1,000, and that he then writes the number 2,004. If you ask why he did this, he responds that he understood that he should add 2 up to the number 1,000, 4 up to 2,000, 6 up to 3,000, etc. (1984c, sec. 185).

     According to Saul Kripke’s dramatized version of the same paradox, a person learns the rule of addition, which works well for additions with numbers below 57. But when he performs additions with larger numbers, the result is always 5. So for him 59 + 67 = 5… Afterward, we discover that he understood ‘plus’ as the rule ‘quus,’ according to which ‘x quus y = x + y if {x, y} < 57, otherwise 5’ (1982: 9). If questioned why he understood addition in this strange way, he answers that he found this the most natural way to understand the rule.

     Now, what these two examples suggest is that a rule can always be interpreted differently from the way it was intended, no matter how many specifications we include in our instructions for using the rule, since these instructions can also be differently interpreted… As Kripke pointed out, there is no fact of the matter that forces us to interpret a rule in a certain way rather than in any other. The consequence is that we cannot be assured that everyone will follow our rules in an expected similar way, or that people will continue to coordinate their actions based on them. And as meaning depends upon following rules, we cannot be certain about the meanings of the expressions we use. How could we be certain, in the exemplified cases, of the respective meanings of ‘add two’ and ‘plus’? However, if we accept that there can be no rules and therefore no meanings, then there could be no riddle since we would not be able to meaningfully formulate the riddle.

     Wittgenstein and later Kripke attempted to find a solution to the riddle. Wittgenstein’s answer can be interpreted as saying that we follow rules blindly, as a result of training (custom) regarding the conventions of our social practices and institutions belonging to our way of life (1984c sec. 198, 199, 201, 219, 241). Kripke’s answer follows a similar logic: according to him, following a rule isn’t justified by truth-conditions derived from their correct interpretation in a correspondential (realist) way, a solution that Wittgenstein tried in his Tractatus. Instead, Kripke thinks that for the later Wittgenstein correspondence is replaced by verification, so that instead of truth-conditions what we have are assertability conditions justified by practical interpersonal utility (1982: 71-74, 77, 108-110). These assertability conditions are grounded on the fact that any other user in the same language community can assert that the rule follower ‘passes the tests for rule following applied to any member of the community’ (1982: 110).

     Notwithstanding, both answers are clearly wanting. They offer a description of how rules work, leaving unexplained why they must work. Admittedly, the simple fact that in our community we have so far openly coordinated our linguistic activity according to rules does not imply that this coordination has to work this way, nor does it imply that it should continue to work this way. Kripke’s answer has in my view an additional burden. It overlooks the fact that assertability conditions must include the satisfaction of truth-conditional correspondential-verificational conditions, only adding to the explanation of the common interpretation of rules an interpersonal social layer.

     For my part, I have always believed that the ‘paradox’ should have a more satisfactory solution. A central point can be seen as in some way already disclosed by Wittgenstein, namely, that we learn rules in a similar way because we share a similar human nature modeled in our form of life.     It seems clear that this makes it easier for us to interpret the rules we are taught in the same manner, suggesting that we must also be naturally endowed with innate, internal corrective mechanisms able to reinforce consistent, conforming behavior. (Costa 1990: 64-66)

     Following this path, we are led to the decisive solution of the riddle, which I think we owe to Craig DeLancey (2004). According to him, we are biologically predisposed to construct and interpret statements in the most economical (or parsimonious) way possible. Or, as I prefer to say, we are innately disposed to put in practice the following principle of simplicity:

 

PS: We should formulate and interpret our rules as the simplest ones.

 

Because of this shared principle derived from our inborn nature as rule followers, we prefer to maintain the interpretation of the rule ‘add 2’ in its usual form, instead of complicating it with the further condition that we should add twice two after each thousand. And because of the same principle, we prefer to interpret the rule of addition as a ‘plus’ instead of a ‘quus’ addition, because with the ‘quus’ addition we would complicate the interpretation by adding the further condition that any sum with numbers above 57 would give as a result the number 5. Indeed, it is the application of this principle of simplicity that is the ‘fact of the matter’ not found by Kripke, which leads us to interpret a rule in one way instead of another. It allows us to harmonize our interpretations of semantic rules, thus solving the riddle. Furthermore, DeLancey clarifies ‘simplicity’ by remarking that non-deviant interpretations are formally more compressible than deviant interpretations like those considered by Wittgenstein and Kripke. Moreover, a Turing machine would need to have a more complex and longer program in order to process these deviant interpretations...

     One might ask: what warrants assuming the long-term consistency of human nature across the entire population or that we are innately equipped to develop such a heuristic principle of simplicity? The obvious answer lies in the appeal to Darwinian evolution. Over long periods of time, a process of natural selection has harmonized our learning capacities around the principle of simplicity and eliminated individuals with deviant, less practical dispositions. Thus, we have a plausible explanation of our capacity to share a sufficiently similar understanding and meaning of semantic rules. If we add to this the assumption that human nature and recurring patterns in the world will not change in the future, we can be confident in the expectation that people will not deviate from the semantic rules they have learned. Of course, underlying this last assumption is Hume’s much more defiant criticism of induction, which might remain a hidden source of concern. But this is a further issue that goes beyond our present concerns (for a plausible approach see the Appendix of the present chapter).[22]

     Summarizing: Our shared interpretation of learned rules only seems puzzling if we insist on ignoring the implications of the theory of evolution, which supports the principle of simplicity. By ignoring considerations like these, we tend to ask ourselves (as Wittgenstein and Kripke did) how it is possible that these rules are and continue to be interpreted and applied in a similar manner by other human beings, losing ourselves within a maze of philosophical perplexities. For a similar reason, modern pre-Darwinian philosophers like Leibniz wondered why our minds are such that we are able to understand each other, appealing to the Creator as producing the necessary harmony among human souls. The puzzle about understanding how to follow rules arises from this same old perplexity.

12. Quine’s objections to analyticity

Since I am assuming that the verifiability principle is an analytic-conceptual statement, before finishing I wish to say a word in defense of analyticity. I am satisfied with the definition of an analytic proposition as the thought-content expressed by a statement whose truth derives from the combination of its constitutive unities of sense. This is certainly the most common and intuitively acceptable formulation. However, W. V-O. Quine would reject it because it seems to be based on an overly vague and obscure concept of meaning.

     The usual answer to this criticism is that there is really nothing overly vague or obscure in the concept of meaning used in our definiens, except from Quine’s own scientistic-reductionist perspective, which tends to confuse expected vagueness with lack of precision and obscurity (See Grice & Strawson 1956: 141-158; Swinburne 1975: 225-243). Philosophy works with concepts such as meaning, truth, knowledge, good… which are in some measure polysemic and vague, as much so as the concepts used in countless attempts to define them. In my judgment, the effort to explain away such concepts only by reason of their vagueness (or supposed obscurity) betrays an impatient positivist-scientistic mental disposition, which is anti-philosophical par excellence (which doesn’t mean to indulge the opposite: a methodology of hyper-vagueness or unjustified obscurity).

     Having let out of consideration the above definition, Quine tried to define an analytic sentence in a Fregean way, as a sentence that is either tautological (true because of its logical constants) or can be shown to be tautological by the replacement of its non-logical terms with cognitive synonyms. Thus, the statement (i) ‘Bachelors are unmarried adult males’ is analytic, because the word ‘bachelor’ is a synonym of the phrase ‘unmarried adult male,’ which allows us by the substitution of synonyms to show that (i) means the same thing as (ii): ‘Unmarried adult males are unmarried,’ which is a tautology. However, he finds the word ‘synonym’ in need of explanation. What is a synonym? Quine’s first answer is that the synonym of an expression is another expression that can replace the first in all contexts salva veritate. However, this answer does not work in some cases. Consider the phrases ‘creature with a heart’ and ‘creature with kidneys.’ They are not synonymous, but are interchangeable salva veritate, since they have the same extension. In a further attempt to define analyticity, Quine makes an appeal to the modal notion of necessity: ‘Bachelors are unmarried males’ is analytic if and only if ‘Necessarily, bachelors are unmarried males.’ But he also sees that the usual notion of necessity does not cover all cases. Phrases like ‘equilateral triangle’ and ‘equiangular triangle’ necessarily have the same extension, but are not synonyms. Consequently, we must define ‘necessary,’ in this case, as the specific necessity of analytic statements, in order for the concept to apply in all possible circumstances... However, the ‘necessity of analyticity’ is an obscure notion, if it really exists. Dissatisfied, Quine concludes that his argument to explain analyticity ‘has the form, figuratively speaking, of a closed curve in space.’ (Quine 1951: 8)

     A problem emerges from Quine’s implicit assumption that a word should be defined with the help of words that do not belong to its specific conceptual field. Thus, for him, the word ‘analyticity’ should not be defined by means of words like ‘meaning,’ ‘synonymy,’ ‘necessity’… which just as much as ‘analyticity’ seem too near and unspecific in their meaning to be trusted in the construction of an adequate definition. Nonetheless, when we consider the point more carefully, we see that the words belonging to a definiens should be sufficiently close in their meanings to the definiendum, simply because in any real definition the terms of a definiens must belong to the same semantic field as its definiendum, notwithstanding the element of vagueness. This is why, in order to define a concept-word from ornithology, we would not use concepts from quantum mechanics, and vice-versa. These conceptual fields are too distant from each other. Because of this, we define ‘arthropod’ as an invertebrate animal having an exoskeleton, all these terms being biological, which does not compromise the definition. And considering the abstractness of the semantic field, a kindred level of vagueness can be expected. Hence, there is nothing especially wrong in defining analyticity using correspondingly vague words belonging to the same conceptual field, like ‘meaning’ and ‘synonymy,’ refraining from further elucidation.

     A more specific and more serious objection is that Quine’s attempt to define synonymy simply took a wrong turn. Since there is probably no proper necessity of analyticity, the lack of synonymy of expressions that necessarily have extensions like ‘equilateral triangle’ and ‘equiangular triangle’ remains unexplained.

     My alternative proposal consists simply in beginning with the dictionary definition according to which:

 

Two words or phrases are synonymous when they have the same or nearly the same meaning as another word or phrase in the same language.[23]

 

Translating this into our terms, this means that any expressions A and B are (cognitively) synonymous if their semantic-cognitive rules (their expressed concepts) are the same or almost the same. This can be tested by adequate definitions (analyses) expressing the criteria for the application of those rules so that when these rules are really the same, the synonymous expressions will be called precise synonyms. However, precise synonyms are difficult to find. Consider, for instance, the words ‘beard’ and ‘facial hair.’ These words are called synonymous because they express a similar semantic-cognitive rule. A ‘beard’ is defined by dictionaries as ‘a growth of hair on the chin and lower cheeks of a man’s face’ and this is considered sufficiently similar to the expression ‘facial hair.’ However, the two terms are not precisely synonymous, because a human being with hair on the forehead has facial hair but no beard. Diversely, the word ‘chair’ and the expression ‘a non-vehicular seat provided with a backrest and made for use by only one person at a time’ can be seen as precise synonymous, because the latter is simply the real definition of the former. The expressions ‘creature with a heart’ and ‘creature with a kidney,’ on the other hand, are not synonymous, because they express different semantic-cognitive rules, the first defined as a creature with an organ used to pump blood, the second defined as a creature with an organ used to clean waste and impurities from blood. Even if approximate in meaning, the expressions ‘equilateral triangle’ and ‘equiangular triangle’ are surely not precisely synonymous for the reason already considered: the first is defined as a triangle whose three sides are equal, while the second is defined as a triangle whose three internal angles are congruent with each other and are each 60°. Hence, we can replace Quine’s flawed definition of analyticity with the following more adequate definition using the concept of precise synonymy:

 

A statement S is analytic (Df): It can generate a tautology by means of substitution of precise cognitive synonyms, namely, of real definitions expressing the same semantic-cognitive criterial rules.

 

The statement ‘The cognitive meaning (e-thought-content) of a declarative sentence X = the verifiability rule for X’ is analytic because the semantic-cognitive rules on each side of the identity sign are identical.

     A complementary point supported by Quine is that, contrary to what is normally asserted, there is no definite distinction between empirical and formal knowledge. What we regard as analytic sentences can always be falsified by greater changes in our more comprehensive system of beliefs. Even sentences of logic such as the excluded middle can be rejected, as occurs in some interpretations of quantum physics.

     Regarding this point, it would not be correct to say that in itself a formal or analytic proposition could be proved false or be falsified by new experience or knowledge. What more precisely can occur is that its domain of application can be restricted or even lost. For example: since the development of non-Euclidean geometries, the Pythagorean Theorem has lost part of its theoretical domain; it is not the only useful geometry anymore. And since the theory of relativity has shown that physical space is better described as Riemannian, this theorem has lost its monopoly on describing physical space. However, this is not the same as to say that the Pythagorean Theorem has been falsified in a strict sense. This theorem remains perfectly true within the theoretical framework of Euclidean geometry, where we can prove it, insofar as we assume the basic rules that constitute this geometry. This remains so, even if Euclidean geometry’s domain of application has been theoretically restricted with the rise of non-Euclidean geometries and even if it has lost its full applicability to real physical space after the development of general relativity theory.

     The case is different when a law belonging to an empirical science is falsified. In this case, the law definitely loses its truth together with the theory to which it belongs, since its truth-value depends solely on its precise empirical application. Newtonian gravitational law, for instance, was falsified by general relativity. It is true that it still has valuable practical applications that do not require the highest level of accuracy. The best one could say in its favor is that it has lost some of its truth, trying to make this idea clear by appealing to multi-valued logic.

13. Conclusion

There is surely much more that can be said about these issues. I believe, however, that the few but central considerations that were offered here were sufficient to convince you that semantic verificationism, far from being a useless hypothesis, comes close to being rehabilitated when investigated with a methodology that does not overlook and therefore does not violate the delicate tissue of our natural language. The fundamental questions of philosophy are as fascinating as difficult because of their subjacent complexity and wideness. Inventing ways to make them easy is to be relieved by illusory answers.

 

 

 

 

 

 

Appendix to Chapter V

The Only Key to Solving the Humean Problem of Induction

 

 

 

It would be impossible to say truly that the universe is a chaos, since if the universe were genuinely chaotic there could not be a language to tell it. A language depends on things and qualities having enough persistence in time to be identified by words and this same persistence is a form of uniformity.

—J. Teichman & C. C. Evans

 

Here I will first reconstruct in the clearest possible way the essentials of Hume’s skeptical argument against the possibility of induction (Hume 1987 Book I, III; 2004 sec. IV, V, VII), viewing it separately from his amalgamated analysis of causality. My aim in doing this is to find a clear argumentative formulation of his argument that allows me to outline what seems to be the only adequate way to react to it in order to re-establish the credibility of inductive reasoning.

1. Formulating a Humean argument

According to Hume, our inductive inferences require support by metaphysical principles of the uniformity of nature. Although induction can move not only from the past to the future, but also from the future to the past and from one spatial region to another, for the sake of simplicity I will limit myself here to the first case. A Humean principle of uniformity from the past to the future can be stated as:

 

PF: The future will resemble the past.

 

If this principle is true, it ensures the truth of inductive inferences from the past to the future. Consider the following very simple example of an inductive argument justifying the (implicit) introduction of PF as a first premise:

 

1. The future will resemble the past. (PF)

2. The Sun has always risen in the past.

3. Hence, the Sun will rise tomorrow.

 

This seems at first glance a natural way to justify the inference according to which if the Sun rose every morning in the past then it will also rise tomorrow, an inference which could be extended as a generalization, ‘The Sun will always rise in the future.’ We make these inferences because we unconsciously believe that the future will be like the past.

     It is at this point that the problem of induction begins to delineate itself. It starts with the observation that the first premise of the argument – a formulation of the principle of the uniformity of nature from the past to the future – is not a truth of reason characterized by the inconsistency of its negation. One could say it is not an analytic thought-content. According to Hume, it is perfectly imaginable that the future could be very different from the past, for instance, that in the future trees could bloom in the depths of winter and snow taste like salt and burn like fire (1748, IV).

     We can still try to ground our certainty that the future will resemble the past on the past permanence of uniformities that once belonged to the future, that is, on past futures. This is the inference that at first glance seems to justify PF:

 

1. Already past futures were always similar to their own pasts.

2. Hence, the future of the present will also resemble its own past.

 

The problem with this inference is that it is also inductive. That is, in order to justify this induction we need to use PF, the principle that the future will resemble the past; but PF itself is the issue. Thus, when we try to justify PF, we need to appeal once more to induction, which will require PF again... Consequently, the above justification is circular.

     From similar considerations, Hume concluded that induction cannot be rationally justified. The consequences are devastating: there is no rational justification either for expectations created by the laws of empirical science or for our own expectations of everyday life, since both are grounded on induction. We have no reason to believe that the floor will not sink under us when we take our next step.

     It is true that we are almost always willing to believe in our inductive inferences. But for Hume, this disposition is only due to our psychological constitution. We are by nature inclined to acquire habits of having inductive expectations. Once we form these expectations, they force us to obey them almost like moths flying towards bright lights. This is an extremely skeptical conclusion, and it is not without reason that only a few philosophers have accepted Hume’s conclusion. Most think that something somewhere must be wrong.

     There have been many interesting attempts to solve or dissolve Hume’s problem; all of them in some way unsatisfactory.[24] I believe my approach, although only sketched out, has the virtue of being on the right track. I want to first present a general argument and then show how it could influence PF.

2. The basic idea

My basic idea has a mildly Kantian flavor, but without its indigestible synthetic a priori. We can sum it up in the view that any idea of a world (nature, reality) that we are able to have must be intrinsically open to induction. I see this as a conceptual truth in the same way as, say, the truth of our view that any imaginary world must in principle be accessible to perceptual experience.

     Before explaining it in more detail, I should note that my view is so close to being self-evident that it would be strange if no one had thought of it earlier, as the citation at the start of this appendix proves. More technically, Keith Campbell followed a similar clue in developing a short argument to show the inevitability of applying inductive procedures in any world-circumstances (1974: 80-83). As he noted, in order to experience a world cognitively – as an objectively structured reality – we must continually apply empirical concepts, which, in turn – if we are to postulate, learn from and use them – require a re-identification of the designata of their applications as identical. However, this is only possible if there is a degree of uniformity in the world that is sufficient to allow these re-identifications. Indeed, if the world were to lose all the regularities implicitly referred to, no concept would be re-applicable and the experience of a world would be impossible.

     Coming back to my basic idea, and understanding the concept of world minimally as any set of empirical entities compatible with each other[25], this idea can be unpacked as follows. First, I consider it an indisputable truism that a world can only be experienced and said to exist if it is at least conceivable.[26] However, we cannot conceive of any world without some degree of uniformity or regularity. Now, since we can only experience what we are able to conceive, it follows that we cannot experience any world completely devoid of regularity. This brings us to the point where it seems reasonable to think that the existence of regularity is all that is necessary for at least some inductive procedure to be applicable. However, if this is the case, then it is impossible for us to conceive of any world of experience that is not open to induction. Consequently, it must be a conceptual truth that if a world is given to us, then some inductive procedure should be applicable to it.

     There is a predictable objection to this idea: why should we assume that we cannot conceive the existence of a chaotic world – a world devoid of regularities and therefore closed to induction? In my view, the widespread belief in this possibility has been a deplorable mistake, and I am afraid that David Hume was chiefly responsible for this.[27] His error was to choose causal regularity as the focus of his discussion, strengthening it with carefully selected examples like those of trees blooming in winter and snow burning like fire. This was misleading, and in what follows, I intend to explain why.

     Causal regularity is what I would call a form of diachronic regularity, that is, one in which a given kind of phenomenon is regularly followed by another kind. We expect the ‘becoming’ (werden) of our world to include regular successions.

     However, induction applies not only to diachronic regularities, but also to something that Hume, with his fixation on causality, did not consider, namely, synchronic regularities. Synchronic regularities are what we could also call structures: states of affairs that endure over time in the constitution of anything we can imagine. The world has not only a ‘becoming’ (werden), but also a ‘remaining’ (bleiben), with its multiple patterns of permanence. And this remaining must also be inductively graspable.

     We can make this last view clear by conceiving of a world without any diachronic regularity, also excluding causal regularities. This world would be devoid of change, static, frozen. It still seems that we could properly call it a world, since even a frozen world must have regularities to be conceivable; it must have a structure filled with synchronic regularities. However, insofar as this frozen world is constituted by synchronic regularities, it must be open to induction: we could foresee that its structural regularities would endure for some time – the period of its existence – and this already allows a very strong degree of inductive reasoning!

     Considerations like this expose the real weakness in Hume’s argument. By concentrating on diachronic patterns and thinking of them as if they were the only regularities that could be inductively treated, it becomes much easier to suppose the possibility of the existence of a world to which induction does not apply or cannot be applicable, a world that nevertheless continues to exist.

     To clarify these points, try to imagine a world lacking both synchronic and diachronic regularities. Something close to this can be grasped if we imagine a world made up of irregular, temporary, random repetitions of a single point of light or sound. However, even if the light or sound occurs irregularly, it will have to be repeated at intervals (as long as the world lasts), which demonstrates that it still displays at least the regularity of a randomly intermittent repetition open to recognition. But what if this world didn’t have even random repetitions? A momentary flash of light… Then it would not be able to be fixed by experience and consequently to be said to exist. The illusion that it could after all be experienced arises from the fact that we already understand points of light or sounds based on previous experiences.

     My conclusion is that a world absolutely deprived of both species of regularity is as such inconceivable, hence inaccessible to experience – a non-world, an anti-world. We cannot conceive of any set of empirical elements without assigning it some kind of static or dynamic structure. But if that’s the case, if a world without regularities is unthinkable, whereas the existence of regularities is all we need for some kind of inductive inference to be applicable, then it is impossible that there is for us a world closed to induction. And since the concept of a world is nothing but the concept of a world for us, there is no world at all that is closed to induction.

     Summarizing the argument: By focusing on causal relationships, Hume invited us to ignore the fact that the world consists of not only diachronic, but also synchronic regularities. If we overlook this point, we are prone to believe that we could conceive of a world inaccessible to inductive inference. If, by contrast, we take into account both general types of regularity to which induction is applicable, we realize that a world which is entirely unpredictable, chaotic, devoid of any regularity is impossible, because any possible world is conceivable and any conceivable world must contain regularities, which makes it intrinsically open to some form of induction.

     One could insist on thinking that at least a world that is chaotic but not entirely chaotic could exist, with a minimum of structure or uniformity, so that it would exist but be insufficient for the application of inductive procedures. However, this is a theoretical impossibility, for induction has a self-adjusting nature, that is, its principles are such that they are always conceivably able to be calibrated to match any degree of uniformity that is given in its field of application. The requirement of an inductive basis, of repeated and varied inductive attempts, can always be further extended, the greater the improbability of the expected uniformity. Consequently, even a system with a minimum of uniformity requiring a maximum of inductive searching would always end up enabling successful induction.

     These general considerations suggest a variety of internal conceptual inferences, such as the following:

 

Conceivable cognitive-conceptual experience of a world ↔ applicability of inductive procedures ↔ existence of regularities in the world ↔ existence of a world ↔ conceivable cognitive-conceptual experience of a world…

 

These phenomena are internally related to each other in order to derive each other at least extensionally, so that their existence already implies these relations. But this means, contrary to what Hume believed, that when properly understood the principles of uniformity should be analytic-conceptual truths, that is, truths of reason applicable in any possible world.

3. Reformulating PF

To show how I would use the just offered proposal to reformulate the principles of uniformity or induction, I will reconsider in some detail PF, the principle that the future will resemble the past. If my suggestion is correct, then it must be possible to turn this principle into an analytic-conceptual truth constituting our only possibilities of conceiving and experiencing the world. – I understand an analytic-conceptual thought-content to be simply one whose truth depends only on the combination of its semantic constituents; its truth isn’t ampliative of our knowledge, in opposition to synthetic propositions, and is such that its denial implies a contradiction or inconsistency (Cf. Ch. V, sec. 12).

     To show how the aforementioned suggestion could be applied to reformulating the principles of uniformity or induction, it is necessary to reformulate PF. If my general thesis is correct, then it must be possible to turn this principle into an analytic-conceptual truth constituting a way of conceiving and experiencing the world. Here is a first attempt to reformulate PF in a clearly analytic form:

 

PF*: The future must have some resemblance to its past.

 

Unlike PF, PF* can easily be accepted as expressing an analytic-conceptual truth, for PF* can be clearly seen as satisfying the above characterization of analyticity. Certainly, it belongs to the concept of the future that it is the future of its own past. It cannot be the future of another past belonging to some alien world. If a future had nothing to do with its past, we could not even recognize it as being the future of its own past, because it could be the future of anything, what seems incoherent. In still clearer words: the future of our actual world W, as FW, can only be the future of the past of W, that is, PW. It cannot be the future of infinitely many possible worlds, W1, W2, W3... that have as their past respectively PW1, PW2, PW3... Thus, it is necessary that there must be something that identifies FW as being the future of PW, and this something can only be some degree of resemblance in the transition.

     Against this proposal, we can try to illustrate by means of examples the possibility of complete changes in the world, only to see that we will always be unsuccessful. Suppose we try to imagine a future totally different from its past, a ‘complete transformation of the world’ as described in the Book of Revelations. It is hard to imagine changes more dramatic than those described by St. John, since he intends to describe the end of the world as we know it. Here is the biblical passage describing the locusts sent by the fifth angel:

 

In appearance the locusts were like horses equipped for battle. And on their heads were what looked like golden crowns; their faces were like human faces and their hair like women’s hair; they had teeth like lions’ teeth and they wore breastplates like iron; the sound of their wings was like the noise of horses and chariots rushing to battle; they had tails like scorpions with stings in them, and in their stings lay their power to plague mankind for five months.[28]

 

At first glance, these changes are formidable. Nonetheless, there is nothing in this report that puts PF* at risk. In fact, closer reflection on the example demonstrates that even PF isn’t seriously challenged. Although these biblical locusts are indeed very strange creatures, they are described as combinations of things already familiar to us. These things are horses, women, hairs, men, heads, teeth, scorpions’ tails with stings, human faces, etc. Both internally and externally, they include a vast quantity of synchronic regularities, of permanent structural associations, together with familiar diachronic associations, like the causal relationship between the noise produced and the movement of wings or the sting of the scorpion and the effects of its poison on humans…

     In fact, were it not for these uniformities, the apocalypse as described by St. John would not be conceivable, understandable and able to be the subject of any linguistic description. The future, at least in proportion to its greater proximity to the present, must maintain sufficient similarity to its past to allow an application of inductive procedures to recognize the continuity of the same world we know today.

     Now one could object that maybe it is possible that at some time in a remote future we could find a dissimilarity so great between the future and our past that it invalidates any of our reasonably applicable inductive procedures – a remote future that would be radically different from its past. Indeed, it seems conceivable that a continuous sequence of small changes could in the course of a very long period of time lead to something, if not completely different, at least extremely different. Nevertheless, this would not discredit PF*, because its formulation is too weak, requiring only that some similarity must remain. However, it also seems that this weakness of PF*, even if not robbing it of its analytic-conceptual character, exposes PF* to charges of disproportionate poverty as a way to assure the reliability of our inductive projections.

     However, precisely this weakness of PF* suggests a way to improve it. It leads us to see that the closer we get to the point of junction between the future and the past, the greater must be the similarity between future and past, both becoming identical at their limit, which is the present. We can approximate this issue by remembering the Aristotelian analysis of change as always assuming the permanence of something that remains identical in a continuous way, without gains or losses (Aristotle 1984, vol 1: Physics, 200b, 33-35); in other words, the intuitive idea is that every change must occur on some basis of permanence.

     This leads us to create another variant of PF, namely, the principle according to which in a process of change the amount of permanence must be inversely proportional to the period of time in which the change occurs. In other words: if there is a sequence of changes that are parts of a more comprehensive change, the changes that belong to a shorter sequence typically presuppose a greater number of permanent structural (and sequential) associations than the sequence constitutive of the more comprehensive change.

     This principle can be illustrated with numerous examples. Consider a simple one: the changes resulting from heating a piece of wax. The change from the solid state to the liquid state presupposes the permanence of the same wax-like material. However, the next change, from liquid wax to carbon ash, presupposes only the permanence of carbon atoms. If the heat then becomes much more intense, carbon will lose its atomic structure, giving place to a super-heated plasma of subatomic particles. We have here a sequence of four time periods: regarding the shortest period of time from t1 to t2, we assume that we will be left with (i) the same wax, made up of (ii) the carbon molecules and atoms, which in turn are composed of (iii) their same subatomic constituents. In the longer period of time from t1 to t3 we assume the identity of only (ii) and (iii): carbon atoms and subatomic particles. And in the still longer period of time from t1 to t4 the only things that remain the same are (iii): subatomic constituents.

     Note that this model is not restricted to changes in the physical material world! As Leibniz saw: Natura non facit saltus. The same examples repeat in every domain that one can imagine, chemical, biological, psychological, social, economic, historical… with the same patterns: the closer the future is to its junction with its past, the more structural identities must be in some way assumed. For example: the process of industrialization. The Industrial Revolution was a period of social and economic changes from an agrarian society to an industrialized society, which suffered an upheaval in the mid-19^th century. As a whole, after its second period it included the refinement of the steam engine, invention of the internal combustion engine, harnessing of electricity, construction of infrastructure such as railways… and, socially, the more complete exodus of families from rural areas to large cities where factories were constructed… However, when we choose to consider a short period in this process, for instance, at the end of the 18^th century, the only outstanding changes were probably the invention of a simple piston engine and a minor exodus from the countryside, most characteristics of society otherwise remaining essentially the same.[29]

     We conclude that it is intrinsic to the very structure of the world of experience – and of possible experience – that changes taking place in a shorter period of time tend to presuppose more permanence than the most comprehensive long-term changes within whose course they occur. Consequently, the future closer to its present should as a rule inevitably be more similar to its past in more aspects than more distant future will be (as already noted, the far distant future may be almost unrecognizably different from the present). At the point of junction between future and past (the present), no difference will be available.

     Regarding induction, this principle assures that inductive predictions will become more likely the closer the future is to the present. On this basis, we can improve the principle PF* as:

 

PF**: As a rule, the closer the future is to the junction point with its own past, the more it will tend to resemble its past, the two being indistinguishable at the point of junction (the present).

 

For a correct understanding of PF**, we must add two specifying sub-conditions:

 

(i)   that this principle should be applied to a future that is sufficiently close to its past and not to an indefinitely distant future.

(ii) to safeguard the possibility of anomalous but conceivable cases in which we find shorter sequential periods where states of affairs of a more distant future are closer to the present than those of the near future.

 

Although I admit that PF** deserves more detailed and precise consideration, it seems to me intuitively obvious that so understood this principle already meets a reasonable standard of analyticity.

     Moreover, it is the truth of PF** which explains why it is natural for us to think that the more distant the future, the less probable our inductive forecasts will be. This is the very familiar case of weather forecasts: they are presently reliable for two or three days, less so for a week or more... It also explains why our inductive generalizations about the future cannot be applied to a very distant future. For instance, through induction we can infer that the Sun will ‘always’ rise, but always must be placed in quotation marks. On the basis of induction, it makes sense to affirm that the sun will rise tomorrow morning or even a thousand years from now. But it defies common sense (and is for cosmological reasons false) to use the same inductive basis to claim that the Sun will still rise every morning in seventeen billion years.

     How PF** applies is circumstantially determined. If the future is sufficiently close to its junction with the past, then the future will be unavoidably similar to its past. The problem, of course, is that we need to establish criteria for judging how close in time the future must be to its past so that PF will still apply. We can speculate as to whether the answer does not depend on the background represented by the domain of regularities in which we are considering the change – a domain of regularities to which a whole system of sufficiently well-entrenched beliefs applies.

     For example: the inductive conclusion that the Sun will rise tomorrow belongs to a domain of regularities that may someday undergo changes predicted by contemporary cosmology. This may include a very distant future in which dramatic changes, such as the death of the Sun, are also predictable based on the astronomically observed fates of similar stars in our universe.

     Of course, it is always possible that the Sun will not rise tomorrow! However, this is only conceivable at the price of an immense loss of other well-entrenched beliefs about astronomical regularities and, subsequently, the loss of the current intelligibility of a considerable portion of the physical world around us. Still, what makes us consider as highly likely the future occurrence of regularities such as that the Sun will rise tomorrow?

     The ultimate answer seems to be based on the inevitable assumption that our world will continue to exist as a system of regularities, at least in the form prescribed by PF**. However, this assumption seems to be a blind gamble! After all, there is nothing preventing our whole world from suddenly disappearing. However, the impression of a paradox evaporates as soon as we consider that this hypothesis is completely unverifiable. If our whole world suddenly disappear and there is no other, how can we know this after we have also disappeared with the world? Now, if the hypothesis is unverifiable, it must be senseless.[30] In contrast, the hypothesis that our world will continue to exist can be verified in the future, hence it is meaningful. Because of this asymmetry, we are free to accept that since we cannot really think that there will be no future at all, the regularities of our world will need to take the form prescribed by PF**, that is, we are inevitably led to admit that certain domains of cohesive regularities will have some permanence.

     The above outlined argument concerns just a single form of induction: from the past to the future. Nevertheless, the attempt to better specify it and to generalize about further developments would be worthwhile, since it suggests a path free of insurmountable hindrances. This may be of some interest regarding a problem that from any other angle seems to remain disorienting and intangible.

 

 

 

Chapter VI

---

[1] Wittgenstein’s best reader at the time, Moritz Schlick, echoes a similar view: ‘Stating the meaning of a sentence amounts to stating the rules according to which the sentence is to be used, and this is the same as stating the way in which it can be verified. The meaning of a proposition is the method of its verification.’ (1938: 340)

[2] See, for a contrast, Carnap’s unfortunate definition of philosophy as ‘the logic of science’ in his 1937, § 72.

[3] C. S. Peirce’s view of metaphysics agrees with what is today the most accepted one (Cf. Loux 2001, ix). On Peirce’s verificationism see also Misak 1995, Ch. 3. As I do, and following Peirce, Cheryl Misak favors a liberalized form of verificationism, opposed to the narrow forms advocated by the Vienna Circle.

[4] See my analysis of the form of semantic-cognitive rules in Chapter III, sec. 12, and considerations regarding the nature of consciousness in Chapter II, sec. 11.

[5] I believe that the germ of the verifiability principle is already present in aphorism 3.11 of the Tractatus Logico-Philosophicus under the title ‘method of projection.’ There he wrote: ‘We use the perceptible sign of a sentence (spoken or written) as a projection of a possible state of affairs. The method of projection is the thinking of the sentence’s sense.’

[6] This is why there is no falsifiability rule, as some authors like Michael Dummett have suggested (1993: 93).

[7] A justified explanation of the resource to structural isomorphism will be given only in Chapter VI, sec. 2-5.

[8] Appendix of Chapter I, sec. 1.

[9] For my account of analyticity, see sec. 12 of the present chapter.

[10] This position was supported by A. J. Ayer, Rudolf Carnap, Herbert Feigl and Hans Reichenbach (Cf. Misak 1995: 79-80).

[11] Ayer’s view wasn’t shared by all positivists. Moritz Schlick, closer to Wittgenstein, defended the view according to which all that the principle of verifiability does is to make explicit the way meaning is assigned to statements, both in our ordinary language and in the languages of science (1936: 342 f.).

[12] This distinction is inspired by Locke’s original distinction between intuitive and demonstrative knowledge. I do not use Locke’s distinction because, as is well known, he questionably applied it to non-analytic knowledge. (Cf. Locke 1975, book IV, Ch. II, § 7)

[13] Obviously, such an example can be decontextualized and therefore cheated in many ways. One could say: red and blue, for instance, can be blended to produce purple on the same surface, which is a bit like both colors… Like everything, examples can also be stolen and then used in the most inappropriate ways.

[14]  From his magnificent short story, ‘El Tintorero Enmascarado Hákim de Merv.’

[15] The difficulty made him propose a more complicated solution that the logician Alonzo Church proved to be equally faulty (Cf. Church 1949).

[16]  I am surely not the first to notice this flaw. See Barry Gower 2006: 200.

[17] Later Quine corrected this thesis, advocating a verifiability molecularism restricted to sub-systems of language, since language has many relatively independent sub-systems. However, our counter-argument will apply to both cases.

[18] I think Galileo’s judges unwittingly did science a great favor by sentencing him to house arrest, leaving him with nothing to do other than concentrate his final intellectual energies on writing his scientific testament, the Discorsi intorno a due nuove scienze.

[19]  Michael Dummett viewed the falsification rule as the ability to recognize under what conditions a proposition is false (Cf. 1996: 62 f.). But this must be the same as the ability to recognize that the proposition isn’t true, namely, that its verifiability rule isn’t applicable, which presupposes that we know its criteria of applicability, being consequently able to recognize their absence.

[20] Another case is the verification of other minds. For an explanatory attempt, see my 2011, Ch. 4.

[21] Today we know that Fermat was only joking since the mathematics of his time did not provide the means to prove his conjecture.

[22] Curiously, in his book Kripke considers the criterion of simplicity, but repudiates it almost fortuitously for the reason that ‘although it allows us to choose between different hypotheses, it can never tell us what the competing hypotheses are’ (1982: 38). However, what the competing hypotheses – call them the rules x and y – ultimately are, is a metaphysically idle question, only answerable by God’s omniscience, assuming that the concept of omniscience makes any sense. The real paradox appears only when we can state it in the form of comparable hypotheses like ‘plus’ versus ‘quus,’ and it is to just such cases that we apply the principle of simplicity.

[23] Oxford Dictionaries.

[24] For example, Hans Reichenbach (1938), D. C. Williams (1942), P. F Strawson (1952), Max Black (1954), Karl Popper (1959)... Original as they may be, when faced with the real difficulties, all these attempts prove disappointing. (For critical evaluation see W. C. Salmon 1966 and Laurence Bonjour 1998, Ch. 7.)

[25] For the sake of the argument, I am abstracting here the subject of experience... Anyway, this would demand an addition of assumed regularities.

[26] After all, conceivability belongs to the grammatical structure of what we understand with the term ‘world.’ The sentence ‘There are worlds that cannot be conceived’ is contradictory, for to know the existence of any inconceivable worlds, we must already have conceived them, at least in some vague, abstract sense.

[27] Strangely enough, the idea of a chaotic world to which induction isn’t applicable has been uncritically assumed as possible in the literature on the problem, from P. F. Strawson to Wesley C. Salmon. This exposes the weight of tradition as a two-edged sword.

[28] Revelation of St. John 9, 7.

[29] One could still object with cases like that of someone who suddenly awakes from a dream… But one forgets the remaining fact that it is the very same person who was dreaming that is now awakened.

[30] In its lack of sense, the question remembers the anthropic principle. The question, ‘Why is possible that we are able to think the world?’ loses its sense as soon as we consider that under infinitely many possible worlds this is one under the few able to produce conscious beings able to pose this pseudo-question.