sábado, 6 de janeiro de 2018


This is a draft for the book PHILOSOPHICAL SEMANTICS to be published in 2018 by CSP


Verificationism is now commonly viewed as a relic of philosophy as practiced in the first half of the 20th 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 repeating the same mistake made by 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 s-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.

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 purely 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] (Wittgenstein 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, precocious desire to develop a purely scientific philosophy had the strongest motivation to develop the verification principle 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… (Peirce 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 our concept of meaning beyond a reasonable limit. For instance: the fact that something cannot be scratched helps to verify the assertion ‘This is a diamond,’ whereas the use of diamonds as abrasives will certainly be of little if any relevance as a proof. 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 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 there is something like a cloud of meaning surrounding the cognitive meaning of a statement S, 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 positivists 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 linguistic horse. In this chapter, I want to show how unwise neglect of some very natural conceptuals 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. Afterwards, I will answer the main objections against the principle, demonstrating that they are much weaker than they seem at first glance.

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

Each sentence (Satz) is a signpost for its verification. (Wittgenstein 1984e: 150)
A sentence (Satz) without any way of verification has no sense (Sinn). (Wittgenstein 1984f: 245)
If two sentences are true or false under the same conditions, they have the same sense (even if they look different). (Wittgenstein 1984f: 244)
To understand the sense of a sentence is to know how the issue of its truth or falsity is to be decided. (Wittgenstein 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.) (Wittgenstein 1984f: 47)
To know the meaning of a sentence, we need to find a well-defined procedure to see if the sentence is true. (Wittgenstein 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. (Wittgenstein 1984f: 226-7)
The meaning of a sentence is its method of verification. (Wittgenstein 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 to be added here. The first is terminological and 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 I called in my discussion of Frege’s semantics the s-thought-content expressed by the declarative sentence, being called by others the descriptive, informative or factual content of the sentence, if not its 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.
  A point to be emphasized is that the verifiability rule includes both the verification and the falsification of the statement, since these rules can be either true or false.[6] The reason is that this 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 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 thought-content – the verifiability rule expressed by the statement – is true. Consider, by contrast, the statement ‘Wittgenstein was bearded’: here the verifiability 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.
  These remarks also lead us to decide against the existence of negative facts: negative statements never assert such phantasmagoric things as negative facts; what it really does is to deny the existence of a positive fact. The true thought expressed by the sentence ‘Teetetus is not flying’ does not properly apply to a negative fact – the fact that he is not flying – but rather it applies to no real fact at all. This is so because ‘Teetetus is not flying’ means the same thing as ‘It is not the case that Teetetus is flying,’ which in turn has the same sense as ‘It is false that Teetetus is flying’. This means only that the here expressed s-thought-content, that is, the verifiability rule, whose application in this case can already be imagined or conceived, does not effectively apply to any real fact in the world but only to a conceivable one. This real fact, the truth-maker, should be the fact in the world of Teetetus flying. But after identifying Teetetus, we see that he is in fact sitting, so that the ascription rule for ‘…is flying’ cannot apply to him. (Due to the flexibility of language, you can say, ‘It is a fact that Teetetus isn’t flying’; but here you are using the word ‘fact’ derivatively in the sense of ‘is true’: ‘It is true that Teetetus isn’t flying’, which means the same thing as ‘It is true that this statement’s verifiability rule is false.’)
  It is important to consider that we can imagine or conceive the application of the verifiability rule of a false statement to be a possible fact, which makes us aware that it could be satisfied in a way that supports its meaningfulness: Plato could imagine his friend Teetetus flying. But this possible fact isn’t a negative, but a positive fact in Plato’s imagination.
  Next, consider the universal negative statement ‘There are no yetis.’ Here as well there is no negative fact to consider. For taking ‘… is a yeti’ as Y, we can formalize this with the sentence ‘There are no yetis’ as ~Ǝx (Yx) or (x) (~Yx). This is the same as {Ya1 v Ya2… v Yan}, where each negative singular statement is false, that is, the ascription rule for characterizing yetis fails to be applied to any given object, since the expected positive fact of the existence of at least one yeti cannot be discovered. ‘There are no yetis’ does not mean ‘There are… no yetis,’ but rather, ‘It is not the case that there are yetis.’ The false statement ‘There are yetis’ remains only a logically and empirically conceivable fact, though in all probability not a constrastivelly real, empirically true and verifiable fact.
  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 accepted as true.

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)


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 here 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 – as far as this is possible – 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 thought-value, which are primary 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.
  Using Wittgensteinian terms, 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 thought-content 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, the satisfaction of the criterial rule seems to have as a minimum condition for satisfaction a 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, and this would constitute the isomorphism with the grounding fact. Since experience is always aspectual and often indirect, this also means that the internal criterial configurations belonging to the rule must also show a structural isomorphism with aspectual configurations of 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 will try 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. On the other hand, secondary criteria (like reading about an event in a magazine) are 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 can be defined as the condition sufficient for a thought-content S to 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 is mistaken, according to which 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 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 was a human being who had as tropes 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 trope 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 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-Aristotle 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 postulate or assume this criterially-based verification as practically certain, which allows us to say we know (K), that Aristotle was bearded, even if we are aware that this is only indirectly proven as highly probable. We can summarize this knowledge in the formula:

K[[IR-Aristotle is applicable to its bearer]AR-bearded is applicable to the 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 really 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 as explained above.

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 is as follows. The principle of verifiability must be either analytic or synthetic. 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.[9] 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.[10]
  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 (s-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 (s-thought-content…) of the declarative sentence X = the verifiability rule for X.

Against this, a critic can react by saying that this claim to analytic identity isn’t transparent. 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) = 11,276 ÷ 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 much 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 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.[11] 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 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 to the suggestion that the principle of verifiability is also 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.[12] 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 clearly 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 could be made plausible:

(1)  Semantic-cognitive rules are criterial rules applicable to objective criteria based on tropical properties.
(2)  Cognitive (descriptive, representational, factual…) meanings (s-thought-contents) expressed by statements are constituted by proper combinations of (referential) semantic-cognitive rules applicable to arrangements and combinations of tropical properties called facts.
(3)  The cognitive meanings of statements (thought-contents) depend on ways of determining their truth by agreement with real facts, which demands the application of the combinations of corresponding semantic-cognitive rules.
(4)  (1, 2, 3) The truth-determination of cognitive meanings or contents of statements lies in the effective application of their proper combinations of semantic-cognitive criterial rules by means of their agreement with the arrangements and combinations of those tropical properties called facts.
(5)  (by definition) Combinations of semantic-cognitive criterial rules determining the truth of statements by their effective application to facts constitute what we have decided to call their verifiability rules.
(6)  (4, 5) The cognitive meanings or contents of statements consist in their verifiability rules.

For me, at least, premises (1), (2), (3), and (5) (which is definitional) sound clearly analytic, although conclusions (4) and mainly (6) do 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 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 s-thoughts, thought-content-rules, propositional contents or simply verifiability rules.
  I know that some stubborn philosophers of language would still vehemently disagree with my reasoning, insisting that they have different intuitions originating from different starting points. I confess to be unable to help them. 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.’[13]

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. (Ayer 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.’ (Ayer 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[14]) 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 s-thought-content or a verifiability rule that is true only if 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.[15]
  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 @#$, being a meaningless combination of symbols, cannot even be considered false in order to materially imply the truth of ‘Snow is white.’
  A. G. Hempel committed a similar mistake when he pointed out that a sentence of the form ‘S v N’, in which S is meaningful, but not N, must be verifiable, in this way making the whole disjunction meaningful (See Ayer ed. 1959: 112). Now we have seen that the real form of this statement is ‘S v @#$.’ However, 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 the 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 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 (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 rules demanded by a statement is able to make it true, then this combination must be its cognitive meaning (this is a principle of verifiability).
(6)  (4 to 6): The statement ‘[It is certainly true that] this piece of metal is magnetized’ is cognitively meaningful (it expresses an s-thought-content-rule).

We can see that in cases like this the variable 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 content. 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 meaningful declarative sentence is the faMILY

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.’ (Quine, 1951: 9)[16]
  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, S3Sn}, 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 spatio-temporal 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 concrete or performative or procedural verificational holism, i.e., a holism regarding the concrete spatio-temporal 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.[17] 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 structural 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:

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. 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

     A1 & A2… & An

But simply:
     (Assuming the prior verification of A1 & A2... & An)

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 are already assumed to have been 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 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 spatio-temporal 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 statements are diverse and differentiated, 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.

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 its 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. Anyway, 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 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 – 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-assertive force (or a force proper to negative judgments, as was once believed), no rule of disidentification of a name, and no rule for the disascription or disapplication of a predicate. The reason 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.’ Indeed, what needs to be satisfied is the verifiability rule for the hole-in-this-shirt, and not the falsifiability rule for ‘this-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.[18]
  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 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 sure 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 formal universal statements in the logical form of empirical universal statements (Chap. III, sec. 11). The empirical universal statement is shown to be conclusively verifiable, since what it claims is nothing beyond a sufficiently high probability. 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

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 – 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 in the narrow formalist canoe of logical positivism, paddling straight ahead, only to tramp against the barriers of natural language with unsuitable requirements. Our assertive 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:

Virtually 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 fossilized 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, i.e. (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. 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.

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, nevertheless 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 is is not able to tells 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 assertive 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).
  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 fossilized skull and reliable carbon dating. 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.
  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 (it is improbable that we will ever 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 real 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 statements in question belong. Statements belonging to a formal field need only be formally verifiable to be fully meaningful: the tautology (PQ ) ↔ (~P v 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.’ This statement is empirically devoid of sense, insofar as it is impossible according to relativity theory. (It might make sense in the fictional context of a Star Trek movie, where spaceships can cruise at multiples of the speed of light.) 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 relevant 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 some expressive meaning, they are devoid of genuine cognitive meaning. Finally, we are free to reformulate statements (1), (2) and (3) as meaningful empirical possibilities. For instance: (2’) ‘It is possible that Napoleon sneezed more than 30 times when he was invading Russia.’ Although confusingly similar 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 physically 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 is 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. (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 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 are scarcely verifiable, like metaphysical statements and indeed like most philosophical statements. They can only be more or less plausible. They aren’t decisively verifiable because we are still unable to state them in adequate ways or make them sufficiently precise, since we lack consensual agreement regarding adequate verifiability rules for ethical statements. Because of this, their verifiability, their cognitive meaningfulness, may be limited or dubious, since we do not reach agreement concerning their applicability, at least for now.

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 choice of a 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 a thought-content 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.
  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 can be proved. Now 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 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 true that G is highly plausible. And this explains 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. 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…[19] 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.
  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 the 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 believe that its sense resides in the failed ways to verify it, which leads to the conclusion that this is a false identity. Thus, in attempting to verify this identity we add 321 to 427, obtaining the result 748. Since 748 is different from 738, we conclude that the identity 321 + 427 = 738 is false. It seems reasonable to conclude that it is such an external operation that gives a kind of cognitive sense to the false identity. The same holds regarding false statements like 3 > 5. They express misrepresentations, incongruities demonstrating inapplicable combinations of 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 regarding 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… Afterwards 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, he 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). 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: 108-110).
  Notwithstanding, both answers are clearly wanting. They offer a description of how rules work, leaving unexplained their normative character, that is, 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 even 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. This is a point that can be easily inferred from the results of the last chapter of the present book.
  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 agreement. (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 establish and interpret a semantic rule in the simplest way possible.

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; and 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[20]).
  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.

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 exponible 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 seem to be inevitably ambiguous 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 scientistic-pragmatic 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 rejected 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, particularly with phrases such as ‘creature with a heart’ and ‘creature with kidneys,’ which are not synonymous, but in many cases are interchangeable salva veritate, since they have the same extension. In a last 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, as Quine puts it, his argument to explain synonymy, though not completely circular, ‘has the form, figuratively speaking, of a closed curve in space.’ (Quine 1951: 8) Moreover, the ‘necessity of analyticity’ is an obscure notion, if it really exists.
  A noteworthy problem is 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 unspecific in their meaning to allow for an adequate definition. Nonetheless, when we consider the point more carefully, we see that the words belonging to a definiens should be relatively 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. To give some examples, 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. 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’ – words that are not impervious to 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 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.[21]

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 at least sufficiently similar, which 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 (in a typical internet dictionary) 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, for instance, has facial hair but no beard. However, the word ‘chair’ and the expression ‘a movable seat provided with a backrest, made for use by only one person at a time’ can be seen as precisely synonymous because the latter is simply the analytical explanation (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 definitions expressing the same semantic-cognitive criterial rules.

A complementary point supported by Quine is that, contrary to what is normally asserted, he does not see any 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 an analytic proposition could be proved false or be falsified by new experience or knowledge. What more precisely can occur is that its domain of external 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 non-Euclidean, 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 literal 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 and try to make this idea clear by appealing to multi-valued logic.

There is surely much more to 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.

[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.’ (Schlick 1938: 340)
[2] See, for a contrast, Carnap’s unfortunate definition of scientific 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. 12.
[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] For an explanation of structural isomorphism see Ch. IV, sec. 3.
[8] Appendix of Chapter I, sec. 1.
[9] This position was supported by A. J. Ayer, Rudolf Carnap, Herbert Feigl and Hans Reichenbach (Cf. Misak 1995: 79-80).
[10] 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.).
[11] 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)
[12] 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… These are only cases of ‘stolen examples.’
[13]  From his magnificent short story, ‘El Tintorero Enmascarado Hákim de Merv.’
[14] The difficulty made him propose a more complicated solution that the logician Alonzo Church proved to be equally faulty (Cf. Church 1949).
[15]  I am surely not the first to notice this flaw. See Barry Gower 2006: 200.
[16] 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.
[17] 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.
[18]  Michael Dummett viewed the falsification rule as the ability to recognize under what conditions a proposition is false. 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 and their absence. (Cf. Dummett 1996: 62 f.)
[19] Today we know that Fermat couldn’t have written this seriously, since the mathematics of his time did not provide the means to prove his conjecture.
[20] 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.
[21] Oxford Dictionaries (internet).