draft
PREVENTING
EXPLOSION BY IMPROVING VALIDITY
“Well
then, don’t draw any conclusion from a contradiction. Make that a rule.”
Wittgenstein
In this short paper I
intend to get rid of the principle of explosion in the most straightforward
manner. I believe we can do this by cutting the Gordian knot that originates
the principle already on its first tie, namely, by emending our traditional
concept of validity. My ultimate intention, I confess, is to defend classical
logic.
I
The idea of the
principle of explosion is that from the
falsehood anything follows (ex falso
sequitur quodlibet), or from the
contradiction anything follows (ex
contradictione sequitur quodlibet). This last formulation is better, since
a contradiction is something usually considered undeniably false, while falsity
is so often deniable.
Now, consider two examples of the
application of this principle:
1.
It is raining and it is not raining, therefore Paris
is the capital of France.
2.
It is raining and it is not raining, therefore I have
a kangaroo in my pocket.
It does not matter that
the conclusion of (1) is true and that of (2) is false: both statements produce
in us the same natural reaction: we tend to see that such arguments are
foolish, unreasonable, illogical. A child would say that. And more than devoid
of meaning, they are absurd or counter-sensical – we understand by
these words something that goes against the meaning-rules of language.
Perhaps the wide acceptance of this
principle in the Middle-Ages (introduced by William of Soissons, in the XIIth
Century) had some appeal in the religious dogmatic milieu. Anyway, the fact is
that for systematic reasons the principle was later seriously endorsed by
classic symbolic logic through thinkers like Gottlob Frege, Bertrand Russell,
and David Hilbert (cf. Bobenrieth
2010).
Logicians are focused, not so much on the
truth of statements, but on the passage of truth from the premises to the
conclusions of arguments. Thus, by the resource of formalization, which deals
only with the logical structures of statements and the abstraction of their
semantic contents, they can perform the magic of transforming what at first
view seems unreasonable into something apparently respectable. This can be the
case when the principle of explosion is formalized. Here is a very clear and
simple way logic allows us to formalize the principle of explosion, where Q
(like P) is in the place of any true or false statement one could conceive:
Argument-form A:
1.
P & ~P Premise (contradiction)
2.
P 1 Simp.
3.
~P 1 Simp.
4.
~P v Q 3 DI
5.
Q 2, 4 DS
A serious problem with
the acceptance of the principle of explosion is that it disqualifies the
extremely useful procedure of reductio ad
absurdum, since from its application one will derive the assumed statement,
as well as its negation.
Paraconsistent logics (including most
relevant logics) has rejected explosion by trying to stop this argument at some
point, often by abandoning the rule of disjunction introduction (DI: A ├ A v B)
and/or the disjunctive syllogism (DS: A v B, ~A ├ B). For dialetheist logic,
which denies the principle of non-contradiction,[1]
it makes full sense to reject disjunctive syllogisms, since A and ~A can both
be true (Priest 2011). There are, however, serious problems with these moves.
The rejection of disjunction introduction or of disjunctive syllogism seems to
be at least as absurd as the acceptance of the principle of explosion.
Nevertheless, since I intend to stop explosion by demanding the conceivability
or possibility that we have all premises true and the conclusion true, some
relation between premises and conclusion must be predicted – though nothing
must be said about the nature of this relation.
II
What I will propose
here is a different and seemingly overly simple way of blocking the principle
of explosion, which is to improve our understanding of a fundamental logical
concept, that of validity. The
concept of validity is central for deductive logic, which in its fundamentals
concerns the investigation of valid arguments, which classical logic would
regard as the only logical ones. Thus, if we show that arguments using the
principle of explosion are invalid, we also show that they are illogical,
confirming our natural intuition. This would not lead us to paraconsistent
logic, which is defined as any logic in which the consequence relation derived
from contradiction is not explosive (Priest 2018), since by improving our
concept of validity I intend to show that there can be no consequence relation
derived from contradiction at all. This is why I think I am not jettisoning
classical logic.
There is a number of definitions of
validity. I chose some of them only as a way to clarify the idea:
1.
An argument is valid if and only if it has the following property:
The truth of its
premises guarantees the truth of its
conclusion.[2]
This first intuitive
definition can be somewhat unfolded, insofar as we explain what we mean with
the word ‘guarantee’ using more precise terms:
2.
An argument is valid if and only if it has the following property:
It would be inconsistent to assert the premises and
deny the conclusion.[3]
3.
An argument is valid if and only if it has the following property:
It is contradictory (impossible) to have the premises all true and its conclusion false.[4]
4.
An argument is valid if and only if it has the following property:
it is (logically) necessary that if its premises are true, its conclusion is also
true.[5]
Since (2) rejecting
inconsistency between the assertion of the premises and the falsity of the
conclusion amounts to the demand that if premises are all true the conclusion
must also be true, (2) amounts to the same as (4), demanding that from true
premises follows a true conclusion by
logical necessity. And since (3) rejecting as contradictory that the
premises are all true and the conclusion false also amounts to the same as (4),
all the definitions above are telling us the same thing.[6]
However, the most distinctive and common way
to define validity, which repeats the same thing that is meant by all the above
definitions is by using the word ‘must’ as the must of logical necessity:
5.
An argument is valid if and only if it has the following property:
If all its premises are
true, its conclusion must be true (or
cannot be false). [7]
Considering definition
(5), defenders of the principle of explosion can argue as follows: the negation
of definition (5) tells us that there is at least one valid argument in which
the premises are true and its conclusion is false. Consider now the
argument-form (A). Since its premise is a contradiction, and a contradiction
cannot be true, one can never find an argumentative instance that clashes against definition (5) of
validity, an argumentative instance in which the premise is true and the
conclusion false. Consequently, argument (A), formulating the principle of
explosion, is valid. This conclusion would be applicable to any other argument
based on necessary falsity, like an argument instantiating the argument-form
(B): ~(P → P) ├ Q or (maybe) 2 + 2 = 5 ├ Q. Since in any such cases there is no
positive example of premises true and conclusion false, they are said to be trivially or vacuously valid – but valid
anyway.
It is not the case that this conclusion
cannot be internally disputed. Concerning definition (1), one can ask (i): “Is
it the case that A satisfies the definition of validity, in which the truth of
its premises guarantees the truth of its conclusions?” Clearly not, since there
is no truth in the premise from (A). And also concerning definitions (2) to (5)
one could say the same. These responses, together with the above argument, can
lead us to argue that the reasons for the acceptance of validity of explosion
are insufficient. One can say (a): Argument-form (A) does satisfy the definition of validity by having no instance that
contradicts it, that is, no instance in which its premise is true and the
conclusion is false. But one can also say (b): Argument-form (A) does not satisfy the definition of validity,
since it cannot show that there is an instance confirming it, an interpretation
in which the premise is true and the conclusion cannot be false. At any rate, I
will accept here interpretation (a), even if mostly for argumentative reasons.
Now, my proposal to block the principle of
explosion is so simple that it can strike us as deceptively naïve. It consists
in the proviso that in order to be valid an argument needs to have at least one
possible true instance. Thus, my first definition of validity that blocks the
way to explosion consists in inserting in it the requirement of a logical
possibility of all premises being true, as follows:
6.
An argument is valid if and only if it has the following property:
If all its premises are
true, and if they can also all be
true, its conclusion must be true.
This definition
preserves the essence of validity suggested in (1)-(5). But (6) is already
sufficient to exclude beyond any doubt the possibility of a contradiction
staying as a premise of a valid argument. This definition blocks the
possibility of the principle of explosion from the start, since explosion
demands contradiction as a feasible premise. Since it also blocks the
possibility of a contradiction as a premise, it also saves axiomatic systems of
allowing contradiction without compromising their validity.
One objection to (6) would concern the ‘can’
of logical possibility. After all, it is not logical possibility that we are
aiming to explain with the concept of validity. The answer is that we are only
considering logical possibility regarding arguments, while the concept of logical
possibility is wider, including sentences. For instance: in classical logic the
sentence P → P is logically necessary, while P → ~P is logically impossible.
Moreover, we can replace the concept of
possibility by the concept of conceivability in definition (6), getting the
following result:
7.
An argument is valid if and only if it has the following property:
If all its premises are
true, and they are also conceivably
all true, its conclusion must be true.
Although I see both (6)
and (7) as adequate, I will choose (7) because of the clarity of the concept of
conceivability. We cannot conceive that it is both raining and not raining,
even if it is only something like a drizzling fog, not in my view because it is
true that it is both raining and not raining at the same time, as dialetheists
would like, but because in this case we suspend our judgment concerning the
truth-value of the assertion or of the negation that it is raining, often
changing the subject (“We are under a drizzling fog”). This is what we usually
do when arriving at undecidable cases.
We can test definitions (6) and (7) in a
series of real arguments by slowly debunking their plausibility. Consider the
following valid and sound argument:
1.
The sky is blue and the grass is green.
2.
Therefore: The sky is blue.
This argument plainly
satisfies definitions (6) and (7) of validity, since we feel that if all
premises are true the conclusion necessarily follows and that the premises are
also all possibly and conceivably true.
Consider now the following valid, but
unsound argument:
1.
All mice are dogs.
2.
All dogs eat cats.
3.
Therefore: all mice eat cats
Although unsound, this
is a valid argument according to definitions (6) and (7). If the premises are true,
the conclusion is necessarily true. Moreover, the truth of the premises is at
least logically possible or conceivably true, leaving aside their factual
impossibility.
Now, consider the following still valid, but
very obviously unsound argument:
1.
A fossil cannot be disappointed in love.
2.
An oyster can be disappointed in love.
3.
Therefore: oysters are not fossils.
Although obviously
unsound, this argument is valid according to definitions (1) to (5), since if
the premises were true, the conclusion would follow necessarily. What about
definitions (6) and (7)? I think that this can still be seen as a valid
argument according to them, since the truth of the premises is at least (with
considerable fantasy!) logically
(though not physically) possible or conceivable.
Now, what could be said about the following modus ponens?
1.
If Babticon is lapticon, then blablapt.
2.
Babticon is lapticon.
3.
Therefore: blablapt.
Note that this is here
not intended as a mere argument-form, but as a concrete argument. For those who
accept the definition (1) to (5) of validity, this argument can be seen as valid, though of course not
sound. The question, clearly, is if this is a valid argument according to
definitions (6) and (7). And the answer is not. For we cannot tell if it is logically
possible that Babticon is lapticon, nor if it is logically conceivable that its
truth implies blablapt… The reason is at hand: when the component-signs of an
argument are senseless, all that we could aim for would be an argument-form,
and an argument-form is not the same as an argument. The components of a real
argument must have semantic content.
I think there is an important lesson in the
foregoing progression. The last argument (and maybe also the second last)
offers an important lesson for logic, since it shows that our argument-forms
must have a ground in possible semantic contents in order to adequately express
logical structures. It is the lack of this ground that makes argument-form (A)
logically valid only in appearance. More precisely: According to our
traditional concept of validity, the possibility of having truth-value is a
sufficient semantic ground for inference. According to my proposal, this is not
enough. Possibility or conceivability of
truth must be added as a semantic ground for deductive logical inference.
The definition of validity demanding not only that if all premises are true the
conclusion must be true, but also demanding the possibility or conceivability
of all premises being true, requires more forcefully that argumentative forms
cannot be conceived in themselves deprived of such additional semantic grounds.
III
A logical reason for
the emergence of the principle of explosion can be the confusion of ‘├’ with
‘→.’ One should not oversee the different roles played by deductive logical
inference and a material implication.[8]
By overseeing this, one would object against definitions (6) and (7),
remembering that the truth-table of material implication, when applied to (P
& ~P) → Q, gives us a tautology. After all, we are all used to the
procedure of transforming any argument in a material implication of the
conjunction of its premises to its conclusion… From this perspective, it seems
that the proposed definition of validity has also undermined material
implication. But this would be an evident mistake, for it is possible to argue
that there is nothing wrong with the truth-table that defines material
implication. (P & ~P) → Q remains a tautology, as much as P → (Q v ~Q). The
mistake arises only when we treat (P & ~P) → Q as if it were an
argument-form, which isn’t possible, since it would demand that we treat P
& ~P as a premise with a truth-claim. However, this cannot be, since in
itself the antecedent of a material implication has no truth-value. Indeed,
while “If the circle is round and it is not round, then I am a circus monkey”
may lack intuitive sense, but “The circle is round and is not round, therefore
I am a circus monkey” seems to be plainly absurd.
Against the material conditional it has been
argued that it cannot be equated to the English conditional. My answer would
be: of course not! But they are also not outside the natural language either.
It is so because the material conditional does not belong to the natural language, not because it should be
equated with some conditional of our natural language, but because it should be
equated with a minimal condition for
natural language conditionals. As with other connectives of classical logic,
material implication deals with the
logically most basal layer of language, the least demanding and therefore
the most universal rule, upon which in usual cases new and more constraining
conditions are added.
One additional constraining layer was
clearly devised by Paul Grice’s pragmatic considerations about conditionals
(1989: Sec. I). According to him, there is nothing wrong in a material
implication like “If pigs fly then the moon is blue,” since normally we add
pragmatic conversational rules constraining mere material implication, like the
tacit rules of normal discourse, according to which (i) speakers should only
assert what they believe to be true and justified, and (ii) speakers should
assert neither more nor less than what they can. The place of these rules
regarding implication is made clearer when we remember that P → Q is the same
as ~P v Q. When we intend this as exclusive disjunction, we should not know in
advance which disjunct is true, since this misses the pragmatic point of
affirming a disjunction, which is the lack of knowledge of what disjunct is
true. So, we have no reason to say “Either pigs don’t fly or the moon is blue,”
since everyone already knows that only the first disjunct is true and the
second false. The same applies to a conditional with false antecedent and true
consequent like “If pigs fly then the moon is white”. We have no reason to
assert the disjunction “Pigs do not fly or the moon is white”, since everyone
already knows that in this inclusive disjunction both disjuncts are true.
It is true that Grice’s considerations do
not work as well regarding the so-called “paradoxes” of material implication
like P → (P → Q), ~P → (P → Q), (P → Q)
v (Q → R)… and still other more complex cases, like Curry’s “paradox”, ((P → (P
→ Q)) → (P → Q)), which are clearly devoid of meaning. Moreover, the lack of demand
of any relationship between antecedent and consequent of a material implication
forces us to accept the truth of statements like “If the sun is hot then the
grass is green” and “(3 > 2) → (5 = 5)”, since in these cases both
antecedent and consequent are true. However, in my view all these cases only seem to be artificial because we are
used to see “if… then” in the English sense of the word, and not in the
minimalist sense given by the material implication.[9]
Since antecedent and consequent are true but unrelated, we can say that their
instantiations are devoid of meaning, but not that they are absurd or
counter-sensical in the sense of going against our meaning rules, which means
that they still belong to the language.
Finally, two particularly important cases in
which material implication is insufficient are those of containment and causality.
If P implies Q in the sense in which Q is contained in P (for instance, P is
the series of natural numbers and Q is the series of even numbers), then, given
that P is true and Q is false, the claim of containment is prima facie[10] false; we are
only granted to have containment when P and Q are together true. Thus, the
expected truth-table for consequential containment (symbolized as ‘co’) of Q
under P should be the same as that of conjunction:
P Q P co→
Q
T T T
T F F
F T F
F F F
Of course, more could
be said about the conditions of containment, but this table selects the basal
condition. Moreover, we see that the second line, saying that the truth of the
antecedent cannot imply the falsity of the consequent, which constitutes the
core of the classic material implication, remains true.
Now, consider the case of causality. If A
causes B (for instance, under fully adequate circumstances a flame causes
heat), then it also seems that always when A is true B must follow. Thus, the
expected truth-table in which B is the causal consequence (symbolized as ‘ca’)
of A should be also that of conjunction:
P Q P ca→ Q
T T T
T F F
F T F
F F F
This constraint is
certainly insufficient, since there are also other conditions: the effect
typically follows after the cause and with some kind of necessity, etc. Anyway,
this seems to be the basal logic condition for causality. Here, as in the
previous case, it cannot be true that the antecedent is true and the consequent
false, which means that containment as well as causality preserve the status of
a material conditional as a basal, necessary though insufficient condition for
most kinds of entailment.
Now I will consider two examples that seem
to obliterate material implication: the first concerning containment, and the
second concerning causation.
Consider first the classically valid
argument of the form: (P → Q) & (R→ S) ├ (P → S) v (R → Q). One English
instantiation can be the following apparently invalid argument:
If John is in London then he is in England,
and if John is in Paris then he is in France.
Hence, if John is in London then he is in
France or if John is in Paris then he is in England.
If one uses the
truth-table of containment, the result will be an invalid argument, which is
intuitively correct. But if one uses the classic argument form, one will get a
valid argument, which seems counter-intuitive. However, it only seems counter-intuitive because we are
not used to the overly weak concept of material implication in classical logic.
This is made clear if we reformulate the conclusion using Grices’ device:
Either John is not in
London or he is in France or either John is not in Paris or he is in England.
Indeed, clumsy as it
seems, the alternative is logically possible: it is possible to be true that
John is not in London and/or he is in France and/or that John is not in Paris
and/or he is in England (using and/or for inclusive disjunction).
A case of causation is the following
classically valid argumentative form: (P & Q) → R ├ (P → R) v (Q → R). A
causal instantiation can be the following:
If
both switch A and switch B are closed, then the light is on.
Hence,
it is either true that if switch A is closed then the light is on, or that if
switch B is closed then the light is on.
In natural language it
seems that the premises can all be true and the conclusion false, since we
expect that the circuits need to be both closed in order to get the light on,
which renders the argument invalid. But this shows itself to be the case only
because we are used to the truth-table for causal consequence with ca→. If we
use the truth-table of material implication, the argument will be accepted as
valid. There is much more that needs to be said in defense of the thesis that a
material conditional is a basal conditional condition of natural language, but
since this is not our main issue, I will stop here.
LITERATURE
Anellis, I. H. (2007). Review of the Handbook of the History of
Logic, vol. 1, in The Review of
Modern Logic, vol. 10, n. 3 & 4, pp. 117-141.
Aristotle (1984). Metaphysics, in The Complete Works of Aristotle – The Revised Oxford Translation,
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University Press.
Beal, J. C. 2010. Logic: The Basics. London: Routledge.
Bobenrieth-Miserda, A.
(2010). “The Origins of the Use of the Argument of Trivialization in the
Twentieth Century”, in History and
Philosophical Logic, vol. 31, 2, pp. 111-121.
Feldman, Richard 1999. Reason and Argument, New Jersey:
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Fisher, Jennifer 2008. On the Philosophy of Logics, Belmont:
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“Logic and Conversation”, in his The Ways
of the Words. Cambridge, MA: Harvard University Press.
Mares, Edwin 2020.
“Relevant Logics”, in Stanford Encyclopedia
of Philosophy, ed. E. N. Zalta.
Newton-Smith, W. H.
1985. Logic: An Introductory Course.
London: Routledge & Kegan-Paul.
Novaes, C. D. 2020.
“Medieval Theories of Consequence”, in Stanford
Encyclopedia of Philosophy, ed. E. N. Zalta.
Priest, G. Tanaka, K.
Weber, Z. (2018): “Paraconsistent Logic”, in Stanford Encyclopedia of Philosophy, ed. E. N. Zalta.
Priest, Graham 2011.
“What is so Bad About Contradictions”, in G. Priest, J.C. Beall, B. Armour-Gab
(eds.), The Law of Non-Contradiction.
Oxford: Clarendon Press.
Read, Stephen 2012. Relevant Logic: A Philosophical Examination
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Restall, Greg 2006. Logic: An Introduction. London:
Routledge.
Salmon, Merrilee 2002. Introduction to Logical and Critical
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Salmon, Wesley 1973. Logic, Hoboken: Prentice Hall.
Shaw, Patrick 1999. Logic and its Limits, Oxford: Oxford
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Tomassi, Paul 1999. Logic, New York: Routledge.
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[1] In Aristotle’s words, “the same attribute cannot at
the same time belong and not belong to the same subject in the same respect”
(1984: Book 4, 1005b-20; see also Tugendhat 1983, Ch. 4).
[2] Feldman (1999: 61).
[3] Strawson (1963: 13), Patrick Shaw (1997: 24).
[4] Gensler (2002: 3), Restall (2006: 11-12).
[5] Feldman (1999: 61).
[6] Marilee
Salmon (2002: 88) notes against the acceptance of such definitions that they
lead us to the conclusion that the expression ‘valid deductive argument’ is
redundant, since the expression ‘invalid deductive argument’ is awkward. But
since her own definition of validity requiring that ‘the premises provide
sufficient level of support’ is too vague, I prefer to bite the bullet and
accept that the definition of validity is an essential constituent of the
definition of what is a deductively logical argument.
[7] See Salmon (1973: Ch. 2, sec. 5), Haack (1978: 14),
(Newton-Smith 1985: 2-3), Tomassi (1999: 5), Beall (2010: 5-8), Read (2012:
2-3, 19), Fisher (2008:6), Copi et all
(2014: 20), and many others.
[8] According to the historian of logic I. H. Anellis
(2007: 132), the principle later called ex
falso sequitur quodlibet had its origins already with the stoic Philon of
Megara (circa 300 BC), who was the
first to defend that the false could imply the truth. Moreover, medieval
philosophers like Jean Buridan have connected the principle of explosion with
the material conditional (Novaes 2020).
[9] The same could
be said about arguments of the form “(P → Q) & ~(R → Q) ├ (R → Q)”, where
the conclusion seems to be false.
[10] I say ‘prima facie’ because the condition to be given will be necessary
but not sufficient. For instance, if P is the series of natural numbers and Q
is the series of even numbers, ~Q can also belong to the natural numbers if it
is, for instance, the series of prime numbers.
