by Corry Shores
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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other distracting mistakes, because I have not finished proofreading.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part I
Propositional Logic
8
First Degree Entailment
8.6
Paraconsistency and the Disjunctive Syllogism
Brief summary:
(8.6.1) On account of truth-value gluts, p ∧ ¬p ⊢ q is not valid in FDE, and thus FDE does not suffer from explosion (which happens when contradictions entail any arbitrary formula and thus a contradiction entails everything). (8.6.2) Both FDE and LP are paraconsistent logics, because in them it is invalid to infer any arbitrary formula from a contradiction. (8.6.3) Disjunctive syllogism (p, ¬p ∨ q ⊨ q) fails in FDE (set p to b and q to 0; b is a designated value but is not preserved), and it fails in LP (set p to i and q to 0.; i is a designated value, but it also is not preserved ). (8.6.4) Arguments for the material and strict conditional that use disjunctive syllogism are thus faulty on account of its invalidity (in FDE and LP). (8.6.5) Because disjunctive syllogism fails for the material conditional in FDE, so too does modus ponens fail for it as well, given their equivalence. This suggests that the material conditional does not adequately represent the real conditional. (8.6.6) Those who argue that disjunctive syllogism is intuitively valid can do so only by showing that truth-value gluts are invalid. They think that by saying one of two disjuncts is false (in a true disjunction) necessities the other disjunct be true. But we can also have the intuition that certain formulas should be both true and false. And suppose one of the disjuncts is ¬p, and suppose that p is both true and false. That does not necessitate that ¬p be just false; for, it would also be both true and false. In other words: “The truth of p does not rule out the truth of ¬p: both may hold” (154). Since ¬p is at least true, it does not necessitate that the other disjunct be true, and so we cannot infer that the other disjunct is true. For, only one needs to be at least true. So if we start with the intuition that there can be truth-value gluts, then disjunctive syllogism is intuitively invalid. (8.6.7) A more convincing defense of the disjunctive syllogism is that we rely on it for reasoning well. Often times we know either of two things can be true; when one proves false, we know it must be the other one. (8.6.8) Even though disjunctive syllogism is invalid, it still functions quite well for normal everyday reasoning. It only fails when there is a truth-value glut. Otherwise, our daily life presents us normally with consistencies, so it will still deliver correct inferences usually. We just need to be careful to distinguish those cases with gluts and remember not to use it then. (8.6.9) There is precedent for this sort of discrimination of situations for appropriate inference uses in mathematics, so we should not feel too uncomfortable with it in cases of logical reasoning. For example, when dealing with finite sets, if one set is a proper subset of another, we can infer that it is smaller. But for infinite sets, we cannot draw that inference. For example, the set of even numbers is a proper subset of the set of natural numbers, but both sets have the same size. (8.6.10) Since we are wiling to accept inference discrimination in mathematics, we can surely accept it in logic, and so we can set aside the objection that we must reject truth-value gluts (or that we need the material conditional) simply because we need disjunctive syllogism to reason properly.
[The Lack of Explosion in FDE]
[FDE and LP as Paraconsistent Logics, Here Defined]
[The Failure of Disjunctive Syllogism in FDE and LP]
[Disjunctive Syllogism as Inoperable in Defenses of the Material and Strict Conditionals]
[The Failure of Modus Ponens for the Material Conditional in FDE]
[The Intuitive Invalidity of Disjunctive Syllogism]
[Disjunctive Syllogism as Possibly Needed for Reasoning]
[Disjunctive Syllogism as Normally Correct Despite Being Invalid]
[The Precedent for Inference Discriminations in Mathematics]
[Putting the Objection Aside]
Summary
[The Lack of Explosion in FDE]
[On account of truth-value gluts, p ∧ ¬p ⊢ q is not valid in FDE, and thus FDE does not suffer from explosion (which happens when contradictions entail any arbitrary formula and thus a contradiction entails everything).]
[Recall from section 8.4.8 that the tableau for p ∧ ¬p ⊢ q in FDE is open and thus it is invalid in FDE. Priest notes now that this is because of truth-value gluts. (Priest here uses semantic entailment: p ∧ ¬p ⊨ q. Recall from section 8.4.4 that the designated values in FDE are 1 and b. ((And recall from section 7.2.2 that designated values are those that are preserved in valid inferences.)) So an inference in FDE is valid only if there is no interpretation that assigns all the premises 1 or b and the conclusion 0 or n. But suppose for p ∧ ¬p ⊨ q that q is 0 and p is b (meaning both values, and thus a glut). That makes ¬p be b also (see section 8.4.2). Then we have all the premises as b and the conclusion as 0, and thus on account of the truth-value glut of b, this formula is not valid.) And recall from section 8.4.11 that p ⊢ q ∨¬q also makes an open tableau in FDE and is thus invalid. Priest notes now that it is so on account of truth-value gaps. (Suppose p is 1 and q is n ((meaning neither value, a gap)). That makes the whole conclusion n ((again, see section 8.4.2)), and thus the premises are 1 but the conclusion is n, making it invalid.) Now recall from section 4.8 that explosion is when contradictions entail any arbitrary formula and thus a contradiction entails everything. As we can see, since p ∧ ¬p ⊨ q is invalid in FDE, that means FDE does not have the problem of explosion.]
As we have seen (8.4.8 and 8.4.11), both of the following are false in FDE: p ⊨ q ∨ ¬q, p ∧ ¬p ⊨ q. This is essentially because there are truth-value gaps (for the former) and truth-value gluts (for the latter). In particular, then, FDE does not suffer from the problem of explosion (4.8).
(154)
[FDE and LP as Paraconsistent Logics, Here Defined]
[Both FDE and LP are paraconsistent logics, because in them it is invalid to infer any arbitrary formula from a contradiction.]
[Priest now gives the criteria for a paraconsistent logic. To be paraconsistent, the inference from p and ¬p to an arbitrary conclusion must be invalid. As we saw in section 8.6.1, p ∧ ¬p ⊨ q is invalid in FDE, thus FDE is a paraconsistent logic. And recall from section 7.4.4 that: p ∧ ¬p ⊭LP q. Thus LP is also a paraconsistent logic.]
A logic in which the inference from p and ¬p to an arbitrary conclusion is not valid is called paraconsistent. FDE is therefore paraconsistent, as is LP (7.4.4).
(154)
[The Failure of Disjunctive Syllogism in FDE and LP]
[Disjunctive syllogism (p, ¬p ∨ q ⊨ q) fails in FDE (set p to b and q to 0; b is a designated value but is not preserved), and it fails in LP (set p to i and q to 0.; i is a designated value, but it also is not preserved ).]
[The disjunctive syllogism is: p, ¬p ∨ q ⊨ q. But in FDE it fails. Suppose that p is both true and false, but q is just false. That means ¬p is both true and false (see section 8.2.6). With q being false, that means ¬p ∨ q is both true and false. Since the premises are all at least true and the conclusion false, that makes the inference invalid (see section 8.2.8, or use the validity criteria from section 8.4.4: the premises can be b but the conclusion 0.) Disjunctive syllogism also fails in LP (similarly set the value of p to i and q to 0. In LP, i is a designated value, but it is not preserved in the disjunctive syllogism. See section 7.4 for more on LP.)]
It is not only explosion that fails in FDE (and LP). The disjunctive syllogism (DS) is also invalid: p, ¬p ∨ q ⊭FDE q. (Relational counter-model: pρ1 and pρ0, but just qρ0.)
(154)
[Disjunctive Syllogism as Inoperable in Defenses of the Material and Strict Conditionals]
[Arguments for the material and strict conditional that use disjunctive syllogism are thus faulty on account of its invalidity (in FDE and LP).]
[Recall from section 1.7.2 that there are technically valid material conditionals that intuitively are invalid on account of the irrelevance of the consequent to the antecedent. (And in section 1.10 we saw a way of arguing for the intuitive validity of the material conditional that converts it into a disjunction (A⊃B becomes ¬A∨B); we then assume A and use disjunctive syllogism to derive B. I did not follow this section well enough, but my guess was that the logical connection between A and B was shown in its validation through disjunctive syllogism. Priest says now that this is a problematic argument, but I do not know why. I am guessing it is because it validates irrelevant conditionals. So maybe the idea now is that because disjunctive syllogism grounds the material conditional, and because the material conditional suffers from irrelevance, then perhaps there is something wrong with the disjunctive syllogism. These are guesses.) And recall from section 4.9.2 how C.I. Lewis uses disjunctive syllogism to argue for the intuitive validity of explosive arguments where any irrelevant conclusion can be inferred from a contradiction. The following comes from our paragraph summary from that section:
C.I. Lewis argues that (A ∧ ¬A) ⥽ B is intuitively valid, because from A ∧ ¬A it is intuitively valid to infer A and ¬A; from ¬A it is intuitively valid to infer ¬A ∨ B, and from A and ¬A ∨ B it is intuitively valid, by disjunctive syllogism, to derive B. [Now, if each step has a connection on the basis of its intuitive validity, that means the final conclusion B should have a connection, by extension, to A ∧ ¬A on the basis of the intuitively valid steps leading from the premise to the final conclusion. So despite objections to the contrary, there is a connection between the antecedent and consequent in (A ∧ ¬A) ⥽ B, according to Lewis.]
(paragraph summary of section 4.9.2. These are not Priest’s words and are probably mistaken.)
Priest’s point seems to be that these defenses for the material and strict conditionals fail, because they use disjunctive syllogism, which is not valid in situations where there are value-gluts. But I am not really sure how that applies, because these defenses assume there cannot be value-gluts, as they use classical logic. Let me quote:]
This is a significant plus. We have seen the DS involved in two problematic arguments: the argument for the material conditional of 1.10, and the Lewis argument for explosion of 4.9.2.We can now see that these arguments do not work, and (at least one reason) why.6
(154)
6. For good measure, the argument of 4.9.3 for the validity of the inference from A to B ∨ ¬B is also invalid in FDE, since p ⊭ (p ∧ q) ∨ (p ∧ ¬q), as may be checked.
(154)
[The Failure of Modus Ponens for the Material Conditional in FDE]
[Because disjunctive syllogism fails for the material conditional in FDE, so too does modus ponens fail for it as well, given their equivalence. This suggests that the material conditional does not adequately represent the real conditional.]
[I do not follow the next idea, but let me work through it. “Note, also, that the DS is just modus ponens for the material conditional” (Priest 154). This is what we found I think in section 4.9.2. There we noted that (not Priest’s words and so do not trust them:) “By modus ponens, from A, A ⥽ B we can infer B. And, A ⥽ B is equivalent ¬A ∨ B. And as we see, by disjunctive syllogism from A, ¬A ∨ B we can infer B.” Next Priest writes “Since this fails, we have another argument against the adequacy of the material conditional to represent the real conditional” (Priest 154). (I am a little confused here. Modus ponens also fails for the conditional in LP (see section 7.4.5). Does this criticism apply also to LP? I will quote.)]
Note, also, that the DS is just modus ponens for the material conditional. Since this fails, we have another argument against the adequacy of the material conditional to represent the real conditional.
(154)
[The Intuitive Invalidity of Disjunctive Syllogism]
[Those who argue that disjunctive syllogism is intuitively valid can do so only by showing that truth-value gluts are invalid. They think that by saying one of two disjuncts is false (in a true disjunction) necessities the other disjunct be true. But we can also have the intuition that certain formulas should be both true and false. And suppose one of the disjuncts is ¬p, and suppose that p is both true and false. That does not necessitate that ¬p be just false; for, it would also be both true and false. In other words: “The truth of p does not rule out the truth of ¬p: both may hold” (154). Since ¬p is at least true, it does not necessitate that the other disjunct be true, and so we cannot infer that the other disjunct is true. For, only one needs to be at least true. So if we start with the intuition that there can be truth-value gluts, then disjunctive syllogism is intuitively invalid.]
[This next idea is quite potent. So disjunctive syllogism fails for FDE and LP. One might then say that this means FDE and LP are flawed. Their reasoning goes as follows. Suppose we have ¬p ∨ q and it is true. This means that at least one of the two conjuncts must be true. It also means, according this argument, that if p were true, that means ¬p is false, and hence, by disjunctive syllogism, q must be true. But as soon as we accept truth-value gluts, then the truth of p will not necessitate that ¬p be simply false. For, p can be true and false, and thus so can ¬p. Then, for our original conjunct, we cannot infer that q is true. It can be false, because ¬p is at least true, even though it is also false. So we begin with the intuition that there can be formulas that are both true and false, we find that this means disjunctive syllogism fails, and thus we can say that disjunctive syllogism is intuitively invalid.]
The failure of the DS has also been thought by some to be a significant minus. First, it is claimed that the DS is intuitively valid. For if ¬p ∨ q is true, either ¬p or q is true. But, the argument continues, if p is true, this rules out the truth of ¬p. Hence, it must be q that is true. But once one countenances the possibility of truth-value gluts, this argument is patently wrong. The truth of p does not rule out the truth of ¬p: both may hold. From this perspective, the inference is intuitively invalid.
(154)
[Disjunctive Syllogism as Possibly Needed for Reasoning]
[A more convincing defense of the disjunctive syllogism is that we rely on it for reasoning well. Often times we know either of two things can be true; when one proves false, we know it must be the other one.]
[A more convincing objection is that we rely on the disjunctive syllogism to reason well: “Thus, we know | that you are either at home or at work. We ascertain that you are not at home, and infer that you are at work – which you are” (154-155).]
A more persuasive objection is that we frequently use, and seem to need to use, the DS to reason, and we get the right results. Thus, we know | that you are either at home or at work. We ascertain that you are not at home, and infer that you are at work – which you are. If the DS is invalid, this form of reasoning would seem to be incorrect.
(154-155)
[Disjunctive Syllogism as Normally Correct Despite Being Invalid]
[Even though disjunctive syllogism is invalid, it still functions quite well for normal everyday reasoning. It only fails when there is a truth-value glut. Otherwise, our daily life presents us normally with consistencies, so it will still deliver correct inferences usually. We just need to be careful to distinguish those cases with gluts and remember not to use it then.]
[Priest then says that even though disjunctive syllogism fails, it can still be legitimate to use it much of the time. It only fails in cases of truth-value gluts. But since the world normally presents us with consistencies, it is fine to still use it in much of our everyday reasoning.]
If the DS fails, then the inference about being at home or work is not deductively valid. It may be perfectly legitimate to use it, none the less. There are a number of ways of spelling this idea out in detail, but at the root of all of them is the observation that when the DS fails, it does so because the premise p involved is a truth-value glut. If the situation about which we are reasoning is consistent – as it is, presumably, in this case – the DS cannot lead us from truth to untruth. So it is legitimate to use it. This fact will underwrite its use in most situations we come across, since consistency is, arguably, the norm.
(155)
[The Precedent for Inference Discriminations in Mathematics]
[There is precedent for this sort of discrimination of situations for appropriate inference uses in mathematics, so we should not feel too uncomfortable with it in cases of logical reasoning. For example, when dealing with finite sets, if one set is a proper subset of another, we can infer that it is smaller. But for infinite sets, we cannot draw that inference. For example, the set of even numbers is a proper subset of the set of natural numbers, but both sets have the same size.]
[Priest’s next point seems to be an analogy to show us that such sorts of discriminations for determining appropriate sorts of inferences is seen already in rigorous sorts of mathematical contexts. His example is that whenever we are working with collections that are finite, it is fine to infer that if a set is a proper subset of another set, then it is smaller than that other set. But we cannot draw this inference when working with infinite sets, because, for example, the set of even numbers is a proper subset of the set of natural numbers, but both sets are of the same size.]
In the same way, if we have some collection, X, one cannot infer from the fact that some other collection, Y, is a proper subset of X that it is smaller.7 But provided that we are working with collections that are finite, this inference is perfectly legitimate: violations can occur only when infinite sets are involved.
(155)
7. For example, the set of all natural numbers is the same size as the set of all even numbers, as can be seen by making the following correlation:
xxxx0xx1xx2xx3xx4xx...
xxxx↕xx↕xx↕xx↕xx↕
xxxx0xx2xx4xx6xx8xx...
(155)
[Putting the Objection Aside]
[Since we are wiling to accept inference discrimination in mathematics, we can surely accept it in logic, and so we can set aside the objection that we must reject truth-value gluts (or that we need the material conditional) simply because we need disjunctive syllogism to reason properly.]
[From this Priest concludes: ]
Thus, this objection can also be set aside.
(155)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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