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 unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part I:
Propositional Logic
7.
Many-Valued Logics
7.7
Truth-value Gluts: Paradoxes of Self-reference
Brief summary:
(7.7.1) We will now consider paradoxes of self-reference as motivation for advocating for truth-value gluts. (7.7.2) One paradox of self-reference is the liar’s paradox. For example, ‘this sentence is false’. “Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false” (129). (7.7.3) Another paradox of self-reference is Russell’s Paradox: “Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false. “ (7.7.4) There are many such arguments that come to a conclusion of the form A∧¬A, and supposing they are sound, that makes the conclusions true and thus means there really are truth-value gluts. (7.7.5) We will now examine briefly a couple claims that these paradoxical arguments are not sound. (7.7.6) Objection 1: All self-referential sentences are meaningless. Reply 1: But, there are many such meaningful sentences, like, ‘this sentence has five words’. (7.7.7) Objection 2: The liar sentence is neither true nor false. Thus our logical assumptions remove excluded middle, and we cannot develop the argument as, “either it is true or false; if false, then thus; if true then false; thus ...”. For, now we have a third situation, that it is neither. (7.7.8) Reply 2: “Extended Paradoxes” still present a contradiction. For example: “This sentence is either false or neither true nor false”. If true, it is either false or neither value. Either way, it is not true, which contradicts our assumption that it is true. If it is either false or neither valued (meaning that it is not true), then its value is what it claims to be, and thus it is true, which contradicts what we assumed. Reply 3: Some paradoxes of self-reference, like Berry’s paradox, do not invoke the law of excluded middle.
[Paradoxes of Self-Reference as Motivation for Gluts]
[The Liar’s Paradox]
[Russell’s Paradox]
[These Paradoxes of Self-Reference as Showing Truth-Value Gluts]
[Turning to Claims that the Paradoxes are not Sound]
[Objection 1: Self-Referential Sentences Are Meaningless. Reply 1: Not So in Many Cases]
[Objection 2: The Liar Sentence Is Neither True nor False]
[Reply 2: Extended Paradoxes Still Produce Contradiction. Reply 3: Not All Paradoxes of Self-Reference Invoke the Law of Excluded Middle]
Summary
[Paradoxes of Self-Reference as Motivation for Gluts]
[We will now consider paradoxes of self-reference as motivation for advocating for truth-value gluts.]
[In the previous section 7.6, we examined a motivation for arguing for truth-value gluts, namely, inconsistent laws. We consider now another motivation: paradoxes of self-reference. There are both old and modern ones.]
A second argument for the existence of truth-value gluts concerns the paradoxes of self-reference. There are many of these; some very old; some very modern. Here are a couple of well-known ones.
(129)
[The Liar’s Paradox]
[One paradox of self-reference is the liar’s paradox. For example, ‘this sentence is false’. “Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false” (129).]
[(ditto)]
THE LIAR PARADOX: Consider the sentence ‘this sentence is false’. Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false.
(129)
[Russell’s Paradox]
[Another paradox of self-reference is Russell’s Paradox: “Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false. “]
[(ditto) (See section P.6 of One and ch.5 of Logic: A Very Short Introduction.)]
RUSSELL’S PARADOX: Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false.
(129)
[These Paradoxes of Self-Reference as Showing Truth-Value Gluts]
[There are many such arguments that come to a conclusion of the form A∧¬A, and supposing they are sound, that makes the conclusions true and thus means there really are truth-value gluts.]
[(ditto)]
These (and many others like them) are both prima facie sound arguments, and have conclusions of the form A∧¬A. If the arguments are sound, the conclusions are true, and hence there are truth-value gluts.
(129)
[Turning to Claims that the Paradoxes are not Sound]
[We will now examine briefly a couple claims that these paradoxical arguments are not sound.]
[(ditto)]
Many people have claimed that the arguments are not, despite appearances, sound. The reasons given are many and complex; let us consider, briefly, just a couple.
(129)
[Objection 1: Self-Referential Sentences Are Meaningless. Reply 1: Not So in Many Cases]
[Objection 1: All self-referential sentences are meaningless. Reply 1: But, there are many such meaningful sentences, like, ‘this sentence has five words’.]
[(ditto)]
Some have argued that any sentence which is self-referential, like the liar sentence, is meaningless. (Hence, such sentences can play no role in logical arguments at all.) This, however, is clearly false. Consider: ‘this sentence has five words’, ‘this sentence is written on page 129 of Part I of An Introduction to Non-Classical Logic’, ‘this sentence refers to itself’.
(129)
[Objection 2: The Liar Sentence Is Neither True nor False]
[Objection 2: The liar sentence is neither true nor false. Thus our logical assumptions remove excluded middle, and we cannot develop the argument as, “either it is true or false; if false, then thus; if true then false; thus ...”. For, now we have a third situation, that it is neither.]
[The second objection is the most popular one. It says that the liar sentence is neither true nor false. This means that we cannot appeal to the law of excluded middle (because it is no longer the case that the sentence is either true nor false. How did we use it previously? I am not really sure. Maybe it goes like this, but I am guessing. We have the sentence, “this sentence is false.” Then we say, “Either it is true or it if false. If it were true, then it is false, and if it is false, then it is true. Either way, it is both true and false.” So maybe, the argument works by having the original proposal that it is either true or false. And maybe the idea now is that were it neither value, then the law of excluded middle does not hold, and so we cannot start off with the assumption “Either it is true or it is false; if true ...”. Or maybe we can start it that way, but it cannot end that way, because we have the third possibility to assess, that it is neither, meaning that we cannot further derive another value from it in addition to it being neither. I am not sure.) “Thus, the paradoxes of self-reference are sometimes used as an argument for the existence of truth-value gaps, too” (129).]
The most popular objection to the argument is that the liar sentence is neither true nor false. In this case, we can no longer appeal to the law of excluded middle, and so the arguments to contradiction are broken. (Thus, the paradoxes of self-reference are sometimes used as an argument for the existence of truth-value gaps, too.)
(129)
[Reply 2: Extended Paradoxes Still Produce Contradiction. Reply 3: Not All Paradoxes of Self-Reference Invoke the Law of Excluded Middle]
[Reply 2: “Extended Paradoxes” still present a contradiction. For example: “This sentence is either false or neither true nor false”. If true, it is either false or neither value. Either way, it is not true, which contradicts our assumption that it is true. If it is either false or neither valued (meaning that it is not true), then its value is what it claims to be, and thus it is true, which contradicts what we assumed. Reply 3: Some paradoxes of self-reference, like Berry’s paradox, do not invoke the law of excluded middle.]
[(ditto)]
This suggestion does not avoid contradiction, however, because of ‘extended paradoxes’.3 Consider the sentence ‘This sentence is either false or neither true nor false.’ If it is true, it is either false or neither. In both cases it is not true. If, on the other hand, it is either false or neither (and so not true), then that is exactly what it claims, and so it is true. In either case, therefore, it is both true and not true.
(130)
3. Moreover, and in any case, not all of the paradoxical arguments invoke the law of excluded middle. Berry’s paradox, for example, does not.
(130)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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