by Corry Shores
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[The following is summary. My commentary is in brackets. Boldface in quotations is mine unless otherwise indicated. Proofreading is incomplete, so please excuse the typos.]
Graham Priest
Doubt Truth To Be a Liar
Part 2 Negation
Ch.4 Contradiction
4.1 Introduction [to Ch.4 Contradiction]
Brief summary:
Throughout the history of logic, there have been a variety of accounts of negation. One matter of contention is what can be inferred from negations that generate contradictions. The different accounts can be categorized according to whether the contradictions entail: {1} nothing (as in medieval connexivist accounts), {2} something but not everything (as in paraconsistent and relevant logics), and {3} everything (as in classical and intuitionistic logics). The view on negation that interests us is a non-trivial dialetheism where explosion does not hold, that is to say, an account in which the following inference is invalid:
α, ¬α ⊢ β, for an arbitrary β
Summary
4.1.1
[Non-trivial dialetheism allows for special instances of negation involved in contradictions which do not allow us to infer any arbitrary formula.]
[In section 2, Priest examined six accounts of truth. He says in section 2.8, the conclusion, that none of these accounts “provides any reason for rejecting dialetheism,” and in fact, “a number of them even point in its direction” (55). He further concludes now in this section that] “there is nothing in the notion of truth that prevents dialetheism from being acceptable” (75). We will now see that dialetheism holds in light of our intuitions about negation. There are however accounts of negation that rule out dialetheism” (75). For example, classical and intuitionistic logics make negation explosive (α, ¬α ⊢ β, for an arbitrary β). As such, they rule out dialetheism, “unless, of course, one is a trivialist” (75). [So probably a dialetheist is one who holds that only certain contradictions are true, and from none of those can we infer any arbitrary formula.] A good view of negation (a dialetheic one) would allow for it to hold for certain contradictions without that leading to triviality.
4.1.2
[Logical constants, like negation and the conditional, have been debated throughout the history of logic.]
Negation is currently considered in logic a “logical constant,” which is a class of logical constants that includes the conditional as well. Throughout the history of logic, the way to analyze such concepts has been heavily debated (75).
4.1.3
[Throughout the history of logic, there have been a variety of accounts of negation, but they can be categorized according to accounts in which contradictions entail: {1} nothing (as in medieval connexivist accounts), {2} something but not everything (as in paraconsistent and relevant logics), and {3} everything (as in classical and intuitionistic logics).]
Negation has seen many rival conceptions in logic. In the 20th century, for example, there were the theories of negation given in the “‘classical’ (Frege/Russell) account and the intuitionist (Brouwer/Heyting) account” (75). Priest notes that there have been rival accounts of negation throughout history. We recall from section 1.13 the three sorts of negation:
those according to which contradictions entail:
1. nothing;
2. something (but not everything);
3. everything.
(75)
Priest says that the third view includes the classical and intuitionist accounts and as well certain later Medieval accounts. The first view includes connexivist accounts from the Middle Ages. And the second view includes contemporary paraconsistent and relevant logics (76).
Graham Priest. 2006. Doubt Truth To Be a Liar. Oxford: Oxford University, 2006.
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