by Corry Shores

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[Priest,* Introduction to Non-Classical Logic*, entry directory]

[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

6

Intuitionistic Logic

6.2

Intuitionism: The Rationale

Brief summary:

(6.2.1) To understand the original rationale for intuitionism, we should note that we can understand strange sentences that we never heard before, like, “Granny had led a sedate life until she decided to start pushing crack on a small tropical island just south of the Equator.” (6.2.2) We can understand such complex unfamiliar sentences on account of compositionality, which says that “the meaning of a sentence is determined by the meanings of its parts, and of the grammatical construction which composes these” (103). (6.2.3) An orthodox view of meaning is that the meaning of a statement is given by its truth conditions (“the conditions under which it is true”). On account of compositionality, statements built up using connectives are determined on the basis of the connectives’ truth-functional operation on the truth conditions of the constituent statements. (6.2.4) The common notion of truth is that it is a correspondence between what a linguistic formulation says and an extra-linguistic reality in which that said circumstance in fact holds. But we think there are mathematical truths and meaningful formulations, yet the idea of an extra-linguistic reality is problematic in mathematics, as we will see. (6.2.5) Mathematical realists hold that there is an extra-linguistic reality corresponding to the truths of mathematical formulations like “2 + 3 = 5;” they think for example that there are “objectively existing mathematical objects, like 3 and 5.” Intuitionists however see this as a sort of mystical view and think rather that we should not apply the correspondence theory of truth to mathematical formulations. (6.2.6) Intuitionism expresses a statement’s meaning on the basis of its proof conditions, which are the conditions under which the sentence is proved. (6.2.7) The proof condition of a simple sentence is whatever we would take to be a sufficient proof [as for example a sufficient mathematical proof for a mathematical formula.] The proof conditions for complex sentences built up using connectives will be similar to the normal conditions only now using the notion of proof (note that ⇁ and ⊐ symbolize negation and the conditional):

A proof of

A∧Bisapair comprisinga proof ofAand a proof ofB.A proof of

A∨Bisaproof ofAoraproof ofB.A proof of ⇁

Aisa proof that there is no proof ofA.A proof of

A⊐BisAconstruction that, given any proof ofA, can be applied to giveaproof ofB.(104)

(6.2.8) These proof conditions cannot validate excluded middle, because there are formulas that cannot be proved nor can it be proven that there is no proof for them.

[The Original Rationale of Intuitionism. Understanding Unfamiliar Sentences]

[Compositionality]

[Truth Conditions and Compositionality]

[The Correspondence Theory of Truth’s Requiring an Extra-Linguistic Reality. The Potential Problem of Such for Mathematics.]

[Mathematical Realism versus Intuitionism]

[Intuitionism and Proof Conditions for Expressing Statement Meanings]

[The Proof Conditions]

[Intuitionism and Excluded Middle]

Summary

[The Original Rationale of Intuitionism. Understanding Unfamiliar Sentences]

[To understand the original rationale for intuitionism, we should note that we can understand strange sentences that we never heard before, like, “Granny had led a sedate life until she decided to start pushing crack on a small tropical island just south of the Equator.” ]

Priest is discussing the “original rationale for intuitionism.” He begins with a sentence that probably we never read before: “Granny had led a sedate life until she decided to start pushing crack on a small tropical island just south of the Equator.” Even though we never heard this sentence before, we can still understand what it means. The question is, then, how can we understand sentences we never heard before?

[Compositionality]

[We can understand such complex unfamiliar sentences on account of compositionality, which says that “the meaning of a sentence is determined by the meanings of its parts, and of the grammatical construction which composes these” (103).]

One explanation for why we understand unfamiliar sentences is that “we understand its individual parts and the way they are put together.” This is on account of *compositionality*: “the meaning of a sentence is determined by the meanings of its parts, and of the grammatical construction which composes these” (103). [There is a posting here on compositionality. Priest discusses compositionality and Frege also in section 12.1 of *Beyond the Limits of Thought*.]

[Truth Conditions and Compositionality]

[An orthodox view of meaning is that the meaning of a statement is given by its truth conditions (“the conditions under which it is true”). On account of compositionality, statements built up using connectives are determined on the basis of the connectives’ truth-functional operation on the truth conditions of the constituent statements.]

Priest then notes an “orthodox view” of meaning that is “usually attributed to Frege” (103). This view says that “the meaning of a statement is given by the conditions under which it is true, its *truth conditions*.” [I am not sure if this is something similar to Tarski conditions.] If we combine this idea with compositionality from section 6.2.2, then we would say that the meaning-providing truth conditions for complex statements that are built up using the logical connectives would be based on the truth conditions of the parts with respect to the truth-functional modifications of the connectives:

An orthodox view, usually attributed to Frege, is that the meaning of a statement is given by the conditions under which it is true, its

truth conditions. Thus, by compositionality, the truth conditions of a statement must be given in terms of the truth conditions of its parts. Thus, for example, ¬Ais true iffAis not true;A∧Bis true iffAis true andBis true; and so on.(103)

[The Correspondence Theory of Truth’s Requiring an Extra-Linguistic Reality. The Potential Problem of Such for Mathematics.]

[The common notion of truth is that it is a correspondence between what a linguistic formulation says and an extra-linguistic reality in which that said circumstance in fact holds. But we think there are mathematical truths and meaningful formulations, yet the idea of an extra-linguistic reality is problematic in mathematics, as we will see.]

[In section 6.2.3, we have noted the role of truth in determining meaning.] Priest next discusses a correspondence type of understanding of truth. Here we would think that truth is a matter of a linguistic formulation saying something about an extra-linguistic reality that in fact holds in that extra-linguistic reality. Yet, Priest notes, this idea of an objective extra-linguistic reality is problematic especially for mathematics, and he next will explain why.

Now, truth, as commonly conceived, is a relationship between language and an extra-linguistic reality. Thus, ‘Brisbane is in Australia’ is true because of certain objective social and geographical arrangements that obtain in the southern hemisphere of our planet. But many have found the notion of an objective extra-linguistic reality problematic – for mathematics, in particular.

(104)

[Mathematical Realism versus Intuitionism]

[Mathematical realists hold that there is an extra-linguistic reality corresponding to the truths of mathematical formulations like “2+3 = 5;” they think for example that there are “objectively existing mathematical objects, like 3 and 5.” Intuitionists however see this as a sort of mystical view and think rather that we should not apply the correspondence theory of truth to mathematical formulations.]

We now ask about extra-linguistic reality in relation to mathematical formulations. Is there such “an extra-linguistic reality that corresponds to the truth of ‘2+3 = 5’?” Mathematical realists claim that in fact there are such “objectively existing mathematical objects, like 3 and 5.” [But such an “object” would seem to not have any physical substantiality of its own.] Others do not take this realist view, because it seems a bit like a mystical notion of such an objectivity. For example, mathematical intuitionists reject the application of the common correspondence theory of truth in mathematics.

(104)

What is the extra-linguistic reality that corresponds to the truth of ‘2+3 = 5’? Some (mathematical

realists) have suggested that there are objectively existing mathematical objects, like 3 and 5. To others, such a view has just seemed like mysticism. These include mathematical intuitionists, who rejected the common conception of truth, as applied to mathematics, for just this reason.(104)

[Intuitionism and Proof Conditions for Expressing Statement Meanings]

[Intuitionism expresses a statement’s meaning on the basis of its proof conditions, which are the conditions under which the sentence is proved.]

[In section 6.2.3 we discussed the notion that the meaning of a statement is given by its truth conditions. And in section 6.2.4 we said that truth conditions would normally be understood in terms of a correspondence between the linguistic expression and an extra-linguistic reality that it accurately states. But when we remove that correspondence to an extra-linguistic reality, we then have the problem of understanding the meaning of a statement, because we can no longer use correspondence truth conditions.] We then have the problem of explaining how in intuitionism the meaning of statements is expressed. Instead of basing it on truth conditions, they rather base it on “the conditions under which it is proved, its *proof conditions* – where a proof is a (mental) construction of a certain kind” (104)

But in this case, how is meaning to be expressed? The intuitionist answer is that the meaning of a sentence is to be given, not by the conditions under which it is true, where truth is conceived as a relationship with some external reality, but by the conditions under which it is proved, its

proof conditions– where a proof is a (mental) construction of a certain kind.(104)

[The Proof Conditions]

[The proof condition of a simple sentence is whatever we would take to be a sufficient proof (as for example a sufficient mathematical proof for a mathematical formula.) The proof conditions for complex sentences built up using connectives will be similar to the normal conditions only now using the notion of proof. Note that ⇁ and ⊐ symbolize negation and the conditional.]

We now explain more formally what these proof conditions are [see section 6.2.6 above]. We first deal with simple sentences, the propositional parameters [atomic sentences]. We suppose that “we know what counts as* *a* *proof of the simplest sentences.” [I do not know yet what that is. Perhaps we are sticking with mathematical formulas, which have their own sort of mathematical proofs. In Nolt’s *Logics *section 16.2 “Intuitionistic Logics,” he might be portraying proofs more broadly to include empirical sorts of data that confirm or deny a claim.] Then, we consider the proof values of formulas that are built up by connectives. We give rules for these connectives (with ⇁ and ⊐ symbolizing negation and the conditional) that are intuitively like what we would expect for the proof-functionality of these connectives:

Thus, supposing that we know what counts as

aproof of the simplest sentences (propositional parameters), the proof conditions for sentences constructed using the usual propositional connectives are as follows. In the following sections, it will make matters easier if we use new symbols for negation and the conditional. Hence, we will now write these as ⇁ and ⊐, respectively.A proof of

A∧Bisapair comprisinga proof ofAand a proof ofB.A proof of

A∨Bisaproof ofAoraproof ofB.A proof of ⇁

Aisa proof that there is no proof ofA.A proof of

A⊐BisAconstruction that, given any proof ofA, can be applied to giveaproof ofB.(104)

[Intuitionism and Excluded Middle]

[These proof conditions cannot validate excluded middle, because there are formulas that cannot be proved, nor can it be proven that there is no proof for them.]

There are some interesting consequences of these proof conditions, with a “notorious” one being that these conditions cannot verify the law of excluded middle. [The basic idea here seems to be the following. There are certain mathematical ideas for which we currently have no proof for, nor do we have a proof that there is no proof for them. Call one such formula *A*. So we have not met the proof conditions for proving the atomic formula *A*. Recall that the proof condition for negation is: “A proof of ⇁*A* is* *a proof that there is no proof of *A*.” This means that we have not met the proof conditions of ⇁*A. *And the proof condition for disjunction is: “A proof of* A *∨ *B* is* *a* *proof of* A *or* *a* *proof of *B*.” So suppose our *B* here is ⇁*A *and we formulate *A *∨ ⇁*A*. We therefore have not satisfied the proof conditions for this particular disjunction, which shows that excluded middle is invalid in intuitionistic logic.]

Note that these conditions fail to verify a number of standard logical principles – most notoriously, some instances of the law of excluded middle:

A∨ ⇁A. For example, a famous mathematical conjecture whose status is currently undecided is the twin prime conjecture: there is an infinite number of pairs of primes, two apart, like 3 and 5, 11 and 13, 29 and 31. Call this claimA.Then there is presently no proof of A; nor is there a proof that there is no proof ofA.Hence, there is no proof ofA∨ ⇁A, which | claim is not, therefore, acceptable. Thus, intuitionism generates a quite distinctive logic.(104-105)

From:

Priest, Graham. 2008 [2001]. *An Introduction to Non-Classical Logic: From If to Is*, 2nd edn. Cambridge: Cambridge University.

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