## 18 Jul 2017

### Priest (1.3) An Introduction to Non-Classical Logic, ‘Semantic Validity’, summary

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

1.
Classical Logic and the Material Conditional

1.3.
Semantic Validity

Brief summary:
An interpretation of an object language is a function, written v,  that assigns truth values to formulas, as for example: ν(p) = 1 and ν(q) = 0. For our classical logic semantics, the interpretation function assigns values for the connectives in the following way:
ν(¬A) = 1 if ν(A) = 0, and 0 otherwise.
ν(A ∧ B) = 1 if ν(A) = ν(B) = 1, and 0 otherwise.
ν(A ∨ B) = 1 if ν(A) = 1 or ν(B) = 1, and 0 otherwise.
ν(A ⊃ B) = 1 if ν(A) = 0 or ν(B) = 1, and 0 otherwise.
ν(A ≡ B) = 1 if ν(A) = ν(B), and 0 otherwise.
A conclusion A is a semantic consequence of a set of the premises Σ (that is, Σ ⊨ A) only if there is no interpretation that makes all the members of Σ true and A false, that is, only if every interpretation that makes all the members of Σ true makes A true as well. ‘Σ ⊭ A’ means there is not semantic consequence. A logical truth or tautology is a formula that is true under every evaluation, written for example as: ⊨ A. This also means it is a semantic consequence of the empty set of premises: φA.

Summary

1.3.1
[The interpretation of our object language in classical logic will assign a truth value to a propositional parameter, formulated for example as: ν(p) = 1 and ν(q) = 0.]

[Recall from section 1.1.5 that an interpretation can be understood “crudely” as “a way of assigning truth values.] Priest will define interpretation in classical logic. We think of it as a function that assigns truth values to propositional parameters.]
An interpretation of the language is a function, ν, which assigns to each propositional parameter either 1 (true), or 0 (false). Thus, we write things such as ν(p) = 1 and ν(q) = 0.
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1.3.2
[Our interpretation schema will assign truth values for the connectives in the normal classical logical way.]

Priest then provides the evaluation schema for assigning truth values to propositional parameters. As we will see, they coincide with the normal truth tables that we know for classical logic.
Given an interpretation of the language, ν, this is extended to a function that assigns every formula a truth value, by the following recursive clauses, which mirror the syntactic recursive clauses:
ν(¬A) = 1 if ν(A) = 0, and 0 otherwise.
ν(A ∧ B) = 1 if ν(A) = ν(B) = 1, and 0 otherwise.
ν(A ∨ B) = 1 if ν(A) = 1 or ν(B) = 1, and 0 otherwise.
ν(A ⊃ B) = 1 if ν(A) = 0 or ν(B) = 1, and 0 otherwise.
ν(A ≡ B) = 1 if ν(A) = ν(B), and 0 otherwise.
(5)

1.3.3
[A conclusion A is a semantic consequence of a set of the premises Σ (that is, Σ ⊨ A) only if it is semantically valid, which means that there is no interpretation that makes all the members of Σ true and A false. This can be also understood that every interpretation that makes all the members of Σ true makes A true as well. Whenever there is not this semantic consequence, we write for example: Σ ⊭ A.]

[Priest next defines semantic consequence. Recall the two types of validity from section 1.1: {1} Semantic validity (symbolized ⊨) which preserves truth: every interpretation that makes the premises true also makes the conclusion true. {2} Proof-theoretic validity (symbolized ⊢)  which is determined by means of a procedure operating on a symbolization of the inference. Now Priest seems to be saying that if a sequent is semantically valid, then the conclusion is a semantic consequence of the premises.]
Let Σ be any set of formulas (the premises); then A (the conclusion) is a semantic consequence of Σ (Σ ⊨ A) iff there is no interpretation that makes all the members of Σ true and A false, that is, every interpretation that makes all the members of Σ true makes A true. ‘Σ ⊭ A’ means that it is not the case that Σ ⊨ A.
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1.3.4
[A logical truth or tautology is a formula that is true under every evaluation, written for example as: ⊨ A. This also means it is a semantic consequence of the empty set of premises: φA.]

Iff every interpretation makes some formula true, then we call it a logical tautology or logical truth. We can write that as: ⊨ A. [I do not completely understand the next idea. When a formula is a tautology, it is a semantic consequence of the empty set of premises. I would guess the idea there is the following. Suppose you have a language where there is a particular formula that is always true on every evaluation. I suppose you could say that given the conditions of the language itself (the structure of its interpretation), you can derive the formula simply from that structure, without need to derive it from any other formula. Let us think of it another way. A conclusion is semantically valid when every interpretation that makes the premises true also makes the conclusion true. In the case of a tautology, we make say with regard to its derivation that the conclusion is already true no matter what premises are given, even if none are given. So there is no interpretation where the premises are true (or even non existent) and the conclusion false. But maybe the idea here is different than those options. I quote.]
A is a logical truth (tautology) (⊨ A) iff it is a semantic consequence of the empty set of premises (φA), that is, every interpretation makes A true.
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Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

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