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
1. Classical Logic and the Material Conditional
1.1. Introduction
Brief summary:
The main purpose of logic is to provide an account of validity, which determines what follows from what. The account is given in a metalanguage for a formal, object language. There are two types of validity: {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. Most contemporary logicians think that semantic validity is more fundamental than proof-theoretic, but it is good nonetheless to provide a proof-theoretic notion of validity to correspond with a semantic notion. A proof-theory is sound when “every proof-theoretically valid inference is semantically valid (so that ⊢ entails ⊨)” and it is complete when “every semantically valid inference is proof-theoretically valid (so that ⊨ entails ⊢)” (4).
Summary
1.1.1
[First chapter half: basics of propositional logic.]
The first part of the chapter reviews classical propositional logic, along with semantic tableaux and basic terminology and notational conventions that will be used in the second part (3).
1.1.2
[Second half: conditional.]
The second half will examine the classical conditional along with its shortcomings.
1.1.3
[Logic is a matter of validity (which determines what follows from what), defined for a formal, object language.]
The main aim of logic is to give an account of validity, which is often defined for a formal language, considered the object language.
The point of logic is to give an account of the notion of validity: what follows from what. Standardly, validity is defined for inferences couched in a formal language, a language with a well-defined vocabulary and grammar, the object language. The relationship of the symbols of the formal language to the words of the vernacular, English in this case, is always an important issue.
(3)
1.1.4
[The account of validity is made in a metalanguage.]
But the account for an object language is made in a metalanguage that is often distinct from the object language. Here our metalanguage will be mathematical English (3).
1.1.5
[Two types of validity: {1} Semantic, which preserves truth: every interpretation that makes the premises true also makes the conclusion true. The metalinguistic symbol for semantic validity is: ⊨]
There are two sorts of validity: semantic and proof theoretic. Semantic validity is one where truth is preserved in the inference. [I am not sure I understand the notion of truth preservation. It might mean something like the following. We suppose we have sufficiently many true premises to make some inference on their basis. We also think that their truth in combination with certain logical structures or rules necessitates that the conclusion be true. So in a sense, in a semantically valid inference, the truth of the premises is preserved in the conclusion. Hence the definition that a semantically valid inference is one where it cannot be that the premises are true and the conclusion not true. In other words, a semantically valid inference is one where the truth of the premises guarantees the truth of the conclusion. Also, we will distinguish different logics by means of their notions of interpretation. Let me quote:]
It is also standard to define two notions of validity. The first is semantic. A valid inference is one that preserves truth, in a certain sense. Specifically, every interpretation (that is, crudely, a way of assigning truth values) that makes all the premises true makes the conclusion true. We use the metalinguistic symbol ‘⊨’ for this. What distinguishes different logics is the different notions of interpretation they employ.
(3)
1.1.6
[{2} Proof-theoretic validity, which is determined by means of a procedure operating on a symbolization of the inference. Its metalinguistic symbol is: ⊢]
Proof-theoretic validity is determined by means of a formal procedure performed on symbolic notation for the inference. Our method here employs tableaux. Priest says that the different tableaux procedures will distinguish the different types of logic (4).
The second notion of validity is proof-theoretic. Validity is defined in terms of some purely formal procedure (that is, one that makes reference only to the symbols of the inference). We use the metalinguistic symbol ‘⊢’ for this notion of validity. In our case, this procedure will (mainly) be one employing tableaux. What distinguish different logics here are the different tableau procedures employed.
(4)
1.1.7
[Semantic validity is more fundamental. A proof-theory is sound when “every proof-theoretically valid inference is semantically valid (so that ⊢ entails ⊨)” and it is complete when “every semantically valid inference is proof-theoretically valid (so that ⊨ entails ⊢)”]
Priest says that the consensus among contemporary logicians is that semantic validity is a more fundamental notion of validity than the proof-theoretic notion. But, he continues, it is useful to provide a proof-theoretic notion of validity corresponding to any semantic one, “in the sense that the two definitions always give the same answers”. A proof-theory is sound when all the proof-theoretically valid inferences [those determined as such by symbolic procedures] are also semantically valid [preserve the truth of the premises]. In that case, ⊢ entails ⊨. And if all semantically valid inferences are also proof-theoretically valid, then the proof-theory is said to be complete. In that case, ⊨ entails ⊢. [I am not sure why in both cases it is the proof-theory that is either sound or complete, and why not instead in the second case we do not say something like the semantic theory is complete, to make the formulation symmetrical.]
If every proof-theoretically valid inference is semantically valid (so that ⊢ entails ⊨) the proof-theory is said to be sound. If every semantically valid inference is proof-theoretically valid (so that ⊨ entails ⊢) the proof-theory is said to be complete.
(3)
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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