Priest (1.2) An Introduction to Non-Classical Logic, ‘The Syntax of the Object Language’, 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

 

1. Classical Logic and the Material Conditional

 

1.2. The Syntax of the Object Language

 

 

 

Brief summary:

Our object language has a formalized syntax with the following notations.

 

Propositional parameters (propositional variables):

p₀, p₁, p₂, ....

 

Connectives:

¬ (negation), ∧ (conjunction), ∨ (disjunction), ⊃ (material conditional), ≡ (material equivalence)

 

Punctuation:

(, )

 

Arbitrary indistinct formulas:

A, B, C, ...

 

Arbitrary distinct formulas:

p, q, r, ...

 

Arbitrary sets of formulas:

Σ, Π, ...,

 

Empty set:

φ

Outer parentheses around complex formulas and curly brackets around finite sets are omitted. Well-formed formulas are either propositional parameters or complex formulas built up upon propositional parameters by means of the connectives.

 

 

 

Summary

 

1.2.1

[Our object language consists of an infinite number of propositional parameters, notated: p₀, p₁, p₂, .... And it has the following connectives: ¬ (negation), ∧ (conjunction), ∨ (disjunction), ⊃ (material conditional), ≡ (material equivalence). Also, it uses for punctuation the comma and the parenthesis: (, )]

 

[Recall from section 1.1 the following notions. The point of logic is to give an account of validity. That account is made in a metalanguage for a formal object language. We will now examine how our object language is symbolized and structured.] [We will assume that our object language is capable of formulating an infinite number of propositions, at least in theory. We will consider any such unspecified but distinguished and designated proposition as being like a variable, called either a propositional parameter or a propositional variable. I an not certain, but perhaps it is a variable in that its propositional content is not explicit. In other words, we do not know what idea the propositional parameter is expressing. But it is not a variable in the sense that its identity or individuality is always left vague. For, we distinguish and designate distinct propositional parameters using subscript numerals. (I am not sure if a propositional parameter stands for a proposition. Perhaps it stands for objects or predicates or I do not know what.) We also use the standard connectives. As well we use the punctuation mark, the comma, perhaps to separate parts in a series, like premises before a conclusion. Another punctuation mark we use seems to be parentheses, perhaps for grouping formulas, but I am not sure.]

The symbols of the object language of the propositional calculus are an infinite number of propositional parameters:¹ p₀, p₁, p₂, ...; the connectives: ¬ (negation), ∧ (conjunction), ∨ (disjunction), ⊃ (material conditional), ≡ (material equivalence); and the punctuation marks: (, ).

(4)

1. These are often called ‘propositional variables’.

(4)

 

 

1.2.2

[Well-formed formulas are either propositional parameters or complex formulas built up upon propositional parameters by means of the connectives.]

 

[What about the well-formed formulas of this object language? They can be built up to any length on the basis of propositional parameters joined by the connectives in the language.]

The (well-formed) formulas of the language comprise all, and only, the strings of symbols that can be generated recursively from the propositional parameters by the following rule:

If A and B are formulas, so are ¬A, (A ∨ B), (A ∧ B), (A ⊃ B), (A ≡ B).

(Priest 4)

 

 

1.2.3

[Capital Roman letters symbolize arbitrary but indistinct formulas, while lower-case letters represent arbitrary but distinct propositional parameters. Upper-case Greek letters stand for arbitrary sets of formulas, and the empty set is written: φ. Outer parentheses around complex formulas and curly brackets around finite sets are omitted. ]

 

[Capital Roman letters stand for arbitrary formulas of the object language, while lower case ones are arbitrary but distinct propositional parameters. (I am not entirely sure I understand the difference. Perhaps the idea is something like the following. We will often see upper-case Roman letters when we are examining certain structural or syntactical things, where it is not important to think of the propositions being one or another specific proposition. But other times we will, in which case we use the lower-case letters. I am not sure how yet, but perhaps they will play a role for example with quantification.) Priest will omit outermost parentheses. Upper-case Greek letters represent sets of formulas, and the empty set is written with lower-case phi. Finite sets are written without the curly brackets.]

I will explain a number of important notational conventions here. I use capital Roman letters, A, B, C, ..., to represent arbitrary formulas of the object language. Lower-case Roman letters, p, q, r, ..., represent arbitrary, | but distinct, propositional parameters. I will always omit outermost parentheses of formulas if there are any. So, for example, I write (A ⊃ (B ∨ ¬C)) simply as A ⊃ (B ∨ ¬C). Upper-case Greek letters, Σ, Π, ..., represent arbitrary sets of formulas; the empty set, however, is denoted by the (lower case) φ, in the standard way. I often write a finite set, {A₁, A₂, ..., A_n}, simply as A₁, A₂, ..., A_n.

(4-5)

 

 

 

 

 

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

 

 

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