Priest (12.2) Introduction to Non-Classical Logic, ‘Syntax [of first classical first-order logic]’, 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 II

Quantification and Identity

 

12

Classical First-order Logic

 

12.2

Syntax

 

 

 

Brief summary [mostly quotation from pp.263-264]:

Our first-order language has the following vocabulary:

• variables: v₀, v₁, v₂, ...
• constants: k₀, k₁, k₂, ...
• for every natural number n > 0, n-place predicate symbols: P⁰_n, P¹_n, P²_n, ...
• connectives: ∧, ∨, ¬, ⊃, ≡
• quantifiers: ∀, ∃
• brackets: (, )

Specifically we may use:

• x, y, z for arbitrary variables

• a, b, c for arbitrary constants

• P_n, Q_n, S_n for arbitrary n-place predicates

• A, B, C for arbitrary formulas

• Σ, Π for arbitrary sets of formulas

Its grammar includes the following:

• Any constant or variable is a term.

The formulas are specified recursively as follows.

• If t₁, ... , t_n are any terms and P is any n-place predicate, Pt_{1 ..} t_n is an (atomic) formula.

• If A and B are formulas, so are the following:

(A ∧ B), (A ∨ B), ¬⁠A, (A ⊃ B), (A ≡ B).

• If A is any formula, and x is any variable, then ∀xA, ∃xA are formulas. I will omit outermost brackets in formulas.

And regarding quantified formulas:

• An occurrence of a variable, x, in a formula, is said to be bound if it occurs in a context of the form ∃x ... x ... or ∀x ... x ....

• If it is not bound, it is free.

• A formula with no free variables is said to be closed.

• A_x(c) is the formula obtained by substituting c for each free occurrence of x in A.

 

 

 

 

Contents

 

12.2.1

[Vocabulary of First-order Logic]

 

12.2.2

[Particular Instantiations of Variables, Constants, Predicates, Formulas, and Formula Sets]

 

12.2.3

[Terms, Atomic Formulas, Complex Formulas, and Quantified Formulas]

 

12.2.4

[Free and Bound Variables and Closed Formulas.]

 

 

Summary

 

12.2

Syntax

 

12.2.1

[Vocabulary of First-order Logic]

 

We are outlining the language for first-order logic. Our “vocabulary” includes the following elements [quoting]:

• variables: v₀, v₁, v₂, ...
• constants: k₀, k₁, k₂, ...
• for every natural number n > 0, n-place predicate symbols: P⁰_n, P¹_n, P²_n, ...
• connectives: ∧, ∨, ¬, ⊃, ≡
• quantifiers: ∀, ∃
• brackets: (, )

We will call ∀ and ∃ the universal and particular quantifiers, respectively. (∃ is often called the existential quantifier. I will return to the nomenclature in the next chapter.)

(263)

[contents]

 

 

12.2.2

[Particular Instantiations of Variables, Constants, Predicates, Formulas, and Formula Sets]

 

[We said above that we have variables v₀, v₁, v₂, ... and constants k₀, k₁, k₂, .... Priest says now that:]

I will use x, y, z for arbitrary variables, and a, b, c for arbitrary constants (possibly with primes or subscripts in each case).

(263)

[The idea might be that x is like a particular v₀, and so on, but I am not sure.] And [similarly] he uses the following other notations:

I will use P_n, Q_n, S_n | for arbitrary n-place predicates.¹ I will omit the subscript in cases where it can be read off from the context. I will use A, B, C for arbitrary formulas, and Σ, Π for arbitrary sets of formulas.

(263-264)

1. I will not use ‘R’ to avoid any confusion with the modal accessibility relation.

(264)

[contents]

 

 

12.2.3

[Terms, Atomic Formulas, Complex Formulas, and Quantified Formulas]

 

[We will now discuss the “grammar” of this language, which I think would mean the ways that the parts can be combined. We call any constant or variable a “term”. We then define formulas (which would contain terms) recursively (meaning, I think, that we can build them up using rules of combination). Intuitively, I think, the three rules are saying the following: {1} When we have an n-place predicate taking a series of terms whose quantity fulfills that of the predicate’s valence (or “arity), then this is an atomic formula. {2} Any formulas combined using the connectives makes a formula. {3} Universally and existentially quantified formulas are formulas.]

• Any constant or variable is a term.

(In general, languages may have other terms as well. We will meet some more in later chapters.)

The formulas are specified recursively as follows.

• If t₁, ... , t_n are any terms and P is any n-place predicate, Pt_{1 ..} t_n is an (atomic) formula.

• If A and B are formulas, so are the following:

(A ∧ B), (A ∨ B), ¬⁠A, (A ⊃ B), (A ≡ B).

• If A is any formula, and x is any variable, then ∀xA, ∃xA are formulas. I will omit outermost brackets in formulas.

(264)

[contents]

 

 

12.2.4

[Free and Bound Variables and Closed Formulas.]

 

[Recall the notions of scope, free variable, and bound variables from Suppes’ Introduction to Logic section 3.5.

DEFINITION. The SCOPE of a quantifier occurring in a formula is the quantifier together with the smallest formula immediately following the quantifier.
(Suppes 53)

(Suppes 53)

 

DEFINITION. An occurrence of a variable in a formula is BOUND if and only if this occurrence is within the scope of a quantifier using this variable.
(Suppes 53)

 

DEFINITION. An occurrence of a variable is FREE if and only if this occurrence of the variable is not bound.
(Suppes 53)

To these notions, we add the idea that a formula with no free variables is closed. We also note another notational convention. Suppose we have formula A, and it has free occurrences of variable x. And suppose further that we substitute all x with c. Then we could write A_x(c) to mean that we have made such a substitution.]

An occurrence of a variable, x, in a formula, is said to be bound if it occurs in a context of the form ∃x ... x ... or ∀x ... x .... If it is not bound, it is free. A formula with no free variables is said to be closed. A_x(c) is the formula obtained by substituting c for each free occurrence of x in A.

(264)

[contents]

 

 

 

From:

 

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

 

 

Also cited:

Suppes, Patrick. 1957. Introduction to Logic. New York: Van Nostrand Reinhold / Litton Educational.

 

 

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