Priest (14.2) An Introduction to Non-Classical Logic, ‘Constant Domain K,’ 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 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 II:

Quantification and Identity

 

14.

Constant Domain Modal Logics

 

14.2

Constant Domain K

 

 

 

 

Brief summary:

(14.2.1) Our quantified modal logic will augment the language of first-order classical logic with the modal operators □ and ◊.

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.

(mostly quotation from pp.263-264, section 12.2)

To all this we add the modal operators □ and ◊:

Intuitively, □A is read as ‘It is necessarily the case that A’; ◊A as ‘It is possibly the case that A’.

If A  is a formula, so are □A and ◊A.

(p.21, sections 2.3.1, 2.3.2).

(14.2.2) “An interpretation for the language is a quadruple ⟨D, W, R, v⟩. W is a (non-empty) set of worlds, and R is a binary accessibility relation on W, as in the propositional case. D is the non-empty domain of quantification, as in classical first-order logic. v assigns each constant, c, of the language a member, v(c), of D, and each pair comprising a world, w, and an n-place predicate, P, a subset of Dⁿ. I will write this as v_w(P). Intuitively, v_w(P) is the set of n-tuples that satisfy P at world w – which may change from world to world. (Thus, ⟨Caesar, Brutus⟩ is in the extension of ‘was murdered by’ at this world, but in a world where Brutus was not persuaded to join the conspirators, it is not.) The language of an interpretation, , is obtained by adding a constant to the language for every member of D” (309). (14.2.3) “Each closed formula, A, is now assigned a truth value, v_w(A), at each world, w. The truth conditions for atomic formulas are as follows: v_w(Pa₁ . . . a_n) = 1 iff ⟨v(a₁), . . . , v(a_n)⟩ ∈ v_w(P) (otherwise it is 0). The truth conditions for the connectives and modal operators are as in the propositional case” and for quantifiers as in first-order logic.

v_w(¬A) = 1 if v_w(A) = 0, and 0 otherwise.

v_w(A ∧ B) = 1 if v_w(A) = v_w (B) = 1, and 0 otherwise.

v_w(A ∨ B) = 1 if v_w(A) = 1 or v_w (B) = 1, and 0 otherwise.

[…]

v_w(◊A) = 1 if, for some w′ ∈ W such that wRw′, v_w′(A) = 1; and 0 otherwise.

v_w(□A) = 1 if, for all w′ ∈ W such that wRw′, v_w′(A) = 1; and 0 otherwise.

v_w(∀xA) = 1 iff for all d ∈ D, v_w(A_x(k_d)) = 1 (otherwise it is 0)

v_w(∃xA) = 1 iff for some d ∈ D, v_w(A_x(k_d)) = 1 (otherwise it is 0)

(pp.21-22, sections 2.3.4, 2.3.5, 309)

(14.2.4) “An inference is valid if it is truth-preserving in all worlds of all interpretations” (309). (14.2.5) “The above semantics define the constant domain modal logic CK, corresponding to the propositional logic K” (309).

 

 

 

 

 

Contents

 

14.2.1

[The Language of Our Quantified Modal Logic]

 

14.2.2

[The Structure of the Quantified Modal Logic]

 

14.2.3

[Truth-Conditions]

 

14.2.4

[Validity]

 

14.2.5

[Modal Logic CK]

 

 

 

 

 

 

Summary

 

14.2.1

[The Language of Our Quantified Modal Logic]

 

[Our quantified modal logic will augment the language of first-order classical logic with the modal operators □ and ◊.]

 

[Our quantified modal logic will augment the language of first-order classical logic. Recall it from section 12.2:

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.

(mostly quotation from pp.263-264, section 12.2)

To all this we add the modal operators □ and ◊:

Intuitively, □A is read as ‘It is necessarily the case that A’; ◊A as ‘It is possibly the case that A’.

If A  is a formula, so are □A and ◊A.

(p.21, sections 2.3.1, 2.3.2)

]

The syntax of quantified modal logic augments the language of first-order classical logic (12.2) with the operators □ and ◊, as propositional modal logic extends classical propositional logic (2.3.1, 2.3.2).

(308)

[contents]

 

 

 

 

 

 

14.2.2

[The Structure of the Quantified Modal Logic]

 

[“An interpretation for the language is a quadruple ⟨D, W, R, v⟩. W is a (non-empty) set of worlds, and R is a binary accessibility relation on W, as in the propositional case (2.3.3). D is the non-empty domain of quantification, as in classical first-order logic (12.3.1). v assigns each constant, c, of the language a member, v(c), of D, and each pair comprising a world, w, and an n-place predicate, P, a subset of Dⁿ. I will write this as v_w(P). Intuitively, v_w(P) is the set of n-tuples that satisfy P at world w – which may change from world to world. (Thus, ⟨Caesar, Brutus⟩ is in the extension of ‘was murdered by’ at this world, but in a world where Brutus was not persuaded to join the conspirators, it is not.) The language of an interpretation, ℑ, is obtained by adding a constant to the language for every member of D, as in 12.3.2” (309).]

 

[Recall from section 2.3.3 the structure of an interpretation in modal logic:

An interpretation for this language is a triple ⟨W, R, v⟩. W is a non-empty set. Formally, W is an arbitrary set of objects. Intuitively, its members are possible worlds. R is a binary relation on W (so that, technically, R ⊆ W×W). Thus, if u and v are in W, R may or may not relate them to each other. If it does, we will write uRv, and say that v is accessible from u. Intuitively, R is a relation of relative possibility, so that uRv means that, relative to u, situation v is possible. υ is a function that assigns a truth value (1 or 0) to each pair comprising a world, w, and a propositional parameter, p. We write this as v_w(p) = 1 (or v_w(p) = 0). Intuitively, this is read as ‘at world w, p is true (or false)’.

(p.21, section 2.3.3)

Since we are now dealing with quantification, will will add our D domain of objects as in section 12.3.1. Recall from there:

An interpretation of the language is a pair, ℑ = ⟨D, v⟩. D is a non-empty set (the domain of quantification); v is a function such that:

• if c is a constant, v(c) is a member of D

• if P is an n-place predicate, v(P) is a subset of Dⁿ

(Dⁿ is the set of all n-tuples of members of D, {⟨d₁, ..., d_n⟩: d₁, ..., d_n ∈ D}. By convention, ⟨d⟩ is just d, and so D¹ is D.)

(p.264, section 12.3.1. Note, the first formulation should look like:

)

The important things to note now are that we are adding the quantification domain to the modal logic structure and that the satisfaction of the predicates can change depending on the world.]

An interpretation for the language is a quadruple ⟨D, W, R, v⟩. W is a (non-empty) set of worlds, and R is a binary accessibility relation on W, as in the propositional case (2.3.3). D is the non-empty domain of quantification, as in classical first-order logic (12.3.1). v assigns each constant, c, of the language a member, v(c), of D, and each pair comprising a world, w, and an n-place predicate, P, a subset of Dⁿ. I will write this as v_w(P). Intuitively, v_w(P) is the set of n-tuples that satisfy P at world w – which may change from world to world. (Thus, ⟨Caesar, Brutus⟩ is in the extension of ‘was murdered by’ at this world, but in a world where Brutus was not persuaded to join the conspirators, it is not.) The language of an interpretation, ℑ, is obtained by adding a constant to the language for every member of D, as in 12.3.2.

(309)

[contents]

 

 

 

 

 

 

14.2.3

[Truth-Conditions]

 

[“Each closed formula, A, is now assigned a truth value, v_w(A), at each world, w. The truth conditions for atomic formulas are as follows: v_w(Pa₁ . . . a_n) = 1 iff ⟨v(a₁), . . . , v(a_n)⟩ ∈ v_w(P) (otherwise it is 0). The truth conditions for the connectives and modal operators are as in the propositional case” and for quantifiers as in first-order logic.]

 

[Truth-values for closed formulas are assigned for each world. Formulas with predicates are true if the predicated terms are in the extension of that predicate for that world, and false otherwise. The truth conditions for connectives and modal operators are the same as for propositional logic, as in sections 2.3.4, 2.3.5:

v_w(¬A) = 1 if v_w(A) = 0, and 0 otherwise.

v_w(A ∧ B) = 1 if v_w(A) = v_w (B) = 1, and 0 otherwise.

v_w(A ∨ B) = 1 if v_w(A) = 1 or v_w (B) = 1, and 0 otherwise.

[…]

v_w(◊A) = 1 if, for some w′ ∈ W such that wRw′, v_w′(A) = 1; and 0 otherwise.

v_w(□A) = 1 if, for all w′ ∈ W such that wRw′, v_w′(A) = 1; and 0 otherwise.

(pp.21-22, sections 2.3.4, 2.3.5)

And the truth conditions for the quantifiers are the same as for first-order logic, as in section 12.3.2:

v_w(∀xA) = 1 iff for all d ∈ D, v_w(A_x(k_d)) = 1 (otherwise it is 0)

v_w(∃xA) = 1 iff for some d ∈ D, v_w(A_x(k_d)) = 1 (otherwise it is 0)

(309)

]

Each closed formula, A, is now assigned a truth value, v_w(A), at each world, w. The truth conditions for atomic formulas are as follows:

v_w(Pa₁ . . . a_n) = 1 iff ⟨v(a₁), . . . , v(a_n)⟩ ∈ v_w(P) (otherwise it is 0)

The truth conditions for the connectives and modal operators are as in the propositional case (2.3.4, 2.3.5). The truth conditions for the quantifiers are as in first-order logic (12.3.2). Thus, for every world, w:

v_w(∀xA) = 1 iff for all d ∈ D, v_w(A_x(k_d)) = 1 (otherwise it is 0)

v_w(∃xA) = 1 iff for some d ∈ D, v_w(A_x(k_d)) = 1 (otherwise it is 0)

(309)

[contents]

 

 

 

 

 

 

14.2.4

[Validity]

 

[“An inference is valid if it is truth-preserving in all worlds of all interpretations” (309).]

 

[(ditto) (See section 2.3.11).]

An inference is valid if it is truth-preserving in all worlds of all interpretations.

(309)

[contents]

 

 

 

 

 

 

14.2.5

[Modal Logic CK]

 

[“The above semantics define the constant domain modal logic CK, corresponding to the propositional logic K” (309).]

 

[(ditto)]

The above semantics define the constant domain modal logic CK, corresponding to the propositional logic K (and not to be confused with the propositional logic of the same name in 10.4a.12).

(309)

[contents]

 

 

 

 

 

 

 

 

From:

 

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