Pirates & Revolutionaries: Priest (21.8) An Introduction to Non-Classical Logic, ‘Identity,’ summary

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[Priest, Introduction to Non-Classical Logic, entry directory]

[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

Many-valued Logics

21.8

Identity

Brief summary:

(21.8.1) We define identity in our many-valued quantified logics as:

(=)(d₁, d₂) ∈ D iff d₁ = d₂

(21.8.2) Under this definition of identity, the following inferences are valid: ⊨ a=a and a=b, Pa ⊨ Pb. (21.8.3) Under this definition, the following inferences are also valid: a=b ⊨ b=a and a=b, b=c ⊨ a=c. (In other words, identity is reflexive (see above), symmetric, and transitive.) It is also substitutable: a =b, A_x(a) ⊨ A_x(b). This holds even when identity is valued i. (21.8.4) “If we are in a logic where i is thought of as neither true nor false, and we enforce the neutrality constraint, then the truth conditions for identity become: if v(a) ∈ E and v(b) ∈ E then v(=)(a, b) ∈ D iff v(a) = v(b) ; if v(a) ∉ E or v(b) ∉ E then v(=)(a, b) = i (which makes sense provided that i ∉ D). Or, if we dispense with the outer domain, and take the denotation function to be a partial function: if v(a) and v(b) are defined then v(=)(a, b) ∈ D iff v(a) = v(b) ; if either v(a) or v(b) is not defined then v(=)(a, b) = i ”(467). (21.8.5) But, if in our logic i is neither true nor false and we also enforce the neutrality constraint, then ⊨ a=a is no longer valid (for, if a is non-existent, then a=a is i, and thus not a designated value). However, a=b, Pa ⊨ Pb and more generally, a=b, A_x(a) ⊨ A_x(b) are valid. (21.8.6) Lastly, Priest notes that “given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a” (467).

Contents

21.8.1

[Defining Identity]

21.8.2

[Some Properties of Identity]

21.8.3

[Other Properties of Identity]

21.8.4

[Truth-Conditions for Neutral Gap Free Logics]

21.8.5

[Certain Valid and Invalid Statements in These Logics]

21.8.6

[Other Valid Formulas]

Summary

21.8.1

[Defining Identity]

[We define identity in our many-valued quantified logics as: (=)(d₁, d₂) ∈ D iff d₁ = d₂.]

[(ditto)]

If we now suppose that one of the predicates in the language is the identity predicate, then the natural truth conditions for this are:

v(=)(d₁, d₂) ∈ D iff d₁ = d₂

(467)

[contents]

21.8.2

[Some Properties of Identity]

[Under this definition of identity, the following inferences are valid: ⊨ a=a and a=b, Pa ⊨ Pb.]

[(ditto)]

It is not difficult to check that ⊨ a=a and a=b, Pa ⊨ Pb. Thus, for the second of these, suppose that in an interpretation a = b is designated. Then v(a) = v(b). So v(P)(v(a)) ∈ D iff v(P)(v(b)) ∈ D.

(467)

[contents]

21.8.3

[Other Properties of Identity]

[Under this definition, the following inferences are also valid: a=b ⊨ b=a and a=b, b=c ⊨ a=c. (In other words, identity is reflexive (see above), symmetric, and transitive.) It is also substitutable: a =b, A_x(a) ⊨ A_x(b). This holds even when identity is valued i.]

[(ditto)]

Similarly, it is not difficult to check that a=b ⊨ b=a and a=b, b=c ⊨ a=c. More generally, a = b, A_x(a) ⊨ A_x(b); for the proof of this, see 21.11.4. Note that this fact in no way depends on identities taking only classical values. Identities may well take the value i in LP or RM₃ (or b in FDE).

(467)

[contents]

21.8.4

[Truth-Conditions for Neutral Gap Free Logics]

[“If we are in a logic where i is thought of as neither true nor false, and we enforce the neutrality constraint, then the truth conditions for identity become: if v(a) ∈ E and v(b) ∈ E then v(=)(a, b) ∈ D iff v(a) = v(b) ; if v(a) ∉ E or v(b) ∉ E then v(=)(a, b) = i (which makes sense provided that i ∉ D). Or, if we dispense with the outer domain, and take the denotation function to be a partial function: if v(a) and v(b) are defined then v(=)(a, b) ∈ D iff v(a) = v(b) ; if either v(a) or v(b) is not defined then v(=)(a, b) = i ”(467).]

[(ditto)]

If we are in a logic where i is thought of as neither true nor false, and we enforce the neutrality constraint, then the truth conditions for identity become:

if v(a) ∈ E and v(b) ∈ E then v(=)(a, b) ∈ D iff v(a) = v(b)

if v(a) ∉ E or v(b) ∉ E then v(=)(a, b) = i

(which makes sense provided that i ∉ D). Or, if we dispense with the outer domain, and take the denotation function to be a partial function:

if v(a) and v(b) are defined then v(=)(a, b) ∈ D iff v(a) = v(b)

if either v(a) or v(b) is not defined then v(=)(a, b) = i

(467)

[contents]

21.8.5

[Certain Valid and Invalid Statements in These Logics]

[But, if in our logic i is neither true nor false and we also enforce the neutrality constraint, then ⊨ a=a is no longer valid (for, if a is non-existent, then a=a is i, and thus not a designated value). However, a=b, Pa ⊨ Pb and more generally, a=b, A_x(a) ⊨ A_x(b) are valid.]

[(ditto) (Note: I am assuming here, probably incorrectly, that in this logic with i as gap that i is not a designated value. We saw in section 7.3 that the logics with gaps have 1 as the only designated value, while, as we saw in section 7.4, the glut ones have i and 1 as the designated values.)]

It is clear that it will not now be the case that ⊨ a=a. (Take v(a) to be not in E, or undefined.) However it is still the case that a=b, Pa ⊨ Pb. If the first premise is true, then v(a) and v(b) are both in E (or defined), and the argument then proceeds as in 21.8.2. Indeed, more generally, a=b, A_x(a) ⊨ A_x(b). The proof is to be found in 21.11.4.

(467)