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by Corry Shores
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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
21
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, Ax(a) ⊨ Ax(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(21.8.5) But, if in our logic i is neither true nor false and we also enforce the neutrality constraint, then ⊨ a=a is no longer valid (for, if a is non-existent, then a=a is i, and thus not a designated value).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, Ax(a) ⊨ Ax(b) are valid. (21.8.6) Lastly, Priest notes that “given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a” (467).
[Defining Identity]
[Some Properties of Identity]
[Other Properties of Identity]
[Truth-Conditions for Neutral Gap Free Logics]
[Certain Valid and Invalid Statements in These Logics]
[Other Valid Formulas]
Summary
[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)
[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)
[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, Ax(a) ⊨ Ax(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, Ax(a) ⊨ Ax(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)
[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)
[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, Ax(a) ⊨ Ax(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, Ax(a) ⊨ Ax(b). The proof is to be found in 21.11.4.
(467)
[Other Valid Formulas]
[Lastly, Priest notes that “given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a” (467).]
[(ditto)]
Note that, given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a, as is easy to check.
(467)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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