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
13.
Free Logics
13.2
Syntax and Semantics
Brief summary:
(13.2.1) Free logics have the same vocabulary as classical first-order logics.
• variables: v0, v1, v2, ...
• constants: k0, k1, k2, ...
• for every natural number n > 0, n-place predicate symbols: P0n, P1n, P2n, ...
• connectives: ∧, ∨, ¬, ⊃, ≡
• quantifiers: ∀, ∃
• brackets: (, )We will call ∀ and ∃ the universal and particular quantifiers, respectively.
(p.263, section 12.2.1)
But in free logics we have the one-place existence predicate ℭ. We can think of ℭa as meaning ‘a exists’. (13.2.2) In free logics, we have our main domain of all objects, D, and we have the “inner domain” E. It is a a subset of D that we think of as being the set of all existent objects. (“An interpretation for the language is a triple ⟨D, E, v⟩, where D is a non-empty set, and E (the ‘inner domain’) is a (possibly empty) subset of D. One can think of D as the set of all objects, and E as the set of all existent objects” (290).) So suppose D contains Sherlock Holmes, the Pegasus, and Julius Caesar. Here, although all of them are in D, only Caesar is in E. (13.2.3) “As in classical logic, v assigns every constant in the language a member of D, and every n-place predicate a subset of Dn. In any interpretation, v(ℭ) = E” (290). (13.2.4) The truth conditions in free logics are the same as for classical logic except that the ones for the quantifiers only concern the domain of existents.
The truth conditions for (closed) atomic sentences are:
v(Pa1 ... an) = 1 iff ⟨v(a1), ..., v(an)⟩ ∈ v(P) (otherwise it is 0)
(p.265, section 12.3.2)
v(¬A) = 1 if v(A) = 0, and 0 otherwise.
v(A ∧ B) = 1 if v(A) = v(B) = 1, and 0 otherwise.
v(A ∨ B) = 1 if v(A) = 1 or v(B) = 1, and 0 otherwise.
v(A ⊃ B) = 1 if v(A) = 0 or v(B) = 1, and 0 otherwise.
v(A ≡ B) = 1 if v(A) = v(B), and 0 otherwise.(p.5, section 1.3.2)
v(∀xA) = 1 iff for all d ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some d ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
(291)
(13.2.5) “An inference is semantically valid if it is truth-preserving in all interpretations, as in classical logic” (291). (13.2.6) If C is some set of constants such that every object in D has a name in C, then:
v(∀xA) = 1 iff for all c ∈ C such that v(ℭc) = 1, v(Ax(c)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some c ∈ C such that v(ℭc) = 1, v(Ax(c)) = 1 (otherwise it is 0)
(291)
[The Existence Predicate]
[The Inner Domain]
[The Interpretation Function of Free Logics]
[Truth Conditions]
[Validity]
[Denotation and Quantification]
Summary
[The Existence Predicate]
[Free logics have the same vocabulary as classical first-order logics. But in free logics we have the one-place existence predicate ℭ. We can think of ℭa as meaning ‘a exists’.]
[Let us recall some notation from section 12.2.1:
• variables: v0, v1, v2, ...
• constants: k0, k1, k2, ...
• for every natural number n > 0, n-place predicate symbols: P0n, P1n, P2n, ...
• connectives: ∧, ∨, ¬, ⊃, ≡
• quantifiers: ∀, ∃
• brackets: (, )We will call ∀ and ∃ the universal and particular quantifiers, respectively.
(p.263, section 12.2.1)
Free logic will use this same vocabulary. Now note especially from this material the part about n-place predicates:
for every natural number n > 0, n-place predicate symbols: P0n, P1n, P2n, ...
(p.263, section 12.2.1)
So here we see that the subscript number indicates how many places there are in the predicate. In free logics, we will have a one-place predicate for existence, which we designate as P01and we symbolize as ℭ. We can think of ℭa as meaning ‘a exists’.]
The vocabulary of free logic is the same as that of classical first-order logic, except that we single out one of the one-place predicates for special treatment. Let this be P01. We will write this as ℭ, and think of it as an existence predicate. Thus, ℭa can be thought of as ‘a exists’.
(290)
[The Inner Domain]
[In free logics, we have our main domain of all objects, D, and we have the “inner domain” E. It is a a subset of D that we think of as being the set of all existent objects. (“An interpretation for the language is a triple ⟨D, E, v⟩, where D is a non-empty set, and E (the ‘inner domain’) is a (possibly empty) subset of D. One can think of D as the set of all objects, and E as the set of all existent objects” (290).) So suppose D contains Sherlock Holmes, the Pegasus, and Julius Caesar. Here, although all of them are in D, only Caesar is in E.]
[(ditto) (Note, see Nolt’s Logics section 15.1.]
An interpretation for the language is a triple ⟨D, E, v⟩, where D is a non-empty set, and E (the ‘inner domain’) is a (possibly empty) subset of D. One can think of D as the set of all objects, and E as the set of all existent objects. Thus, one might think of D as containing objects such as Sherlock Holmes, Pegasus and Julius Caesar. Only the last of these would be in E.
(290)
[The Interpretation Function of Free Logics]
[“As in classical logic, v assigns every constant in the language a member of D, and every n-place predicate a subset of Dn. In any interpretation, v(ℭ) = E” (290).]
[(ditto)]
As in classical logic, v assigns every constant in the language a member of D, and every n-place predicate a subset of Dn. In any interpretation, v(ℭ) = E.
(290)
[Truth Conditions]
[The truth conditions in free logics are the same as for classical logic except that the ones for the quantifiers only concern the domain of existents.]
[Priest says that the truth conditions for closed sentences as the same as in classical logic, which we saw in section 12.3.2. But we change the rules for the quantifiers. (Below I compile all the rules together.)
The truth conditions for (closed) atomic sentences are:
v(Pa1 ... an) = 1 iff ⟨v(a1), ..., v(an)⟩ ∈ v(P) (otherwise it is 0)
The truth conditions for the connectives are as in the propositional case (1.3.2).
(p.265, section 12.3.2)
v(¬A) = 1 if v(A) = 0, and 0 otherwise.
v(A ∧ B) = 1 if v(A) = v(B) = 1, and 0 otherwise.
v(A ∨ B) = 1 if v(A) = 1 or v(B) = 1, and 0 otherwise.
v(A ⊃ B) = 1 if v(A) = 0 or v(B) = 1, and 0 otherwise.
v(A ≡ B) = 1 if v(A) = v(B), and 0 otherwise.(p.5, section 1.3.2)
v(∀xA) = 1 iff for all d ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some d ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
(291)
As we can see, the truth conditions for the particular and universal quantifiers only concern the items in the domain of existents.]
The truth conditions for closed sentences in the language of an interpretation, ℑ, are given in exactly the same way as in classical logic (12.3), except for those of the quantifiers, which are as follows:
v(∀xA) = 1 iff for all D ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some D ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
(291)
[Validity]
[“An inference is semantically valid if it is truth-preserving in all interpretations, as in classical logic” (291).]
[(ditto) (See section 12.3.3).]
An inference is semantically valid if it is truth-preserving in all interpretations, as in classical logic.
(291)
[Denotation and Quantification]
[“If C is some set of constants such that every object in D has a name in C, then: v(∀xA) = 1 iff for all c ∈ C such that v(ℭc) = 1, v(Ax(c)) = 1 (otherwise it is 0) ; v(∃xA) = 1 iff for some c ∈ C such that v(ℭc) = 1, v(Ax(c)) = 1 (otherwise it is 0)” (291).]
[(ditto)]
Note that we have the free analogue of 12.3.5. If C is some set of constants such that every object in D has a name in C, then:
v(∀xA) = 1 iff for all c ∈ C such that v(ℭc) = 1, v(Ax(c)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some c ∈ C such that v(ℭc) = 1, v(Ax(c)) = 1 (otherwise it is 0)
The proof is, again, a simple corollary of the Denotation Lemma, and is given in 13.7.14. The result carries over to all logics with a domain of quantification circumscribed by an existence predicate, and I will not keep mentioning the fact.
(291)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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