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
15.
Variable Domain Modal Logics
15.3
Variable Domain K and its Normal Extensions
Brief summary:
(15.3.1) “[A] variable domain interpretation is a quadruple ⟨D, W, R, v⟩. D, W, R and v are the same as in the constant domain case, with the exception that for every w ∈ W, v maps w to a subset of D, that is, | v(w) ⊆ D. v(w) is the domain at world w. I will write it as Dw. Note that for any n-place predicate, P, vw(P) ⊆ Dn (not Dnw), and vw(ℭ) is always Dw” (330-331). (15.3.2) “The truth conditions for atomic sentences, truth-functional and modal operators, are as in the constant domain case. Those for the quantifiers (as is to be expected) are” (included in the following):
vw(¬A) = 1 if vw(A) = 0, and 0 otherwise.
vw(A ∧ B) = 1 if vw(A) = vw (B) = 1, and 0 otherwise.
vw(A ∨ B) = 1 if vw(A) = 1 or vw (B) = 1, and 0 otherwise.
[…]
vw(◊A) = 1 if, for some w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
vw(□A) = 1 if, for all w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
(pp.21-22, sections 2.3.4, 2.3.5)
vw(∃xA) = 1 iff for some d ∈ Dw, vw(Ax(kd)) = 1
vw(∀xA) = 1 iff for all d ∈ Dw, vw(Ax(kd)) = 1
(331)
(15.3.3) “Semantic validity is defined in terms of truth preservation at all worlds of all interpretations, as in the constant domain case” (331). (15.3.4) “These semantics give the variable domain version of the propositional logic K, VK” (331). (15.3.5) “Adding constraints on the accessibility relation produces the extensions VKρ, VKρσ, etc.” (331).
[The Structure of Variable Domain Interpretations]
[The Truth Conditions]
[Semantic Validity]
[VK]
[Adding Constraints to VK]
Summary
[The Structure of Variable Domain Interpretations]
[“[A] variable domain interpretation is a quadruple ⟨D, W, R, v⟩. D, W, R and v are the same as in the constant domain case, with the exception that for every w ∈ W, v maps w to a subset of D, that is, | v(w) ⊆ D. v(w) is the domain at world w. I will write it as Dw. Note that for any n-place predicate, P, vw(P) ⊆ Dn (not Dnw), and vw(ℭ) is always Dw” (330-331).]
[First 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 vw(p) = 1 (or vw(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, ..., dn⟩: d1, ..., dn ∈ 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:
)
Now recall from section 14.2.2 that for the structure of constant domain quantified normal modal logics we add 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 vw(P). Intuitively, vw(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.
(p.309, section 14.2.2)
It seems that for our variable domain interpretations, we have something like the following, but I might get this wrong. For each world, the interpretation function v will assign to each world a subset of the domain, with that subset being the existent things of that world belonging to the existence predicate. So perhaps the other things in the domain that are not in this subset are like non-existent things in that world.]
Bearing this in mind, a variable domain interpretation is a quadruple ⟨D, W, R, v⟩. D, W, R and v are the same as in the constant domain case, with the exception that for every w ∈ W, v maps w to a subset of D, that is, | v(w) ⊆ D. v(w) is the domain at world w. I will write it as Dw. Note that for any n-place predicate, P, vw(P) ⊆ Dn (not Dnw), and vw(ℭ) is always Dw.3
(330-331)
3. Using v to specify the domain of world w in this way is entirely artifactual, but it allows constant and variable domain interpretations to have a common form. In some contexts there are good reasons to keep v separate from the rest of the structure. In that case, we have to add an extra component, δ, to an interpretation, such that δ(w) is the domain of world w. Alternatively, we can take D itself to be a function from worlds to sets, so that D(w) is now the domain of world w. This means that we lose D in the old sense though. We still have a set D′ = ∪{D(w): w ∈ W}. But if this replaces our old D, it ensures that every object exists at some world. Better not to build this extra assumption into the semantics.
(331)
[The Truth Conditions]
[“The truth conditions for atomic sentences, truth-functional and modal operators, are as in the constant domain case. Those for the quantifiers (as is to be expected) are: vw(∃xA) = 1 iff for some d ∈ Dw, vw(Ax(kd)) = 1 ; vw(∀xA) = 1 iff for all d ∈ Dw, vw(Ax(kd)) = 1” (331).]
[(ditto). Here are the truth conditions for atomic sentences, truth-functional and modal operators, are as in the constant domain case, given in section 14.2.3:
vw(¬A) = 1 if vw(A) = 0, and 0 otherwise.
vw(A ∧ B) = 1 if vw(A) = vw (B) = 1, and 0 otherwise.
vw(A ∨ B) = 1 if vw(A) = 1 or vw (B) = 1, and 0 otherwise.
[…]
vw(◊A) = 1 if, for some w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
vw(□A) = 1 if, for all w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
(pp.21-22, sections 2.3.4, 2.3.5)
]
The truth conditions for atomic sentences, truth-functional and modal operators, are as in the constant domain case (14.2.3). Those for the quantifiers (as is to be expected) are:
vw(∃xA) = 1 iff for some d ∈ Dw, vw(Ax(kd)) = 1
vw(∀xA) = 1 iff for all d ∈ Dw, vw(Ax(kd)) = 1
(331)
[Semantic Validity]
[“Semantic validity is defined in terms of truth preservation at all worlds of all interpretations, as in the constant domain case” (331).]
[(ditto)]
Semantic validity is defined in terms of truth preservation at all worlds of all interpretations, as in the constant domain case.
(331)
[VK]
[“These semantics give the variable domain version of the propositional logic K, VK” (331).]
[(ditto)]
These semantics give the variable domain version of the propositional logic K, VK.
(331)
[Adding Constraints to VK]
[“Adding constraints on the accessibility relation produces the extensions VKρ, VKρσ, etc.” (331).]
[(ditto) (See section 3.2.3.)]
Adding constraints on the accessibility relation produces the extensions VKρ, VKρσ, etc.
(331)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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