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 I:
Propositional Logic
11a.
Appendix: Many-valued Modal Logics
11a.2
General Structure
Brief summary:
(11a.2.1) A “propositional many-valued logic is characterised by a structure ⟨V, D, {fc : c ∈ C}⟩, where V is the set of semantic values, D ⊆ V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values in V to propositional parameters; the values of all formulas can | then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation” (242-243). (11a.2.2) We will assume that the set of truth-values of V are ordered from lesser to greater (or to equal than: ≤) and that “every subset of the values has a greatest lower bound (Glb) and least upper bound (Lub) in the ordering” (242). (11a.2.3) Our many-valued modal logic adds to many-valued logic “the monadic operators, □ and ◊ in the usual way” (242). (11a.2.4) “An interpretation for a many-valued modal logic is a structure ⟨W, R, SL, v⟩, where W is a non-empty set of worlds, R is a binary accessibility relation on W, SL is a structure for a many-valued logic, L, and for each propositional parameter, p, and world, w, v assigns the parameter a value, vw(p), in V” (242). (11a.2.5) The value of a connective on formulas in a certain world is determined by the value generated by the corresponding connective function operating recursively on the truth values of those formulas in that particular world: “if c is an n-place connective vw(c(A1, . . . , An)) = fc(vw(A1), . . . , vw(An))” (242). (11a.2.6) The truth-conditions for the modal operators are:
vw(□A) = Glb{vw′(A) : wRw′}
vw(◊A) = Lub{ vw′(A) : wRw′}
(11a.2.7) Validity is defined in the following way: “Σ ⊨ A iff for every interpretation, ⟨W, R, SL, v⟩, and for every w ∈ W, whenever vw(B) ∈ for every B ∈ Σ, vw(A) ∈ ” (242). (11a.2.8) Our many-valued modal logic is called KL, and we can apply the accessibility relation constraints on it to derive stronger logics, like KLρ, KLσ, KLρτ, and so forth.
[The Structure]
[Certain Assumptions: V-Ordering and Bounds]
[Adding Monadic Operators □ and ◊]
[The Structure of Many-Valued Modal Logic]
[Connective Evaluation]
[Modal Operator Evaluation]
[Validity]
[KL and Its Constraints]
Summary
[The Structure]
[A “propositional many-valued logic is characterised by a structure ⟨V, D, {fc : c ∈ C}⟩, where V is the set of semantic values, D ⊆ V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values in V to propositional parameters; the values of all formulas can | then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation” (242-243).]
[We will now call to mind the structure for many-valued semantics from section 7.2. It is:
⟨V, D, {fc : c ∈ C}⟩
V is the set of assignable truth values. is the set of designated values, which are those that are preserved in valid inferences (like 1 for classical bivalent logic). C is the set of connectives. c is some particular connective. And fc is the truth function corresponding to some connective, and it operates on the truth values of the formula in question. In a classical bivalent logic,
V = {1, 0}
D = {1}
C = {¬, ∧, ∨, ⊃, ≡} (but we have redefined ≡)
fc; c ∈ C = {f¬, f∧, f∨, f⊃}
with the connective functions being formulated in the classical way. So for example:
f¬ is a one-place function such that f¬(0) = 1 and f¬(1) = 0;
f∧ is a two-place function such that f∧(x, y) = 1 if x = y = 1, and f∧(x, y) = 0 otherwise [...]
| f¬ | |
| 1 | 0 |
| 0 | 1 |
| f∧ | 1 | 0 |
| 1 | 1 | 0 |
| 0 | o | o |
(pp.120-121, section 7.2.2)
With this structure in mind, we can vary the components, like adding truth-values, adding designated values, and changing the functions for connectives, to generate various other sorts of logics. (See section 7.3 for K3 and Ł3 and section 7.4 for LP and RM3.) We could also do this for continuum valued logics (see section 11.4).]
As we observed in 7.2, semantically, a propositional many-valued logic is characterised by a structure ⟨V, D, {fc : c ∈ C}⟩, where V is the set of semantic values, D ⊆ V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values in V to propositional parameters; the values of all formulas can | then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation.
(242-243)
[Certain Assumptions: V-Ordering and Bounds]
[We will assume that the set of truth-values of V are ordered from lesser to greater (or to equal than: ≤) and that “every subset of the values has a greatest lower bound (Glb) and least upper bound (Lub) in the ordering” (242).]
[(ditto)]
It is standard for V to come with an ordering, ≤. We will assume in what follows that this is so. We also assume that every subset of the values has a greatest lower bound (Glb) and least upper bound (Lub) in the ordering.
(242)
[Adding Monadic Operators □ and ◊]
[Our many-valued modal logic adds to many-valued logic “the monadic operators, □ and ◊ in the usual way” (242).]
[(ditto)]
The language of a many-valued modal logic is the same as that of the many-valued logic, except that it is augmented by the monadic operators, □ and ◊ in the usual way.
(242)
[The Structure of Many-Valued Modal Logic]
[“An interpretation for a many-valued modal logic is a structure ⟨W, R, SL, v⟩, where W is a non-empty set of worlds, R is a binary accessibility relation on W, SL is a structure for a many-valued logic, L, and for each propositional parameter, p, and world, w, v assigns the parameter a value, vw(p), in V” (242).]
[Recall again from section 11a.2.1 the structure of logics: ⟨V, D, {fc : c ∈ C}⟩. By having more than the normal two truth-values in V, we obtain a many-valued logic. When the V in this structure produces a many-valued logic, (along with the other changes in the other parts of the structure needed for this purpose), then we can call the whole formulation ⟨V, , {fc : c ∈ C}⟩ SL, for “a structure for a many-valued logic, L” (242). Now recall the structure for modal logics from section 2.3.3:
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)
So now we want to combine the modal logic structure ⟨W, R, v⟩ with the many-valued logic structure ⟨V, , {fc : c ∈ C}⟩, which is abbreviated as SL, and thus we now have: ⟨W, R, SL, v⟩.]
An interpretation for a many-valued modal logic is a structure ⟨W, R, SL, v⟩, where W is a non-empty set of worlds, R is a binary accessibility relation on W, SL is a structure for a many-valued logic, L, and for each propositional parameter, p, and world, w, v assigns the parameter a value, vw(p), in V.
(242)
[Connective Evaluation]
[The value of a connective on formulas in a certain world is determined by the value generated by the corresponding connective function operating recursively on the truth values of those formulas in that particular world: “if c is an n-place connective vw(c(A1, . . . , An)) = fc(vw(A1), . . . , vw(An))” (242).]
[Recall from section 11a.2.1 above how we reviewed the semantics for the connective functions. For example, the connective function for conjunction was:
f∧ is a two-place function such that f∧(x, y) = 1 if x = y = 1, and f∧(x, y) = 0 otherwise
(p.120, section 7.2.2)
We now do something similar but now including worlds. So, the value of a connective of formulas in a certain world is determined by the value generated by the corresponding connective function operating on the truth values of those formulas in that particular world.]
The truth conditions for the many-valued connectives at a world simply deploy the functions fc. Thus, if c is an n-place connective vw(c(A1, . . . , An)) = fc(vw(A1), . . . , vw(An))). (So if c is conjunction, vw(A ∧ B) = f∧(vw(A), vw(B)).)
(242)
[Modal Operator Evaluation]
[The truth-conditions for the modal operators are: vw(□A) = Glb{vw′(A) : wRw′} ; vw(◊A) = Lub{ vw′(A) : wRw′}.]
[I will probably missummarize the next ideas, so it is best to skip to the quotation. We will formulate the truth-conditions for the modal operators in our many-valued modal logic. Let us consider two cases, one where we have fuzzy values and another where we have FDE values. We want to know the value of □A in world 1. Suppose there is another world 2, and they have access to themselves and each other. In world 1, A has the truth value 0.5, and in world 2 A has the value 0.25. From what I can tell, □A in world 1 would be valued 0.25, but I am not sure. I am guessing that we take all the values for A in all the worlds, and we look for the least value. The real formula is: vw(□A) = Glb{vw′(A) : wRw′}, and I am not entirely sure what it means. Suppose instead that we have FDE values, and in world 1 A is 1 and in world 2 A is both 1 and 0. I would guess that □A would then be 0. But I am not sure what to say if in world 2 A has neither value. I would guess that □A would then also have neither value, but I am not sure how it works with the scaling of values outside fuzzy logic. Now let us consider ◊A, but we will keep the above value assignments. In the fuzzy interpretation, it would seem that ◊A in world 1 would be 0.5, because the truth-condition is: vw(◊A) = Lub{vw′(A) : wRw′}. But I am guessing. And for the many-valued interpretation, I would guess that ◊A in world 1 would be 1. I am sorry that I am simply guessing here. The quote follows.]
The natural generalisation of the two-valued truth conditions for the modal operators is as follows:1
vw(□A) = Glb{vw′(A) : wRw′}
vw(◊A) = Lub{vw′(A) : wRw′}
(242)
1. Semantically, □ and ◊ are forms of (respectively) universal and particular quantifiers over worlds. The following truth conditions are the obvious analogues of the truth conditions for these quantifiers in many-valued logic. (See Part II, 21.3.)
(242)
[Validity]
[Validity is defined in the following way: “Σ ⊨ A iff for every interpretation, ⟨W, R, SL, v⟩, and for every w ∈ W, whenever vw(B) ∈ for every B ∈ Σ, vw(A) ∈ ” (242).]
[Validity seems to still be defined as preservation of the designated values in all worlds and interpretations.]
Validity is naturally defined as follows:
Σ ⊨ A iff for every interpretation, ⟨W, R, SL, v⟩, and for every w ∈ W, whenever vw(B) ∈ for every B ∈ Σ, vw(A) ∈ .
(242)
[KL and Its Constraints]
[Our many-valued modal logic is called KL, and we can apply the accessibility relation constraints on it to derive stronger logics, like KLρ, KLσ, KLρτ, and so forth.]
[Recall the two-valued modal semantics we examined in section 2.3. We said in section 2.1 that this modal logic is called K (for Kripke). Priest says that our many-valued version is an analog to it, and we call our many-valued modal logic KL. Next recall from section 3.2.3 the constraints on the accessibility relation that generate variations of a modal logic:
ρ (rho), reflexivity: for all w, wRw.
σ (sigma), symmetry: for all w1, w2, if w1Rw2, then w2Rw1.
τ (tau), transitivity: for all w1, w2, w3, if w1Rw2 and w2Rw3, then w1Rw3.
η (eta), extendability: for all w1, there is a w2 such that w1Rw2.
(p.36, section 3.2.3)
Priest says now that we can apply these constraints to KL as well.]
This gives the analogue of the two-valued modal logic K. Call it KL. Stronger logics can be obtained by the addition of constraints on the accessibility relation, such as reflexivity (ρ), symmetry (σ), transitivity (τ), giving the logics KLρ, KLσ, KLρτ, etc. (See ch.3.)
(242)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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