Expressive Three-valued Truth Functions

The expressive truth functions of two-valued logic have all been characterized, as have the expressive unary truth functions of finitely-many-valued logic. This paper introduces some techniques for identifying expressive functions in three-valued logics. 1  We are going to explore a property of truth functions known as “expressiveness.” Before subjecting readers to the long-winded explanation of what that property is, I would like to indicate why the property is worth investigating. Pick any n-valued sentential logic. Suppose p is a sentential variable. Consider, for the moment, those sentences with occurrences of no sentential variables other than p. Call these the p-sentences. Let Dk be the set of p-sentences that receive a designated value when p receives the value k. Dk is the set of p-sentences that are true in some way when p is true or untrue in manner k. Such a set is sometimes called a . Suppose, for any values k and j, that Dk ⊆ Dj only if k = j. This means the theories D1, . . . ,Dn are pairwise distinct: Dk 6= Dj if k 6= j. It also means they are maximally satisfiable in the set of p-sentences: ifφ is a p-sentence not inDk, then, nomatter what value p has, some member of Dk ∪ {φ} will be untrue in some way. Since each p-sentence has one of only nn possible truth tables, each theory Dk is finite modulo logical equivalence (i.e., modulo identity of truth tables). Take a representative from each logical equivalence class and form new sets ∆1(p), . . . , ∆n(p) by throwing out all the members of each Dk except any of those representatives that might be present. (So each member of Dk is represented by exactly one equivalent sentence in ∆k(p).) ∗Grateful thanks to the two referees who helped make this paper much more readable. Stephen Pollard, “Expressive Three-valued Truth Functions”, Australasian Journal of Logic (4) 2006, 226–243 http://www.philosophy.unimelb.edu.au/ajl/2006 227 Now return to the full language of our logic. Given any sentence φ, form the set ∆k(φ) by replacing every occurrence of p in every member of ∆k(p) with an occurrence of φ. We can show that an interpretation will assign a designated value to each member of ∆k(φ) if and only if it assigns k to φ. That is, the members of ∆k(φ) jointly affirm that φ has value k. The right-left direction is easy: an interpretation assigning k to φ will assign a designated value to each member of ∆k(φ). Here is a proof of the converse. F(p) and G(p) will be p-sentences. F(φ) will be the result of replacing each occurrence of p in F(p) with an occurrence of φ. f will be the unary truth function whose graph is the same as the truth table for F(p). M(φ)will be the value an arbitrary interpretationM assigns to the sentenceφ. Now supposeG(p) ∈ Dk. Let F(p) be the member of ∆k(p) equivalent to G(p). Then F(φ) ∈ ∆k(φ). Suppose M assigns a designated value to each member of ∆k(φ). Then M(F(φ)) is designated, as is f(M(φ)). So F(p) receives a designated value when p receives the valueM(φ). That is, F(p) ∈ Dj where j = M(φ). So G(p) ∈ Dj. More generally, Dk ⊆ Dj. So k = j = M(φ). A key assumption here was that Dk ⊆ Dj only if k = j. We have seen that if D1, . . . ,Dn have this property, if they are pairwise distinct and maximally satisfiable in the set of p-sentences, then, for each value k and sentence φ, we can use finitely many sentences to assert that φ has value k. Here are two reasons to take an interest in this. Forms of assertion and denial. Say that one denies a sentence when one attributes some form of untruth to it. In assessing the expressive capacities of some language, one might inquire about how many forms of denial it affords. If D1, . . . ,Dn are distinct and maximally satisfiable, then a partial answer is, “The language provides at least as many forms of denial as there are undesignated values.” After all, to attribute an undesignated value j to a sentence φ, one need only assert eachmember of∆j(φ). The same holds for forms of assertion. One can affirm thatφ is true in some unspecified way by assertingφ. One can affirm that φ is true in some particular way by asserting each member of ∆k(φ) for some designated k. Students of many-valued resolution might find it helpful to recall the intended meaning of signed formulas or meta-language literals. (See, for example, [1].) A literal pk is meant to assert that sentence p has value k. IfD1, . . . ,Dn are distinct and maximally satisfiable, then the content of each such literal is captured by finitely many object-language sentences. We now consider one reason this might be convenient. Formalizability. Suppose we can translate each meta-language literal using finitely many sentences of the object-language. Then a generalization of a wellknown algorithm yields a deductive system that is sound and complete with respect to our logic. Beall and van Fraassen [2, pp. 182–185] provide a nice exposition. (See also [3] and [8].) They presuppose, for each value k, a predicate Fk such that Fk(φ) receives a designated value in an interpretation if and only if φ receives the value k in that interpretation. That is, they assume there is a single sentence announcing thatφ has value k. It presents no problem, though, Stephen Pollard, “Expressive Three-valued Truth Functions”, Australasian Journal of Logic (4) 2006, 226–243 http://www.philosophy.unimelb.edu.au/ajl/2006 228 if several (but still only finitely many) sentences are needed to disclose the value of φ. A few easy modifications allow us to use the members of ∆k(φ) in place of the sentence Fk(φ). Now what does this have to do with expressiveness? Suppose f is an nvalued truth function with our (as yet undefined) property: suppose f is expressive. Actually, “expressive” is shorthand for “expressive with respect to a set of designated values.” So suppose f is expressive with respect to the designated values of our logic. Suppose, further, that the graph of f is the same as the truth table of some sentence in our logic. If f is unary, thenD1, . . . ,Dn will be maximally satisfiable in the set of p-sentences and, as long as D1, . . . ,Dn are pairwise distinct, our logic will have the nice properties just discussed. This is the case no matter what n is. (See Theorem 4.3 of [8].) But suppose now that n = 3, our values being 1, 2, and 3 with just 1 designated. Suppose f is expressive with respect to {1}. If f(1 . . . 1) 6= 1, then, again, D1, . . . ,Dn will be maximally satisfiable in the set of p-sentences and, as long as D1, . . . ,Dn are all distinct, our logic will have the nice properties just discussed. (See Theorem 6 below.) There are various results of this sort. The general idea is that we can confirm that a logic has certain desirable properties (say, an elegant formalization with a straightforward Henkin-style completeness proof ) if we can show that an expressive function of one kind or another is definable in the logic. This makes it desirable to have techniques for identifying expressive functions. 2   Since expressiveness began life as a property of closure spaces, it will be helpful to review a bit of closure space theory. Pick some universe of discourse S. A   C on this universe is a set of subsets of S closed under intersection. That is, ⋂ W ∈ C whenever W ⊆ C (letting ⋂ ∅ = S). The members of C are known as the  . A subset W of C  a closed set B if and only if B = ⋂ W. B is  if and only if it is reduced by some subset of C \ {B}. If A ⊆ S, then Cl(A), the  of A, is the intersection of all the members of C that contain A. In a  closure space, if x ∈ S andA ⊆ S, then x will belong to Cl(A) only if it belongs to the closure of some finite subset of A. If B ∈ C, then B is   if and only if B sits just below S in the lattice of closed sets: that is, S is the only member of C that properly contains B. Now, at last, we come to our first definition of expressiveness. A finitary closure space is  if and only if all its irreducible sets are maximally consistent. (There is a more general definition that applies to non-finitary closure spaces, but we can make do with the simpler, more restricted version since we consider only finitary logics. For the more general version, see [5, p. 121].) Return, now, to n-valued logic. Say that a sentence φ is a  Stephen Pollard, “Expressive Three-valued Truth Functions”, Australasian Journal of Logic (4) 2006, 226–243 http://www.philosophy.unimelb.edu.au/ajl/2006 229 of a set of sentences A if and only if every interpretation assigning a designated value to each member of A also assigns a designated value to φ. Let C consist of the sets of sentences closed under consequence. (Those sets whose consequences are already members.) Then C is a finitary closure space. (See [9, pp. 142–144]; [10, 11].) For each interpretation M, let the  DM be the set of sentences assigned a designated value by M. Each irreducible set is a theory. (S, the set of all sentences, is reducible because S = ⋂ ∅. Every other closed set is the intersection of the theories that contain it.) So C is expressive if and only if each of its irreducible theories is maximally consistent. We are interested in the circumstances under which this occurs. In particular, we are interested in whether the definability of certain truth functions might guarantee expressiveness. 3   Suppose our logic is two-valued. Suppose the function g returns a T when we feed it nothing but F’s and returns F for at least one sequence of inputs. Say, for example, that g(FFFF) = T and g(TFFT) = F. Suppose there is a sentence θ in our language whose truth table is exactly the graph of g. So θ(φψψφ) is T when φ and ψ are both F, while θ(φψψφ) is F when φ is T and ψ is F. Then we can show that our logic has a property a bit stronger than expressiveness: each of its consistent theories (each of its theories other than S) is maximally consistent. To confirm this, supposeDM 6= S. IfM assigns F to only one sentence, thenDM is maximally consistent and we are done. Suppose M assigns F to at least two sentences. SupposeDN properly containsDM. Let φ be a sentence that M considers false but N considers true. Let ψ be any other sentence that M considers false. Then M thinks θ(φψψφ) is true. N thinks a sentence is true wheneverM thinks it is true. So N thinks θ(φψψφ) is true. But this meansN cannot think ψ is false. ψ was an arbitrary falsehood ofM. So N thinks all ofM’s falsehoods are truths. So N thinks everything is true. That is, DN = S. So S is the only theory that properly contains DM. So S is the only closed set that properly contains DM. So DM is maximally consistent. Every two-valued logic with a sentence expressing g is expressive. This leads us to say that g itself is expressive in the sense of “expressiveness guaranteeing” or “expressiveness producing.” To be pedantic, g is expressive with respect to the choice of {T } as the set of designated values (a qualification that can be omitted without fear of confusion in the two-valued case). We are interested in characterizing expressive n-valued truth functions in general. The two-valued ones have all been characterized [7], as have all the unary ones for each choice of n [8]. We are going to investigate some of the remaining functions, with special emphasis on the three-valued ones. First, however, a brief detour that will further motivate this project. Stephen Pollard, “Expressive Three-valued Truth Functions”, Australasian Journal of Logic (4) 2006, 226–243 http://www.philosophy.unimelb.edu.au/ajl/2006 230 4  Intuitionist and classical connectives do not always mix well. If you have a million connectives obeying the rules for intuitionist negation and even one of them satisfies the classical principle of double negation elimination, then they all do. If you have a million connectives obeying the rules for the intuitionist conditional and even one of them satisfies Peirce’s law, then they all do. Our investigations into expressive functions help us to identify deductive properties that give connectives this power of assimilation. For example, there is an algorithm that takes us from the truth table for an expressive two-valued function like g to a pair of sequents that will “deintuitionize” each intuitionist conditional and negation. To see an example of how this works, consider again our two-valued logic and our function g. We first introduce a new connective with the stipulation that M(φ ψ) = g(M(φ)M(ψ)M(ψ)M(φ)). If A and B are sets of sentences, we say that A |= B if and only if B intersects each theory that contains A. (That is, an interpretation assigning T to each member of A will assign T to at least one member of B.) Then we can verify the following. • {φ, (φ ψ)} |= {ψ} • ∅ |= {φ,ψ, (φ ψ)} Now go to any deductive system featuring the corresponding sequents. • φ, (φ ψ) ` ψ • ` φ,ψ, (φ ψ) Suppose this system also features a connective ¬ obeying the intuitionistically valid principle that ψ1, . . . , ψk ` ¬φ whenever φ,ψ1, . . ., ψk ` ¬φ. Then we can argue as follows. 1. φ, (φ ¬φ) ` ¬φ 2. (φ ¬φ) ` ¬φ 3. ` φ, ¬φ, (φ ¬φ) 4. ` φ, ¬φ Since this last line is a version of excluded middle, we see that has assimilated or “de-intuitionized” ¬. has a similar effect on any connective satisfying modus ponens and the deduction theorem. This means that anyone who wants to speak an intuitionist language and a language expressing g will have to prevent the latter from infiltrating the former. This holds for any expressive two-valued truth function. Things are just a bit more complicated when we add more truth-values. Here is an example. Go to a three-valued logic where 2 and 3 are designated, Stephen Pollard, “Expressive Three-valued Truth Functions”, Australasian Journal of Logic (4) 2006, 226–243 http://www.philosophy.unimelb.edu.au/ajl/2006 231 but 1 is not. Suppose the function h has a matrix of the following form. (The idea is that an empty cell can be filled by any value drawn from {1, 2, 3}.)