Suszko’s Problem: Mixed Consequence and Compositionality

Abstract

Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko's problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that (intersective) mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those theorems to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various structural properties (reflexivity, transitivity, none, or both). We strengthen these results into exact rank results for non-permeable logics (roughly, those which distinguish the role of premises and conclusions). We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties.by Tarski and the Polish school (see Bloom et al., 1970;Tarski, 1930;Wójcicki, 1973Wójcicki, , 1988, namely: substitution-invariance, monotonicity, reflexivity and transitivity. Those properties play a central role in Suszko's reduction. We then articulate what we mean by a semantics for a logic, and state various semantic properties which we will prove to be counterparts of those structural properties in section 3. Syntactic notions LogicThroughout the paper, we work within a multiple-premise multiple-conclusion logic, following the tradition of Gentzen (1935), Scott (1974) and Shoesmith and Smiley (1978). One of the reasons for doing so is simplicity in that premises and conclusions are handled symmetrically. This choice is not neutral and some of our results are likely to differ in a multi-premise single-conclusion setting. We leave that issue aside, and define a logic as follows:Definition 2.1 (Logic). A logic is a triple L, C, , with L a language (set of formulae), C a (possibly empty) set of connectives, and a consequence relation, what we call a formula-relation (i.e. a subset of P (L) × P (L), also called a set of arguments, where ...

From many-valued consequence to many-valued connectives

“…The last one, atomic expressiveness, is the least demanding. We call it atomic expressiveness because it holds for instance if the language has atomic formulae whose semantic values are meant to cover the whole space of truth-values and with maximal variation across the different atomic formulae, as in the columns of a truth-table (this situation corresponds to a 'valuational' semantics in the sense of Chemla and Egré, 2019). Maximal expressiveness and constant expressiveness play a useful role in the rest of the paper, and by default we assume that they hold.…”

“…An important feature of the Gentzen rules is their analytic character, namely the fact that the meaning of a connective, whether in premise position or in conclusion position, is explained fully in terms of the (possibly empty) conjunction of sequent rules involving the component subformulae of the formula build from that connective. In Chemla and Egré (2019), we put forward the notion of a regular connective to describe logical connectives obeying such constraints:…”

“…Beyond Gentzen's approach, regularity matters because of the following result from Chemla and Egré (2019):…”

Metainferential duality

“…With all these pieces in place, let us now proceed to the discussion of duality, and to the assessment of how it resonates in the context of the non-classical logics that we just introduced. 5 A word on how q-and p-matrices generalize the usual notion of a logical matrix is in order. In a usual logical matrix V, D, O the truth-values of the matrix, i.e.…”

“…In the same line, there is another interesting way of thinking about the self-duality of the inferences of ST and TS, which is a more abstract way of taking the idea of negation duality as applied to the semantics and may also be related to Kleene's ideas. As suggested by an anonymous reviewer, let us say (as in [5]) that a set of truth-values A is in a p-relation R to some other set B provided either some a i ∈ A does not belong to D + or some b j ∈ B belongs to D − -and similarly for a q-relation. 11 In this vein, we could define the dual relation R to R as determined by the fact that either some a i ∈ A is such that n(a i ) does not belong to the set n(D + ) made of negations of values in D + , or some b j ∈ B is such that n(b j ) belongs to the set n(D − ) of negations of values in D − .…”