Suszko's Thesis and Many-valued Logical Structures

In this article, we try to formulate a definition of ‘‘many-valued logical structure’’. For this, we embark on a deeper study of Suszko’s Thesis ($\mathbf{ST}$) and show that the truth or falsity of $\mathbf{ST}$ depends, at least, on the precise notion of semantics. We propose two different notions of semantics and three different notions of entailment. The first one helps us formulate a precise definition of inferentially many-valued logical structures. The second and the third help us to generalise Suszko Reduction and provide adequate bivalent semantics for monotonic and a couple of nonmonotonic logical structures. All these lead us to a closer examination of the played by language/metalanguage hierarchy vis-á-vis $\mathbf{ST}$. We conclude that many-valued logical structures can be obtained if the bivalence of all the higher-order metalogics of the logic under consideration is discarded, building formal bridges between the theory of graded consequence and the theory of many-valued logical structures, culminating in generalisations of Suszko’s Thesis.

💡 Research Summary

The paper tackles the fundamental question of what “many‑valuedness” really means in logic, focusing on Suszko’s Thesis (ST) – the claim that there are only two logical values, true and false. The authors argue that the truth of ST hinges on the precise notion of semantics employed. They introduce two distinct semantic frameworks and three kinds of entailment, using these to give a rigorous definition of inferentially many‑valued logical structures.

First, a logical structure is presented either as a consequence relation (L, ⊢) or as a consequence operator (L, W). Traditional matrix semantics are shown to be equivalent to a more general semantic tuple (M, {|= i}₍ᵢ∈I₎, S, P(L)) that includes a set of “worlds” M, world‑specific satisfaction relations, and a transition relation S between worlds. This generalized semantics yields an induced entailment relation ⊢ₛ, and a logical structure is said to be adequate with respect to a semantics if its entailment coincides with ⊢ₛ.

The authors then define four families of consequence operators – q‑, p‑, sκ‑, and rκ‑operators – each giving rise to a different class of logical structures. They prove that monotonic logical structures split into four (possibly overlapping) subclasses, each of which admits a minimal three‑valued semantics (Theorems 3.8, 3.15, 3.31). Moreover, they show that these three‑valued semantics are optimal: no smaller‑valued semantics can capture the same entailment (Theorems 3.10, 3.17, 3.33). This establishes a formal hierarchy of inferential many‑valuedness, distinguishing it from the traditional algebraic many‑valuedness.

In Section 4 the classic Suszko Reduction – which collapses any logic to a bivalent one – is generalized. By adopting a new semantics (Definition 4.1) the authors construct adequate bivalent semantics not only for monotonic logics but also for certain non‑monotonic systems (Theorems 4.21, 4.22). This demonstrates that the failure of ST in many contexts can be remedied either by altering the semantics or by relinquishing bivalence at higher meta‑logical levels.

Section 5 introduces κ‑valued logical structures of order λ (Definition 5.2). The authors argue that if the bivalence of all higher‑order metalogics up to a certain level is discarded, one can systematically build logical structures with any prescribed cardinal κ of logical values. This bridges the theory of graded consequence with many‑valued logical structures and yields a generalized form of Suszko’s Thesis: ST holds only when the meta‑logic is forced to be bivalent; otherwise, many‑valued structures naturally arise.

The conclusion emphasizes that the principle of bivalence is not an absolute logical law but a meta‑theoretical assumption. By carefully distinguishing logical values from algebraic values and by allowing higher‑order metalogics to be non‑bivalent, the paper provides a comprehensive framework that both clarifies the status of Suszko’s Thesis and opens avenues for further research on non‑monotonic logics, graded consequence, and the interplay between κ‑valued logics and probabilistic or fuzzy systems.