Selcuk Kaan Tabakci

Moderate inferentialists defend the thesis that inferences determine the truth-conditional meanings of the logical operators. An important technical and philosophical challenge to this position is the so-called categoricity problem. This is the problem that logics can be soundly interpreted by non-standard models which undermines the idea that inferences lead, unambiguously, to model-theoretic meanings. Several strategies to solve this issue can be found in the literature. One approach consists in strengthening the proof-systems in such a way that the logics in question cannot be soundly interpreted by the non-standard models. Another restricts the class of admissible models of a logic by inferentialistically acceptable principles to rule out non-standard models. This paper discusses attempts in line with this latter solution strategy to the categoricity problem. Our goal is to investigate restrictions on models that are both inferentialistically well-motivated and applicable to a wide variety of logical systems. This is not an easy task, since several inferentialistically unobjectionable restrictions do not generalize beyond particular logical systems, whereas other general and successful principles lack inferentialistically acceptable motivations. This paper proposes a taxonomy of several classes of principles and investigates their scope and chances of success in the context of the project of the moderate inferentialist about logical meanings.

Even though the strong relationship between proof-theoretic and model-theoretic notions in one’s logical theory can be shown by soundness and completeness proofs, whether we can define the model-theoretic notions by means of the inferences in a proof system is not at all trivial. For instance, provable inferences in a proof system of classical logic in the logical framework do not determine its intended models as shown by Carnap (Formalization of logic, Harvard University Press, Cambridge, 1943), i.e., there are non-Boolean models that satisfy its provable inferences. In the literature, this is known as the Categoricity problem or Carnap’s problem. In this paper, we will discuss the Categoricity problem (or Carnap’s problem) for three-valued logics K3 and LP. We will provide three different restrictions on admissible models that will deliver us categoricity results, some of which draw from the solutions provided for the Categoricity problem for classical logic in Belnap and Massey (Stud Log 49(1):67–82, 1990) and Bonnay and Westerståhl (Erkenntis 81(4):721–739, 2016). We will then argue that two of those solutions are philosophically well-motivated: (1) restricting the admissible models where negation is interpreted as a Strong Kleene truth-function, and (2) restricting the admissible models where a complex formula is assigned the third value when its immediate subformulas are assigned the third value.

Frame semantics for negation on the Australian Plan accommodates many different negations, but it falls short on accommodating subminimal negation when the language contains conjunction and disjunction. In this paper, I will present a multi-relational frame semantics –multi-incompatibility frame semantics– that can accommodate subminimal negation. I will first argue that multi-incompatibility frames are in accordance with the philosophical motivations behind negation on the Australian Plan, namely its modal and exclusion-expressing nature. Then, I will prove the soundness and completeness results of a subminimal logic that consists of the multi-incompatibility semantics and a proof system with operational rules that characterize subminimal negation, conjunction and disjunction. Lastly, I will prove some key correspondence theorems that relate frame conditions to certain principles that are associated with stronger negations, which will give rise to a new kite of negations that includes subminimal negation.