Under Review

As discussed in Tabakci (2022), the weakest constructive negation, subminimal negation, can be interpreted on the Australian Plan. In this paper, we will show that subminimal negation as interpreted on the Australian Plan can be a basis for a family of relevant constructive logics. We first add subminimal negation to the positive fragment of the logic B and provide soundness and completeness for our base subminimal relevant logic BN. Then, we will provide correspondence results for some negation rules and axioms that can extend BN relevantly and constructively. We will then show that some of these negation axioms can be added to the positive fragment of R-mingle without causing a violation of the variable sharing principle, and hence, we will show that a wide range of constructive relevant logics can be interpreted on the Australian Plan. 

In Preparation

Do logicians talk past each other when they defend different logics, i.e., does the meaning of logical vocabulary change when logics differ?  In this paper, we will provide a model-theoretic inferentialist approach to this question by comparing the model-theoretic semantics expressed by some logics. In particular, we will determine whether there is meaning preservation across different Strong Kleene Logics and Classical Logic by comparing the valuational semantics determined by them. Our first result shows that some of our Strong Kleene Logics and Classical Logic, known as ST and TS in the literature, share the meaning of their connectives. However, our second result shows that the connective meaning expressed by LP and K3 differ from each other and all the other three logics. 


The problematic connective tonl was introduced by Prior (1960) to demonstrate that solely stipulating sets of inference rules is not sufficient for a logical operator to be meaningful. Following Belnap (1962), we can diagnose the problem caused by tonk in terms of the incompatibility of its inference rules with the Tarskian deducibility relation. As argued by Ripley (2014), this means that the inference rules of tonk can confer meaning in proof systems that gives rise to non-Tarskian deducibility relations, in particular, non-transitive or non-reflexive deducibility relations. In this paper, we will analyze whether we can have a meaningful tonk when non-Tarskian deducibility relations are individuated by their metainferences. 


As shown by Comaseña (2008), there is an inconsistency between the universal composition principle and intuitive modal claims such as "there could be even number of objects" or "For any natural number n, if there are n objects, possibly there n+1 objects". In this paper, we first formalize the inconsistency proofs given by Comaseña (2008) in Classical Mereology, and then argue that one either needs to give up Weak Supplementation principle, reflexivity of parthood, Leibniz's Law or the definition of fusion in order to reconcile universal composition principle with such modal claims, . We then revise the classical theory of mereology by introducing indeterminate fusion. We also argue that this theory is particularly useful to model the behavior of indeterminately existing objects, since it allows fusion to occur indeterminately.