Logic and Philosophy of Logic

This paper makes first steps toward a systematic investigation of how pertinence to topic contributes to determine deductively valid reasoning along with preservation of designated values. I start from the interpretation of Weak Kleene Logic WKL as a reasoning tool that preserves truth and topic pertinence, which is offered by Jc Beall. I keep Beall’s motivations and I argue that WKL cannot meet them in a satisfying way. In light of this, I propose an informal definition of a topic-sensitive logic (...) and I proceed to turn it into a formal one by deploying the topic-algebraic framework from the Topic-Sensitive Intentional Modalities project by Francesco Berto. I apply the framework in order to define semantically a ‘classical topic-sensitive logic’ CTSL that meets Beall’s motivations and proves topic-sensitive. Then, I prove results that connect CTSL and any possible topic-sensitive logic to the tradition of containment logics and provide a unification tool for a wide range of recent proposals in philosophical logic. A topic-theoretical interpretation of WKL is then offered without prejudice to the fact that the logic is not topic-sensitive. Finally, the paper discusses some conceptual issues and research perspectives. (shrink)

The aim of this work is to investigate the problem of Logical Omniscience in epistemic logic by means of truthmaker semantics. We will present a semantic framework based on $$\varvec{W}$$ W -models extended with a partial function, which selects the body of knowledge of the agents, namely the set of verifiers of the agent’s total knowledge. The semantic clause for knowledge follows the intuition that an agent knows some information $$\varvec{\phi }$$ ϕ, when the propositional content that $$\varvec{\phi }$$ ϕ (...) is contained in her total knowledge. We will argue that this idea mirrors the philosophical conception of immanent closure by Yablo (2014), giving to our proposal a strong philosophical motivation. We will discuss the philosophical implications of the semantics and we will introduce its axiomatization. (shrink)

Bochvar algebras consist of the quasivariety $\mathsf {BCA}$ playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [4] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one proper extension (apart from classical logic), algebraized by the (...) subquasivariety $\mathsf {NBCA}$ of $\mathsf {BCA}$. Furthermore, we address the problem of (passive) structural completeness ((P)SC) for each of them, showing that $\mathsf {NBCA}$ is SC, while $\mathsf {BCA}$ is not even PSC. Finally, we prove that both $\mathsf {BCA}$ and $\mathsf {NBCA}$ enjoy the amalgamation property (AP). (shrink)

Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form $$p\wedge \Diamond \lnot p$$ (‘p, but it might be that not p’) appears to be a contradiction, $$\Diamond \lnot p$$ does not entail $$\lnot p$$, which would follow in classical logic. Likewise, the classical laws of distributivity and disjunctive syllogism fail for epistemic modals. Existing attempts to account for these facts generally either under- or over-correct. (...) Some theories predict that $$ p\wedge \Diamond \lnot p$$, a so-called epistemic contradiction, is a contradiction only in an etiolated sense, under a notion of entailment that does not always allow us to replace $$p\wedge \Diamond \lnot p$$ with a contradiction; these theories underpredict the infelicity of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that intuitively fail, like distributivity and disjunctive syllogism, but also rules like non-contradiction, excluded middle, De Morgan’s laws, and disjunction introduction, which intuitively remain valid for epistemic modals. In this paper, we aim for a middle ground, developing a semantics and logic for epistemic modals that makes epistemic contradictions genuine contradictions and that invalidates distributivity and disjunctive syllogism but that otherwise preserves classical laws that intuitively remain valid. We start with an algebraic semantics, based on ortholattices instead of Boolean algebras, and then propose a possibility semantics, based on partial possibilities related by compatibility. Both semantics yield the same consequence relation, which we axiomatize. We then show how to lift an arbitrary possible worlds model for a non-modal language to a possibility model for a language with epistemic modals. The goal throughout is to retain what is desirable about classical logic while accounting for the non-classicality of epistemic vocabulary. (shrink)

We show that the properties of [relative] semisimplicity and congruence 3-permutability of a [quasi]variety with equationally definable [relative] principal congruences (EDP[R]C) can be characterized syntactically. We prove that a quasivariety with EDPRC is relatively semisimple if and only if it satisfies a finite set of quasi-identities that is effectively constructible from any conjunction of equations defining relative principal congruences in the quasivariety. This in turn allows us to obtain an ‘axiomatization’ of relatively filtral quasivarieties. We also show that a variety (...) is 3-permutable and has EDPC if and only if there is a single pair of quaternary terms satisfying two simple equations, and whose equality defines principal congruences in the variety. Finally, we combine both results to obtain a neat characterization of semisimple, 3-permutable varieties with EDPC, which is applied to solve a problem posed by Blok and Pigozzi in the third paper of their series on varieties with EDPC. (shrink)

De Rijke introduced a unary interpretability logic $$\textbf{il}$$, and proved that $$\textbf{il}$$ is the unary counterpart of the binary interpretability logic $$\textbf{IL}$$. In this paper, we find the unary counterparts of the sublogics of $$\textbf{IL}$$.

We develop intuitionistic public announcement logic over intuitionistic \({\textbf{K}}\), \({{\textbf{K}}}{{\textbf{T}}}\), \({{\textbf{K}}}{{\textbf{4}}}\), and \({{\textbf{S}}}{{\textbf{4}}}\) with distributed knowledge. We reveal that a recursion axiom for the distributed knowledge is _not_ valid for a frame class discussed in [ 12 ] but valid for the restricted frame class introduced in [ 20, 26 ]. The semantic completeness of the static logics for this restricted frame class is established via the concept of pseudo-model.

Recent research on algebraic models of _quasi-Nelson logic_ has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a _nucleus_. Among these various algebraic structures, for which we employ the umbrella term _intuitionistic modal algebras_, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations (...) arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of _weak implicative semilattices_, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus. (shrink)

We provide an alternative proof of the decidability of the equational theory of lattices. The proof presented here is quite short and elementary.

The main aim of this paper is to introduce the logics of evidence and truth $$LET_{K}^+$$ and $$LET_{F}^+$$ together with sound, complete, and decidable six-valued deterministic semantics for them. These logics extend the logics $$LET_{K}$$ and $$LET_{F}^-$$ with rules of propagation of classicality, which are inferences that express how the classicality operator $${\circ }$$ is transmitted from less complex to more complex sentences, and vice-versa. The six-valued semantics here proposed extends the 4 values of Belnap-Dunn logic with 2 more values (...) that intend to represent (positive and negative) reliable information. A six-valued non-deterministic semantics for $$LET_{K}$$ is obtained by means of Nmatrices based on swap structures, and the six-valued semantics for $$LET_{K}^+$$ is then obtained by imposing restrictions on the semantics of $$LET_{K}$$. These restrictions correspond exactly to the rules of propagation of classicality that extend $$LET_{K}$$. The logic $$LET_{F}^+$$ is obtained as the implication-free fragment of $$LET_{K}^+$$. We also show that the 6 values of $$LET_{K}^+$$ and $$LET_{F}^+$$ define a lattice structure that extends the lattice L4 defined by the Belnap-Dunn four-valued logic with the 2 additional values mentioned above, intuitively interpreted as positive and negative reliable information. Finally, we also show that $$LET_{K}^+$$ is Blok-Pigozzi algebraizable and that its implication-free fragment $$LET_{F}^+$$ coincides with the degree-preserving logic of the involutive Stone algebras. (shrink)

In the paper we examine the problem of logical systems that are extensions of Classical Propositional Logic with new, intensional connectives of agency: monadic and dyadic _bringing it about that_. These systems are usually studied within the neighbourhood semantics. Here we propose a different strategy. We study all of the accepted laws and rules of logic of agency and define a translation of the agency operators into connectives interpreted in relating semantics. After this translation we can make a reduction to (...) more basic semantic properties that are required by the particular groups of laws and axioms. Finally, we define proper semantic structures and prove that they are complete with respect to all possible logical systems determined by the combinations of the axioms and rules. (shrink)

We extend meet-combination of logics for capturing the consequences that are common to both logics. With this purpose in mind we define meet-combination of consequence systems. This notion has the advantage of accommodating different ways of presenting the semantics and the deductive calculi. We consider consequence systems generated by a matrix semantics and consequence systems generated by Hilbert calculi. The meet-combination of consequence systems generated by matrix semantics is the consequence system generated by their product. On the other hand, the (...) meet-combination of consequence systems generated by Hilbert calculi is the consequence system generated by their interconnection. We investigate preservation of several properties. Capitalizing on these results we show that interconnection provides an axiomatization for the product. Illustrations are given for intuitionistic and modal logics, Łukasiewicz logic and some paraconsistent logics. (shrink)

We investigate whether ordinary quantification over objects is an extensional phenomenon, or rather creates non-extensional contexts; each claim having been propounded by prominent philosophers. It turns out that the question only makes sense relative to a background theory of syntax and semantics (here called a grammar) that goes well beyond the inductive definition of formulas and the recursive definition of satisfaction. Two schemas for building quantificational grammars are developed, one that invariably constructs extensional grammars (in which quantification, in particular, thus (...) behaves extensionally) and another that only generates non-extensional grammars (and in which quantification is responsible for the failure of extensionality). We then ask whether there are reasons to favor one of these grammar schemas over the other, and examine an argument according to which the proper formalization of deictic utterances requires adoption of non-extensional grammars. (shrink)

It is widely thought that chance should be understood in reductionist terms: claims about chance should be understood as claims that certain patterns of events are instantiated. There are many possible reductionist theories of chance, differing as to which possible pattern of events they take to be chance-making. It is also widely taken to be a norm of rationality that credence should defer to chance: special cases aside, rationality requires that one’s credence function, when conditionalized on the chance-making facts, should (...) coincide with the objective chance function. It is a shortcoming of a theory of chance if it implies that this norm of rationality is unsatisfiable. The primary goal of this paper is to show, on the basis of considerations concerning computability and inductive learning, that this shortcoming is more common than one would have hoped. (shrink)

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa (...) $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness. (shrink)

We formulate and explore two basic axiomatic systems of type-free subjective probability. One of them explicates a notion of finitely additive probability. The other explicates a concept of infinitely additive probability. It is argued that the first of these systems is a suitable background theory for formally investigating controversial principles about type-free subjective probability.

The aim of this paper is to provide a complete Natural Kind Semantics for an Essentialist Theory of Kinds. The theory is formulated in two-sorted first order monadic modal logic with identity. The natural kind semantics is based on Rudolf Willes Theory of Concept Lattices. The semantics is then used to explain several consequences of the theory, including results about the specificity (species–genus) relations between kinds, the definitions of kinds in terms of genera and specific differences and the existence of (...) negative kinds. First, I show under which conditions the Hierarchy principle, which has been subjected to counterexamples in the literature, holds. I also show that a different principle about the species–genus relations between kinds, namely Kant’s Law, follows from the essentialist theory. Second, I introduce two new operations for kinds and show that they can be used to provide traditional definitions of kinds in terms of genera and specific differences. Finally, I show that these operations of specific difference induce, for each kind, a uniquely specified contrary kind and a uniquely specified subcontrary kind, which can be used as semantic values for non-classical predicate negations of kind terms. (shrink)

Two of the most well-known belief contraction operators are partial meet contractions (PMCs) and kernel contractions (KCs). In this paper we propose two new classes of contraction operators, namely the class of generalized partial meet contractions (GPMC) and the class of generalized kernel contractions (GKC), which strictly contain the classes of PMCs and of KCs, respectively. We identify some extra conditions that can be added to the definitions of GPMCs and of GKCs, which give rise to some interesting subclasses of (...) those classes of functions, namely the classes of extensional and of uniform GPMCs/GKCs. In the context of contractions on belief sets the classes of partial meet contractions, uniform GPMCs and extensional GPMCs are all identical. Nevertheless, when considered as operations on belief bases, the class of uniform GPMCs coincides with the class of partial meet contractions, but the extensional GPMCs constitute a new kind of belief base contraction functions whose characterizing postulate of irrelevance of syntax is extensionality—the same postulate of irrelevance of syntax which occurs in the classical axiomatic characterization of partial meet contractions for belief sets—rather than the postulate of uniformity—which is the irrelevance of syntax postulate used in the axiomatic characterization for partial meet contractions on belief bases. Analogous results are obtained regarding the classes of extensional and of uniform GKCs. We present the interrelations in the sense of inclusion among all the new classes of operators presented in this paper and several well known classes of PMCs and of KCs. (shrink)

In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication $\phi \rightarrow \psi $ with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article is about the property more correctly named ‘deductive interpolation’ (see (...) Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent. (shrink)