Logic and Philosophy of Logic

We explore the interplay between $\omega $ -categoricity and pseudofiniteness for groups, and we conjecture that $\omega $ -categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class 2, and give a proof that several of the known examples of $\omega $ -categorical p-groups satisfy the conjecture. In particular, we show by a direct counting argument that for any odd prime p the ( $\omega $ -categorical) model companion of the theory of nilpotent class (...) 2 exponent p groups, constructed by Saracino and Wood, is not pseudofinite, and that an $\omega $ -categorical group constructed by Baudisch with supersimple rank 1 theory is not pseudofinite. We also survey some scattered literature on $\omega $ -categorical groups over 50 years. (shrink)

In the original Ulam-Rényi game with _m_ lies/errors, Player I chooses a secret number \({\bar{x}}\) in a finite search space _S_, and Player II must guess \({\bar{x}}\) by adaptively asking Player I a minimum number of binary questions. Up to _m_ answers may be mendacious/erroneous or may be distorted before reaching Player II. In his monograph “Fault-Tolerant Search Algorithms. Reliable Computation with Unreliable Information”, F. Cicalese provides a comprehensive account of many models of the game and their applications in error-correcting (...) coding with noiseless feedback. Since for \(m>0\) lies the game is not called off by contradictory answers, and repeated answers to the same question are more informative than single answers, the underlying logic of the game with _m_ lies is not boolean. Indeed, the logic of Rényi-Ulam games is Łukasiewicz infinite-valued logic Ł \(_\infty \). In this paper we consider Ulam-Rényi games with variable numbers of lies over infinite search spaces. We characterize the MV-algebras and the unital Specker \(\ell \) -groups given by the logic of these games. (shrink)

Proof-theoretic semantics (P-tS) is the approach to meaning in logic based on _proof_ (as opposed to truth). There are two major approaches to P-tS: proof-theoretic validity (P-tV) and base-extension semantics (B-eS). The former is a semantics of arguments, and the latter is a semantics of logical constants. This paper demonstrates that the B-eS for _intuitionistic propositional logic_ (IPL) encapsulates the declarative content of a version of P-tV based on the elimination rules. This explicates how the B-eS for IPL works, and (...) shows the completeness of this version of P-tV. (shrink)

We investigate the tower spectrum in the generalized Baire space, i.e., the set of lengths of towers in $\kappa ^\kappa $. We show that both small and large tower spectra at all regular cardinals simultaneously are consistent. Furthermore, based on previous work by Bağ, the first author and Friedman, we prove that globally, a small tower spectrum is consistent with an arbitrarily large spectrum of maximal almost disjoint families. Finally, we show that any non-trivial upper bound on the tower spectrum (...) in $\kappa ^\kappa $ is consistent. (shrink)

The concept of location—of something being located at, or occupying, a region or a place—has been an important topic of philosophical investigation over the past fifteen years or so. Yet all the theories of location that have been put forward so far are unsatisfactory, because they fail to have the conceptual resources to describe certain basic locational phenomena. I introduce and partly develop a novel theory of location that does better in this respect than its predecessors. Its most distinctive feature (...) is the way in which it handles so-called cases of multilocation: according to the theory, objects capable of being multilocated occupy regions via proxies, and—in a slogan, and very roughly—to be multilocated is to have several proxies located at different regions. (shrink)

The philosophy of superdeterminism is based on a single scientific fact about the universe, namely that cause and effect in physics are not real. In 2020, accomplished Swedish theoretical physicist, Dr. Johan Hansson published a physics proof using Albert Einstein’s Theory of Special Relativity that our universe is superdeterministic meaning a predetermined static block universe without cause and effect in physics. The unity of our universe originates from its creation from the same nothingness under the zero energy universe theory. However, (...) nothingness cannot actually cause a universe to exist on its own under superdeterminism due to the absence of cause and effect in physics. Moreover, nothingness is the absence of anything real and cannot logically be the reality underlying our universe. Consequently, our universe must exist from the reality of the logic or Logos underlying our physical reality resulting from the origination of our universe from the same nothingness or common creation. (shrink)

We present a logical framework that enables us to define a formal theory of computational trust in which this notion is analysed in terms of epistemic attitudes towards the possible objects of trust and in relation to existing evidence in favour of the trustworthiness of these objects. The framework is based on a quantified epistemic and justification logic featuring a non-standard handling of identities. Thus, the theory is able to account for the hyperintensional nature of computational trust. We present a (...) proof system and a frame semantics for the logic, we prove soundness and completeness results and we introduce the syntactical machinery required to define a theory of trust. (shrink)

It is widely recognized that Frege's systematic conception of science has a major impact on his work. I argue that central to this conception and its impact is Frege's Simplicity Requirement that a scientific system must have as few primitive truths as possible. Frege states this requirement often, justifies it in several ways, and appeals to it to motivate important aspects of his broader views. Acknowledging its central role illuminates several aspects of his work in new ways, including his treatment (...) of truth values, logical concepts, and the dependence relations that induce a ‘natural order’ of truths. (shrink)

George Boole published the pamphlet The Mathematical Analysis of Logic in 1847. He believed that logic should belong to a universal mathematics that would cover both quantitative and nonquantitative research. With his pamphlet, Boole signalled an important change in symbolic logic: in contrast with his predecessors, his thinking was exclusively extensional. Notwithstanding the innovations introduced he accepted all traditional Aristotelean syllogisms. Nevertheless, some criticisms have been raised concerning Boole’s view of Aristotelean logic as the solution of algebraic equations. In order (...) to circumvent such criticisms, we show here how Boole’s conclusions may be deduced from the axioms and inference rules proposed in his pamphlet, from which Aristotle’s deductions in Prior Analytics (including those that are completed through an impossibility) can be stated as theorems. We clarify his method for dealing with both universal and particular premises by means of his symbol v and demystify some criticisms concerning the way he expressed particular premises. Boole’s conclusions for some of those premises for which according to Aristotle there is no deduction are also discussed. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to logic in The Mathematical Analysis of Logic. (shrink)

This article gives a holistic re-examination of the semantic content and syntactic structure of the concept of relative identity: it suggests and explains an expanded and enhanced dual-track characterization of relative identity. It is expanded in this sense: its due coverage is not narrowly restricted to the equal-status case of identity statements (the symmetric case for identity simplex) but also includes the category-assimilating case (the asymmetric case for identity complex), both of which are unified by the shared semantic core content (...) of relative identity. It is enhanced in this sense: it is intended to give a more refined characterization of relative identity for the sake of enhancing our understanding of the structure and content of real-life relative-identity statements in people’s basic employment of language in view of the relation between thought, language, and the world. (shrink)

This paper proposes and partially defends a novel philosophy of arithmetic—finitary upper logicism. According to it, the natural numbers are finite cardinalities—conceived of as properties of properties—and arithmetic is nothing but higher-order modal logic. Finitary upper logicism is furthermore essentially committed to the logicality of finitary plenitude, the principle according to which every finite cardinality could have been instantiated. Among other things, it is proved in the paper that second-order Peano arithmetic is interpretable, on the basis of the finite cardinalities’ (...) conception of the natural numbers, in a weak modal type theory consisting of the modal logic $\mathsf {K}$, negative free quantified logic, a contingentist-friendly comprehension principle, and finitary plenitude. By replacing finitary plenitude for the axiom of infinity this result constitutes a significant improvement on Russell and Whitehead’s interpretation of second-order Peano arithmetic, itself based on the finite cardinalities’ conception of the natural numbers. (shrink)

Recently, it has been proposed to understand a logic as containing not only a validity canon for inferences but also a validity canon for metainferences of any finite level. Then, it has been shown that it is possible to construct infinite hierarchies of ‘increasingly classical’ logics—that is, logics that are classical at the level of inferences and of increasingly higher metainferences—all of which admit a transparent truth predicate. In this paper, we extend this line of investigation by taking a somehow (...) different route. We explore logics that are different from classical logic at the level of inferences, but recover some important aspects of classical logic at every metainferential level. We dub such systems meta-classical non-classical logics. We argue that the systems presented deserve to be regarded as logics in their own right and, moreover, are potentially useful for the non-classical logician. (shrink)

We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell’s paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be used to (...) reason informally about incomplete and inconsistent phenomena, and is sufficiently similar to ${\mathrm {ZFC}}$ to enable the development of interesting mathematics.We propose an axiomatic system ${\mathrm {BZFC}}$, obtained by analysing the ${\mathrm {ZFC}}$ -axioms and translating them to a four-valued setting in a careful manner, avoiding many of the obstacles encountered by other attempted formalizations. We introduce the anti-classicality axiom postulating the existence of non-classical sets, and prove a surprising results stating that the existence of a single non-classical set is sufficient to produce any other type of non-classical set.Our theory is naturally bi-interpretable with ${\mathrm {ZFC}}$, and provides a philosophically satisfying view in which non-classical sets can be seen as a natural extension of classical ones, in a similar way to the non-well-founded sets of Peter Aczel [1].Finally, we provide an interesting application concerning Tarski semantics, showing that the classical definition of the satisfaction relation yields a logic precisely reflecting the non-classicality in the meta-theory. (shrink)

In the early nineteenth century, a series of articles by Laplace and Poisson discussed the importance of ‘directness’ in mathematical methodology. In this thesis, we argue that their conception of a ‘direct’ proof is similar to the more widely contemplated notion of a ‘pure’ proof. More rigorous definitions of mathematical purity were proposed in recent publications by Arana and Detlefsen, as well as by Kahle and Pulcini: we compare Laplace and Poisson’s writings with these modern definitions of purity and show (...) how the modern definitions fail to grasp some more nuanced aspects. (shrink)

Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, (...) Glymour and Halvorson. (shrink)

Logical frameworks that are sensitive to features of sentences’ subject-matter—like Berto’s topic-sensitive intentional modals (TSIMs)—demand a maximally faithful model of the topics of sentences. This is an especially difficult task in the case in which topics are assigned to intensional formulae. In two previous papers, a framework was developed whose model of intensional subject-matter could accommodate a wider range of intuitions about particular intensional conditionals. Although resolving a number of counterintuitive features, the work made an implicit assumption that the subject-matter (...) of an intensional conditional is a function of the subject-matters of its subformulae. This assumption—which I will call a principle of topic sufficiency—runs counter to some natural intuitions concerning topic. In this paper, we will investigate topic sufficiency and offer a semantic account that is state-sensitive, providing an implementation through the introduction of topic-sensitive logics related to William Parry’s prototypical $\mathsf {PAI}$. (shrink)

We formulate Hilbert’s epsilon calculus in the context of expansion proofs. This leads to a simplified proof of the epsilon theorems by disposing of the need for prenexification, Skolemisation, and their respective inverse transformations. We observe that the natural notion of cut in the epsilon calculus is associative.

This paper proves the finite axiomatizability of transitive modal logics of finite depth and finite width w.r.t. proper-successor-equivalence. The frame condition of the latter requires, in a rooted transitive frame, a finite upper bound of cardinality for antichains of points with different sets of proper successors. The result generalizes Rybakov’s result of the finite axiomatizability of extensions of $\mathbf {S4}$ of finite depth and finite width.

In this note we study a counterpart in predicate logic of the notion of logical friendliness, introduced into propositional logic in [15]. The result is a new consequence relation for predicate languages with equality using first-order models. While compactness, interpolation and axiomatizability fail dramatically, several other properties are preserved from the propositional case. Divergence is diminished when the language does not contain equality with its standard interpretation.

Theory of Mind (ToM), the cognitive capacity to attribute internal mental states to oneself and others, is a crucial component of social skills. Its formal study has become important, witness recent research on reasoning and information update by intelligent agents, and some proposals for its formal modelling have put forward settings based on Epistemic Logic (EL). Still, due to intrinsic idealisations, it is questionable whether EL can be used to model the high-order cognition of ‘real’ agents. This manuscript proposes a (...) framework that takes visibility, memory, perspective and communication as the main factors underpinning mental attributions. This is more in-line with findings in cognitive science and allows us to model well-known False-Belief Tasks that are typically used in testing people's ToM. We discuss some of the framework's technical features, arguing why it does justice to empirical observations. (shrink)