In this paper we present a characterization of hyper-connexivity by means of a relating semantics for Boolean connexive logics. We also show that the minimal Boolean connexive logic is Abelardian, strongly consistent, Kapsner strong and antiparadox. We give an example showing that the minimal Boolean connexive logic is not simplificative. This shows that the minimal Boolean connexive logic is not totally connexive.

Semantic information is usually supposed to satisfy the veridicality thesis: p qualifies as semantic information only if p is true. However, what it means for semantic information to be true is often left implicit, with correspondentist interpretations representing the most popular, default option. The article develops an alternative approach, namely a correctness theory of truth (CTT) for semantic information. This is meant as a contribution not only to the philosophy of information but also to the philosophical debate on the nature (...) of truth. After the introduction, in Sect. 2, semantic information is shown to be translatable into propositional semantic information (i). In Sect. 3, i is polarised into a query (Q) and a result (R), qualified by a specific context, a level of abstraction and a purpose. This polarization is normalised in Sect. 4, where [Q + R] is transformed into a Boolean question and its relative yes/no answer [Q + A]. This completes the reduction of the truth of i to the correctness of A. In Sects. 5 and 6, it is argued that (1) A is the correct answer to Q if and only if (2) A correctly saturates Q by verifying and validating it (in the computer science’s sense of verification and validation ); that (2) is the case if and only if (3) [Q + A] generates an adequate model (m) of the relevant system (s) identified by Q; that (3) is the case if and only if (4) m is a proxy of s (in the computer science’s sense of proxy ) and (5) proximal access to m commutes with the distal access to s (in the category theory’s sense of commutation ); and that (5) is the case if and only if (6) reading/writing (accessing, in the computer science’s technical sense of the term) m enables one to read/write (access) s. Sect. 7 provides some further clarifications about CTT, in the light of semantic paradoxes. Section 8 draws a general conclusion about the nature of CTT as a theory for systems designers not just systems users. In the course of the article all technical expressions from computer science are explained. (shrink)

There are two dimensions to Fine’s truthmaker semantics. One involves a claim about the nature of propositions: propositions are not structural and nothing but sets of their possible truthmakers, and the other talks about the relation between truthmaking and Boolean operations. In this paper, I show that a claim by Fine in the latter dimension—that truthmaking is distributed over “or”—faces a counterexample. I will then go on to argue that one possible way to do away with the counterexample (...) is to restrict truthmakers to fundamentals, namely entities that are not grounded in anything else. This would, nevertheless, pose a problem for the first dimension of truthmaker semantics: certain distinct propositions would fail to be distinct. (shrink)

I propose a unified solution to two puzzles: Ross's puzzle and free choice permission. I begin with a pair of cases from the decision theory literature illustrating the phenomenon of act dependence, where what an agent ought to do depends on what she does. The notion of permissibility distilled from these cases forms the basis for my analysis of 'may' and 'ought'. This framework is then combined with a generalization of the classical semantics for disjunction — equivalent to (...) class='Hi'>Boolean disjunction on the diagonal, but with a different two-dimensional character — that explains the puzzling facts in terms of semantic consequence. (shrink)

Syntax and semantics of Propositional Logic are conventionally regarded as two distinct notions. We consider solely one notion, syntax, based on syntactic description of semantics of a 3CNF Boolean formula, generated via the Tseytin transformation. We use this novel approach to determine a sufficient condition for unsatisfiability.

Natural language contains simple lexical items for some but not all Boolean operators. English, for example, contains conjunction and, disjunction or, negated disjunction nor, but no word to express negated conjunction *nand nor any other Boolean connective. Natural language grammar can be described by a logic that expresses what the lexicon can express by its primitives, and the rest compositionally. Such logic for propositional connectives is described here as a bilateral extension of update semantics. The basic intuition (...) is that a context can be updated by assertion or by rejection, and by one or multiple propositions at once. These distinctions suffice to characterize the logic of the lexicon. (shrink)

We attempt here to trace the evolution of Frege’s thought about truth. What most frames the way we approach the problem is a recognition that hardly any of Frege’s most familiar claims about truth appear in his earliest work. We argue that Frege’s mature views about truth emerge from a fundamental re-thinking of the nature of logic instigated, in large part, by a sustained engagement with the work of George Boole and his followers, after the publication of Begriffsschrift and the (...) appearance of critical reviews by members of the Boolean school. (shrink)

Modifying the descriptive and theoretical generalizations of Relativized Minimality, we argue that a significant subset of weak island violations arise when an extracted phrase should scope over some intervener but is unable to. Harmless interveners seem harmless because they can support an alternative reading. This paper focuses on why certain wh-phrases are poor wide scope takers, and offers an algebraic perspective on scope interaction. Each scopal element SE is associated with certain operations (e.g., not with complements). When a wh-phrase scopes (...) over some SE, the operations associated with that SE are performed in its denotation domain. The requisite operations may or may not be available in a domain, however. We present an empirical analysis of a variety of wh-phrases. It is argued that the wh-phrases that escape all weak islands (i.e., can scope over any intervener) are those that range over individuals, the reason being that all Boolean operations are defined for their domain. Collectives, manners, amounts, numbers, etc. all denote in domains with fewer operations and are thus selectively sensitive to scopal interveners—a “semantic relativized minimality effect”. (shrink)

The article develops a correctness theory of truth (CTT) for semantic information. After the introduction, in section two, semantic information is shown to be translatable into propositional semantic information (i). In section three, i is polarised into a query (Q) and a result (R), qualified by a specific context, a level of abstraction and a purpose. This polarization is normalised in section four, where [Q + R] is transformed into a Boolean question and its relative yes/no answer [Q + (...) A]. This completes the reduction of the truth of i to the correctness of A. In sections five and six, it is argued that (1) A is the correct answer to Q if and only if (2) A correctly saturates (in a Fregean sense) Q by verifying and validating it (in the computer science’s sense of “verification” and “validation”); that (2) is the case if and only if (3) [Q + A] generates an adequate model (m) of the relevant system (s) identified by Q; that (3) is the case if and only if (4) m is a proxy of s (in the computer science’s sense of “proxy”) and (5) proximal access to m commutes with the distal access to s (in the category theory’s sense of “commutation”); and that (5) is the case if and only if (6) reading/writing (accessing, in the computer science’s technical sense of the term) m enables one to read/write (access) s. The last section draws a general conclusion about the nature of CTT as a theory for systems designers not just systems users. (shrink)

This book surveys research in quantification starting with the foundational work in the 1970s. It paints a vivid picture of generalized quantifiers and Boolean semantics. It explains how the discovery of diverse scope behavior in the 1990s transformed the view of quantification, and how the study of the internal composition of quantifiers has become central in recent years. It presents different approaches to the same problems, and links modern logic and formal semantics to advances in generative syntax. (...) A unique feature of the book is that it systematically brings cross-linguistic data to bear on the theoretical issues, discussing French, German, Dutch, Hungarian, Russian, Japanese, Telugu (Dravidian), and Shupamem (Grassfield Bantu), and pointing to formal semantic literature involving quantification in around thirty languages. -- -/- 1. What this book is about and how to use it; 2. Generalized quantifiers and their elements: operators and their scopes; 3. Generalized quantifiers in non-nominal domains; 4. Some empirically significant properties of quantifiers and determiners; 5. Potential challenges for generalized quantifiers; 6. Scope is not uniform and not a primitive; 7. Existential scope versus distributive scope; 8. Distributivity and scope; 9. Bare numeral indefinites; 10. Modified numerals; 11. Clause-internal scopal diversity; 12. Towards a compositional semantics of quantifier words. (shrink)

Formal semantic analyses often take words to be minimal building blocks for the purposes of compositionality. But various recent theories of morphology and syntax have converged on the view that there is no demarcation line corresponding to the word level. The same conclusion has emerged from the compositional semantics of superlatives. In the spirit of extending compositionality below the word level, this paper explores how a small set of particles (Japanese KA and MO, Chinese DOU, and Hungarian VALA/VAGY, MIND, (...) and IS) form quantifier words and serve as connectives, additive and scalar particles, question markers, and existential verbs. The main question is whether the meanings of these particles across the varied environments are highly regular, or they are lexicalized with a variety of different meanings that bear a family resemblance. This paper does not reach definitive conclusions, but it raises analytical possibilities using Boolean semantics and the semantics of alternatives. It also draws attention to systematic similarities and some differences between MO and DOU that have not been studied in the literature. (shrink)

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms (...) of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)

Bradley offers a quick and convincing argument that no Boolean semantic theory for conditionals can validate a very natural principle concerning the relationship between credences and conditionals. We argue that Bradley’s principle, Preservation, is, in fact, invalid; its appeal arises from the validity of a nearby, but distinct, principle, which we call Local Preservation, and which Boolean semantic theories can non-trivially validate.

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, (...) a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics. (shrink)

Aristotle’s philosophy is considered with respect to one central concept of his philosophy, viz. opposition. Far from being a mere side-effect of syllogistic, it is argued in the present paper that opposition helps to articulate ontology and logic through an account of what can be or cannot be in a systematic and structural way. The paper is divided into three main parts. In Section I, the notion of Being is scrutinized through Aristotle’s theory of categories. In Section II, the notion (...) of Non-Being is connected to Aristotle’s theory of oppositions. In Section III, the notion of essence is revisited in order to bring about a holist theory of meaning by individuating through opposite properties. In conclusion, the legacy of Aristotle is depicted as balanced between a powerful reflection around Being and a restrictive ontology of substance. (shrink)

This paper develops a theory of propositional identity which distinguishes necessarily equivalent propositions that differ in subject-matter. Rather than forming a Boolean lattice as in extensional and intensional semantic theories, the space of propositions forms a non-interlaced bilattice. After motivating a departure from tradition by way of a number of plausible principles for subject-matter, I will provide a Finean state semantics for a novel theory of propositions, presenting arguments against the convexity and nonvacuity constraints which Fine (2016, 2017a,b) (...) introduces. I will then move to compare the resulting logic of propositional identity (PI) with Correia’s (2016) logic of generalised identity (GI), as well as the first degree fragment of Angell’s (1989) logic of analytic containment (AC). The paper concludes by extending PI to include axioms and rules for a subject-matter operator, providing a much broader theory of subject-matter than the principles with which I will begin. (shrink)

Hyperboolean algebras are Boolean algebras with operators, constructed as algebras of complexes (or, power structures) of Boolean algebras. They provide an algebraic semantics for a modal logic (called here a {\em hyperboolean modal logic}) with a Kripke semantics accordingly based on frames in which the worlds are elements of Boolean algebras and the relations correspond to the Boolean operations. We introduce the hyperboolean modal logic, give a complete axiomatization of it, and show that it (...) lacks the finite model property. The method of axiomatization hinges upon the fact that a "difference" operator is definable in hyperboolean algebras, and makes use of additional non-Hilbert-style rules. Finally, we discuss a number of open questions and directions for further research. (shrink)

Simple taste predications come with an acquaintance requirement: they require the speaker to have had a certain kind of first-hand experience with the object of predication. For example, if I tell you that the creme caramel is delicious, you would ordinarily assume that I have actually tasted the creme caramel and am not simply relying on the testimony of others. The present essay argues in favor of a 'lightweight' expressivist account of the acquaintance requirement. This account consists of a recursive (...) semantics and an account of assertion; it is compatible with a number of different accounts of truth and content, including contextualism, relativism, and purer forms of expressivism. The principal argument in favor of this account is that it correctly predicts a wide range of data concerning how the acquaintance requirement interacts with Boolean connectives, generalized quantifiers, epistemic modals, and attitude verbs. (shrink)

Analytical philosophy is ruled by the alliance of logic, linguistics and mathematics since its beginnings in the syllogistic calculus of terms and premises in Aristotle's Analytica protera, in the theories of medieval logic that dealt with what are Proprietatis Terminorum (significatio, suppositio, appellatio), in the theological apologetics of argumentation with the combinatorics of symbols by Raymundus Llullus in the work Ars Magna, Generalis et Ultima (1305-08), in what is presented as Theologia Combinata (cf. Tomus II.p.251) in Ars Magna Sciendi sive (...) Combinatoria by Athanasius Kircher (1669), in analytical combinatorics based in Leibniz's dissertation Ars Combinatoria (1666), where language is understood as a lingua characteristica and as a calculus ratiocinator, in the combinatorics of classes that have the meanings 1 or 0 given in the idea of Boolean functions (1854), in Frege's work Begriffschrift, eine der arithmeticeschen nachgebildete Fomelsprache des reinen Denkens (Concept letter, the language of formulas of pure thought made according to the model in arithmetical language), in Cantor's study of set theory / Grundlagen einer allgemeinen Mannigfaltigkeitstheorie (1883) and up to algorithms of fuzzy logic as a "means of approximate thinking that is in the human prior" (Zadeh, 1990) and fuzzy linguistics in computer semantics (Burghard B. Rieger, 1993). These are all the philosophical ideas of mathematicians and logicians that founded what we call logical pragmatism here as a demand that logic be absolutely used in every way in which it will dominate the grammar of language until logic itself becomes the philosophical grammar of the mind!!! (shrink)

This paper decomposes the Liar Paradox into its semantic atoms using Meaning Postulates (1952) provided by Rudolf Carnap. Formalizing truth values of propositions as Boolean properties of these propositions is a key new insight. This new insight divides the translation of a declarative sentence into its equivalent mathematical proposition into three separate steps. When each of these steps are separately examined the logical error of the Liar Paradox is unequivocally shown.

By elaborating on the results presented in Lógica cuántica, Nmatrices y adecuación I, here we discuss the notions of adequacy and truth functionality in quantum logic from the point of view of a non-deterministic semantics based on Nmatrices. We present a proof of the impossibility of providing a functional semantics for the quantum lattice. An advantage of our proof is that it is independent of the number of truth values involved, generalizing previous works. Due to the impossibility of (...) defining adequate interpreting sets for the conjunction and disjunction (as was shown in our previous work), it follows that a homomorphism between the lattice of quantum projections and a Boolean algebra of n-elements cannot exist. (shrink)

We begin with the hypothetical assumption that Tarski’s 1933 formula ∀ True(x) φ(x) has been defined such that ∀x Tarski:True(x) ↔ Boolean-True. On the basis of this logical premise we formalize the Truth Teller Paradox: "This sentence is true." showing syntactically how self-reference paradox is semantically ungrounded.

Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of (...) growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed. (shrink)

The aim of this paper is to make sense of the Keynes–Johnson octagon of oppositions. We will discuss Keynes' logical theory, and examine how his view is reflected on this octagon. Then we will show how this structure is to be handled by means of a semantics of partition, thus computing logical relations between matching formulas with a semantic method that combines model theory and Boolean algebra.

Abstract. Gettier’s famous examples intended to show that knowledge cannot always be equated with justified true belief. The Gettier problem can also be considered as a problem for topological epistemic logic: If knowledge and justified belief are conceived as topological operators K and B on topological spaces (to be considered as universes of possible worlds), one may ask whether it happens that there is a proposition A such that KA ≠ A & BA or not. If this is the case, (...) the epistemological logic defined by the topological operators K and B may be said to be non-traditional since there is for them a “Gettier proposition” where knowledge does not coincide with justified true belief. As far as we know, for the first time, this topological Gettier problem is discussed in Steinvold’s PhD dissertation (Steinsvold 2012). In Baltag, Bezhanishvili, Özgün, and Smets (2013), Steinvold’s co-derived set account is criticized since it can be easily “Gettierized”. Baltag et alii claim (without proof) that their topological account of Stalnaker’s logic KB of knowledge and belief does not have this flaw. In the following we will give some topological conditions that determine whether a topological KB model does or does not cope with the Gettier challenge. First, it will be shown that every KB model defines a (topologically slightly simplified) model that is rather similar to it but a is a model of traditional JTB (knowledge = justified true belief) logic of knowledge and belief. The existence of such doppelgangers of all KB models may be read as a qualified and partial topological rehabilitation of traditional JTB epistemology. Second, we consider the Gettier problem for a special class of models of Stalnaker’s combined logic of knowledge and belief KB, namely, for T0 Alexandroff spaces. We prove that, that almost all Alexandroff spaces do not face Gettier counterexamples. Third, we prove that the Stone spaces of Boolean algebras of regular closed subsets of Hausdorff (or T2) spaces X do not face Gettier counterexamples if the X are not pseu–docompact, and these Stone spaces do face Gettier counterexamples for compact spaces X. Succinctly formulated, Stalnaker’s KB logic of knowledge and belief can avoid Gettier problems for some models, while it falls back to traditional pre-Gettier JTB epistemology for other models. In contrast, co-derived set semantics is doomed to fail always in the sense that its models always fall prey to Gettier counterexamples. (shrink)

After some generalities about connections between functions and relations in Sections 1 and 2 recalls the possibility of taking the semantic values of ‐ary Boolean connectives as ‐ary relations among truth‐values rather than as ‐ary truth functions. Section 3, the bulk of the paper, looks at correlates of these truth‐value relations as applied to formulas, and explores in a preliminary way how their properties are related to the properties of “logical relations” among formulas such as equivalence, implication (entailment) and (...) contrariety (logical incompatibility), concentrating for illustrative purposes on binary logical relations such as those just listed. To avoid an excess of footnotes, some points have been deferred to an Appendix as “Longer Notes”. (shrink)

The logico-algebraic study of Lewis's hierarchy of variably strict conditional logics has been essentially unexplored, hindering our understanding of their mathematical foundations, and the connections with other logical systems. This work aims to fill this gap by providing a comprehensive logico-algebraic analysis of Lewis's logics. We begin by introducing novel finite axiomatizations for varying strengths of Lewis's logics, distinguishing between global and local consequence relations on Lewisian sphere models. We then demonstrate that the global consequence relation is strongly algebraizable in (...) terms of a specific class of Boolean algebras with a binary operator representing the counterfactual implication; in contrast, we show that the local consequence relation is generally not algebraizable, although it can be characterized as the degree-preserving logic over the same algebraic models. Further, we delve into the algebraic semantics of Lewis's logics, developing two dual equivalences with respect to particular topological spaces. In more details, we show a duality with respect to the topological version of Lewis's sphere models, and also with respect to Stone spaces with a selection function; using the latter, we demonstrate the strong completeness of Lewis's logics with respect to sphere models. Finally, we draw some considerations concerning the limit assumption over sphere models. (shrink)

The standard rule of single privative modification replaces privative modifiers by Boolean negation. This rule is valid, for sure, but also simplistic. If an individual a instantiates the privatively modified property (MF) then it is true that a instantiates the property of not being an F, but the rule fails to express the fact that the properties (MF) and F have something in common. We replace Boolean negation by property negation, enabling us to operate on contrary rather than (...) contradictory properties. To this end, we apply our theory of intensional essentialism, which operates on properties (intensions) rather than their extensions. We argue that each property F is necessarily associated with an essence, which is the set of the so-called requisites of F that jointly define F. Privation deprives F of some but not all of its requisites, replacing them by their contradictories. We show that properties formed from iterated privatives, such as being an imaginary fake banknote, give rise to a trifurcation of cases between returning to the original root property or to a property contrary to it or being semantically undecidable for want of further information. In order to determine which of the three forks the bearers of particular instances of multiply modified properties land upon we must examine the requisites, both of unmodified and modified properties. Requisites underpin our presuppositional theory of positive predication. Whereas privation is about being deprived of certain properties, the assignment of requisites to properties makes positive predication possible, which is the predication of properties the bearers must have because they have a certain property formed by means of privation. (shrink)

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for negations. The approach unifies in a philosophically motivated picture the following (...) results: nothing can be called a negation properly if it does not satisfy Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds. (shrink)

The de Morgan laws characterize how negation, conjunction, and disjunction interact with each other. They are fundamental in any semantics that bases itself on the propositional calculus/Boolean algebra. This paper is primarily concerned with the second law. In English, its validity is easy to demonstrate using linguistic examples. Consider the following: (3) Why is it so cold in here? We didn’t close the door or the window. The second sentence is ambiguous. It may mean that I suppose we (...) did not close the door or did not close the window, but I am not sure which. This `I am not sure which’ reading is irrelevant to us because it has disjunction scoping over negation. But the sentence may equally well mean (and indeed this is the preferred reading) that we didn’t close the door and did not close the window. This `neither’ reading bears out de Morgan law (2). (shrink)

This paper introduces the category of b-frames as a new tool in the study of complete lattices. B-frames can be seen as a generalization of posets, which play an important role in the representation theory of Heyting algebras, but also in the study of complete Boolean algebras in forcing. This paper combines ideas from the two traditions in order to generalize some techniques and results to the wider context of complete lattices. In particular, we lift a representation theorem of (...) Allwein and MacCaull to a duality between complete lattices and b-frames, and we derive alternative characterizations of several classes of complete lattices from this duality. This framework is then used to obtain new results in the theory of complete Heyting algebras and the semantics of intuitionistic propositional logic. (shrink)

The present thesis focuses on counterfactuals. Specifically, we will address new questions and open problems that arise for the standard semantic accounts of counterfactual conditionals. The first four chapters deal with the Lewisian semantic account of counterfactuals. On a technical level, we contribute by providing an equivalent algebraic semantics for Lewis' variably strict conditional logics, which is notably absent in the literature. We introduce a new kind of algebra and differentiate between local and global versions of each of Lewis' (...) variably strict conditional logics. We study the algebraic properties of Lewis' logics and the structure theory of our newly introduced algebras. Additionally, we employ a new algebraic construction, based on the framework of Boolean algebras of conditionals, to provide an alternative semantics for Lewisian counterfactual conditionals. This semantic account allows us to establish new truth conditions for Lewisian counterfactuals, implying that Lewisian counterfactuals are definable conditionals, and each counterfactual can be characterized as a modality of a corresponding probabilistic conditional. We further extend these results by demonstrating that each Lewisian counterfactual can also be characterized as a modality of the corresponding Stalnaker conditional. The resulting formal semantic framework is much more expressive than the standard one and, in addition to providing new truth conditions for counterfactuals, it also allows us to define a new class of conditional logics falling into the broader framework of weak logics. On the philosophical side, we argue that our results shed new light on the understanding of Lewisian counterfactuals and prompt a conceptual shift in this field: Lewisian counterfactual dependence can be understood as a modality of probabilistic conditional dependence or Stalnakerian conditional dependence. In other words, whether a counterfactual connection occurs between A and B depends on whether it is "necessary" for a Stalnakerian/probabilistic dependence to occur between A and B. We also propose some ways to interpret the kind of necessity involved in this interpretation. The remaining two chapters deal with the probability of counterfactuals. We provide an answer to the question of how we can characterize the probability that a Lewisian counterfactual is true, which is an open problem in the literature. We show that the probability of a Lewisian counterfactual can be characterized in terms of belief functions from Dempster-Shafer theory of evidence, which are a super-additive generalization of standard probability. We define an updating procedure for belief functions based on the imaging procedure and show that the probability of a counterfactual A > B amounts to the belief function of B imaged on A. This characterization strongly relies on the logical results we proved in the previous chapters. Moreover, we also solve an open problem concerning the procedure to assign a probability to complex counterfactuals in the framework of causal modelling semantics. A limitation of causal modelling semantics is that it cannot account for the probability of counterfactuals with disjunctive antecedents. Drawing on the same previous works, we define a new procedure to assign a probability to counterfactuals with disjunctive antecedents in the framework of causal modelling semantics. We also argue that our procedure is satisfactory in that it yields meaningful results and adheres to some conceptually intuitive constraints one may want to impose when computing the probability of counterfactuals. (shrink)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. These (...) practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (shrink)

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic property of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide valid the deductive inference. Conclusions of sound arguments are derived from truth preserving finite string transformations applied to true premises.

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic value of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide the valid deductive inference. Sound deductive conclusions are the result of these finite string transformation rules.

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the (...) class='Hi'>Boolean logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)--just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition. (shrink)

Configurational Comparative Methods (CCMs) aim to learn causal structures from datasets by exploiting Boolean sufficiency and necessity relationships. One important challenge for these methods is that such Boolean relationships are often not satisfied in real-life datasets, as these datasets usually contain noise. Hence, CCMs infer models that only approximately fit the data, introducing a risk of inferring incorrect or incomplete models, especially when data are also fragmented (have limited empirical diversity). To minimize this risk, evaluation measures for sufficiency (...) and necessity should be sensitive to all relevant evidence. This article points out that the standard evaluation measures in CCMs, consistency and coverage, neglect certain evidence for these Boolean relationships. Correspondingly, two new measures, contrapositive consistency and contrapositive coverage, which are equivalent to the binary classification measures specificity and negative predictive value, respectively, are introduced to the CCM context as additions to consistency and coverage. A simulation experiment demonstrates that the introduced contrapositive measures indeed help to identify correct CCM models. (shrink)

Deontic logic is devoted to the study of logical properties of normative predicates such as permission, obligation and prohibition. Since it is usual to apply these predicates to actions, many deontic logicians have proposed formalisms where actions and action combinators are present. Some standard action combinators are action conjunction, choice between actions and not doing a given action. These combinators resemble boolean operators, and therefore the theory of boolean algebra offers a well-known athematical framework to study the properties (...) of the classic deontic operators when applied to actions. In his seminal work, Segerberg uses constructions coming from boolean algebras to formalize the usual deontic notions. Segerberg’s work provided the initial step to understand logical properties of deontic operators when they are applied to actions. In the last years, other authors have proposed related logics. In this chapter we introduce Segerberg’s work, study related formalisms and investigate further challenges in this area. (shrink)

Externalism is widely endorsed within contemporary philosophy of mind and language. Despite this, it is far from clear how the externalist thesis should be construed and, indeed, why we should accept it. In this entry I distinguish and examine three central types of externalism: what I call foundational externalism, externalist semantics, and psychological externalism. I suggest that the most plausible version of externalism is not in fact a very radical thesis and does not have any terribly interesting implications for (...) philosophy of mind, whereas the more radical and interesting versions of externalism are quite difficult to support. (shrink)

I argue that semantics is the study of the proprietary database of a centrally inaccessible and informationally encapsulated input–output system. This system’s role is to encode and decode partial and defeasible evidence of what speakers are saying. Since information about nonlinguistic context is therefore outside the purview of semantic processing, a sentence’s semantic value is not its content but a partial and defeasible constraint on what it can be used to say. I show how to translate this thesis into (...) a detailed compositional-semantic theory based on the influential framework of Heim and Kratzer. This approach situates semantics within an independently motivated account of human cognitive architecture and reveals the semantics–pragmatics interface to be grounded in the underlying interface between modular and central systems. (shrink)

This chapter develops a semantic theory for maps and situates it within the broader geography of meaning and semiotic significance. The discussion focuses on three central aspects of map semantics: the use of space, line marking, and linguistic tags. It is argued that the treatment of space in maps must be based on geometrical projection from a viewpoint rather than the traditional analysis in terms of spatial isomorphism. The chapter then shows how to integrate the projection-based semantics of (...) maps and the symbolic semantics of markers and tags, within the framework of possible-world semantics. The resulting account addresses issues of accuracy, viewpoint, orientation, and exhaustivity. (shrink)

Truthmaker semantics is a recent development in formal and philosophical semantics, with similar motivation and scope to possible worlds semantics. The technical background is rather different, however, and results in a more fine-grained hyperintensional notion of content, allowing us to distinguish between classically equivalent propositions. After briefly introducing the main ideas, this entry will describe the technical apparatus of state spaces and the central notions of content and partial content. It will then outline applications of truthmaker (...) class='Hi'>semantics in language, logic, and other areas of philosophy. (shrink)

nated discussion of the semantics of such verbs. I will call this view.

Two semantic paradoxes, the Liar and Curry’s paradox, are analysed using a newly developed conception of procedural semantics (semantics according to which the truth of propositions is determined algorithmically), whose main characteristic is its departure from methodological realism. Rather than determining pre-existing facts, procedures are constitutive of them. Of this semantics, two versions are considered: closed (where the halting of procedures is presumed) and open (without this presumption). To this end, a procedural approach to deductive reasoning is (...) developed, based on the idea of simulation. As is shown, closed semantics supports classical logic, but cannot in any straightforward way accommodate the concept of truth. In open semantics, where paradoxical propositions naturally ‘belong’, they cease to be paradoxical; yet, it is concluded that the natural choice—for logicians and common people alike—is to stick to closed semantics, pragmatically circumventing problematic utterances. (shrink)

I develop and defend a truthmaker semantics for the relevant logic R. The approach begins with a simple philosophical idea and develops it in various directions, so as to build a technically adequate relevant semantics. The central philosophical idea is that truths are true in virtue of specific states. Developing the idea formally results in a semantics on which truthmakers are relevant to what they make true. A very natural notion of conditionality is added, giving us relevant (...) implication. I then investigate ways to add conjunction, disjunction, and negation; and I discuss how to justify contraposition and excluded middle within a truthmaker semantics. (shrink)

In this paper I will be concerned with the question of the extent to which semantics can be thought of as a purely formal exercise, which we can engage in in a way that is neutral with respect to how our formal system is to be interpreted. I will be arguing, to the contrary, that the features of the formal systems which we use to do semantics are closely linked, in several different ways, to the interpretation that we (...) give to those formal systems. The occasion for this question, and the main example that I will use to illustrate my answer to it, is the close relationship between the formal systems employed in recent statements of apparently competing accounts of epistemic modals with the dynamic, expressivist, and relativist theoretical paradigms. The structure of the paper will be straightforward. In part 1, I will briefly introduce four theories of epistemic modals – one dynamic theory, two expressivist theories, and one relativist theory. Then in part 2 I’ll show that one expressivist theory is formally equivalent to the dynamic theory, that the other is formally equivalent to the relativist theory, and that the two expressivist theories are themselves essentially notational variants. I’ll use these facts to pose our central question: if these theories have so much formally in common, then doesn’t that suggest that we can separate the task of constructing a formal semantics from the task of deciding between competing interpretations of it? Finally, in part 3 I’ll answer that question in the negative. There are at least three reasons why formal semantics cannot be separated from questions of interpretation that are illustrated by the theories I introduce in part 1. (shrink)

Quine's views on semantic holism and observation statements appear to be incompatible. My paper is an attempt to alleviate this tension.

My paper examines the popular idea, defended by Kripke, that meaning is an essentially normative notion. I consider four common versions of this idea and suggest that none of them can be supported, either because the alleged normativity has nothing to do with normativity or because it cannot plausibly be said that meaning is normative in the sense suggested. I argue that contrary to received opinion, we don’t need normativity to secure the possibility of meaning. I conclude by considering the (...) repercussions of rejecting semantic normativity on three central issues: justification, communication, and naturalism. (shrink)

In the chapter “Information and Content” of their Impossible Worlds, Berto and Jago provide us with a semantic account of information in deductive reasoning such that we have an explanation for why some, but not all, logical deductions are informative. The framework Berto and Jago choose to make sense of the above-mentioned idea is a semantic interpretation of Sequent Calculus rules of inference for classical logic. I shall argue that although Berto and Jago’s idea and framework are hopeful, their definitions (...) do not capture what is intended. This is so because the definitions are solely based on the logical complexity of an argument and they fail to capture the richness of the non- logical content of that argument. Then I will suggest some amendments to address this problem. Finally, I will extend the application of the definitions to first-order logic. It shall be observed that in some classical deductions, applying contraction may lead to hiding some epistemic impossibilities. This happens when formulae which contribute different propositions to the proof get contracted at the quantified level. (shrink)