“If the tower is any taller than 320 ms, it may collapse,” Eiffel thinks out loud. Although understanding this counterfactual poses no trouble, the most successful interventionist semantics struggle to model it because the antecedent can come about in infinitely many ways. My aim is to provide a semantics that will make modeling such counterfactuals easy for philosophers, computer scientists, and cognitive scientists who work on causation and causal reasoning. I first propose three desiderata that will guide my (...) theory: it should be general, yet conservative, yet useful. Next, I develop a formalization of events in the form of an algebra. I identify an event with all the ways in which it can be brought about and provide rules for determining the referent of an arbitrary event description. I apply this algebra to counterfactuals expressed using underdeterministic causal models-models that encode non-probabilistic causal indeterminacies. Specifically, I develop semaphore interventions, which represent how the target system may be modified from without in a coordinated fashion. This, in turn, allows me to bring about any event within a single model. Finally, I explain the advantages of this semantics over other interventionist competitors. (shrink)

Since the introduction of the computer in the early 1950's, the investigation of artificial intelligence has followed three chief avenues: the discovery of self-organizing systems; the building of working models of human behavior, incorporating specific psychological theories; and the building of "heuristic" machines, without bias in favor of humanoid characteristics. While this work has used philosophical logic and its results may illustrate philosophical problems, the artificial intelligence program is by now an intricate, organized specialty. This book, therefore, has a quite (...) specialized audience of its own although it can be very valuable to those philosophers who are interested and competent in using this pioneering material. Five scientific papers report attempts to solve five different kinds of problems. Bertram Raphael describes an attempt to build a memory structure that converts the information input into a systematic model by "understanding" the informational statements as they are made. Daniel Bobrow's machine can set up algebraic equations from informal verbal statements. M. Ross Quillan asks: "What sort of representational format can permit the 'meanings' of words to be stored?" Thomas Evans' machine, Analogy, serves as a model for "pattern-recognition" rather than the "common-property" method of semantic memory. Fischer Black has developed a logical deduction mechanism for question-answering which keeps track of where we are and avoids endless deduction. The editor and John McCarthy contribute more general chapters, providing the historical background of cybernetics, and dealing with the problem of formalizing a concept of causality. Minsky ends the volume with his view that our convictions on dualism, consciousness, free will, and the like are used in the attempt to explain the complicated interactions between parts of our model of ourselves.--M. B. M. (shrink)

Metaphysical semantics is supposed to connect the nonfundamental to the fundamental in a distinctively “linguistic” way, explaining how nonfundamental truths can be grounded in fundamental facts , and so inducing a radically eliminative vision of the nonfundamental as mere talk. I wonder how the story goes when a single nonfundamental truth can be grounded in many different fundamental facts. For instance, the truth that Moore has hands can presumably be grounded in many different distributions of fields, arrangements of (...) particles, vibrations of strings, or whatnot. I am asking how metaphysical semantics can accommodate this sort of multiple realizability. (shrink)

The Interpolation Theorem, first formulated and proved by W. Craig fifty years ago for predicate logic, has been extended to many other logical frameworks and is being applied in several areas of computer science. We give a short overview, and focus on the theory of software systems and modules. An algebra of theories TA is presented, with a nonstandard interpretation of the existential quantifier . In TA, the interpolation property of the underlying logic corresponds with the quantifier combination property . (...) It is shown how the Modularization Theorem, the Factorization Lemma and the Normal Form Theorem for module expressions can be proved in TA. (shrink)

One desideratum for a theory of fundamentality is to give us the conceptual tools to articulate fruitful metaphysical distinctions between the assortment of ‘realist’ and ‘anti-realist’ positions in a given domain such as meta-ethics. The ability to articulate such distinctions gives us a way to assess rival theories of fundamentality, such as Fine’s grounding theory and Sider’s metaphysical semantic theory. Indeed, Sider has argued that metaphysical semantic theories have an edge with respect to this desideratum and takes (...) this as an important reason to prefer those theories over rival grounding theories. This paper takes a closer look at how the rival theories compare with respect to questions of realism and defends the grounding theorist’s ability to meet the desideratum. (shrink)

Rationalism offers an account of moral properties as a subset of the properties which serve to rationalize right actions, and these properties are fit to be the referents of our moral terms. That fitness can be exploited in constructing an externalist theory of reference determination for these terms. The resulting externalist theory draws support from standard responses to Moral Twin-Earth scenarios. The relevance of these responses to moral semantics has recently beenvigorously challenged by Dowell and by Schroeter and Schroeter. (...) The social character of meaning relations, which can explain the openness of questions about an analysis, may thereby also make Twin-Earth judgements beside the point. But the resources available to translators go beyond semantic competence and it is these resources that nonetheless make the Moral Twin-Earth responses relevant. (shrink)

In this paper I argue that, to conceive transcendence, Levinas retrieves the Platonic concept of “separation” and deploys it in three ways: metaphysically, semantically, and affectively. Levinas finds in the interaction between being and the Good beyond being of Republic VI 509b a certain “formal structure of transcendence”—one in which a term is conditioned by another while remaining absolutely separated from it. This formal structure is subsequently deployed metaphysically, in the relation between creator and creature; semantically, in the relation between (...) meaning and sense; and affectively, in the relation between the desiring self and its desired aim. (shrink)

This monograph offers a comprehensive study of contingent a priori truths. Building onto a theoretical framework developed by the philosopher and logician Saul Kripke, the author also presents a new approach to these truths. The first part of the book details the many theories on contingent a priori truths. The coverage examines the cases of Kripke and David Kaplan, Donnellan and the de re requirement, Evans and weak contingency, as well as Plantinga, Salmon, Soames, and the pseudo a priori. Overall, (...) it provides a systematic discussion and critical review of all these many positions. Next, the author develops an alternative approach. His working hypothesis is that performative verbs must play a central role in Kripke's examples, even if they do not show up at the surface structure of the corresponding sentences. This opens up an entirely new way of looking at Kripke's cases and of treating them by exploring some aspects of the theory of illocutionary acts. His discussion also examines brute facts and institutional facts, indexicals and performatives, as well as Frege's theory of definitions. Providing an authoritative exploration into contingent a priori truths, this book will be of interest to students, academics, and researchers in philosophy and logic. (shrink)

In classical logic, the proposition expressed by a sentence is construed as a set of possible worlds, capturing the informative content of the sentence. However, sentences in natural language are not only used to provide information, but also to request information. Thus, natural language semantics requires a logical framework whose notion of meaning does not only embody informative content, but also inquisitive content. This paper develops the algebraic foundations for such a framework. We argue that propositions, in order (...) to embody both informative and inquisitive content in a satisfactory way, should be defined as non-empty, downward closed sets of possibilities, where each possibility in turn is a set of possible worlds. We define a natural entailment order over such propositions, capturing when one proposition is at least as informative and inquisitive as another, and we show that this entailment order gives rise to a complete Heyting algebra, with meet, join, and relative pseudo-complement operators. Just as in classical logic, these semantic operators are then associated with the logical constants in a first-order language. We explore the logical properties of the resulting system and discuss its significance for natural language semantics. We show that the system essentially coincides with the simplest and most well-understood existing implementation of inquisitive semantics, and that its treatment of disjunction and existentials also concurs with recent work in alternative semantics. Thus, our algebraic considerations do not lead to a wholly new treatment of the logical constants, but rather provide more solid foundations for some of the existing proposals. (shrink)

An analysis of the structural properties of collections or pluralities, homogeneous objects like water, and the semantics and philosophy of events.

This paper deals with one kind of Kripke-style semantics, which we shall call algebraic Kripke-style semantics, for relevance logics. We first recall the logic R of relevant implication and some closely related systems, their corresponding algebraic structures, and algebraic completeness results. We provide simpler algebraic completeness proofs. We then introduce various types of algebraic Kripke-style semantics for these systems and connect them with algebraic semantics.

We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive (...) algebras determine Medvedev’s logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces). In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open elements of this space. This connection yields a new topological semantics for inquisitive logic. (shrink)

We show that the variety of equivalential algebras with regularization gives the algebraic semantics for the -fragment of intuitionistic propositional logic. We also prove that this fragment is hereditarily structurally complete.

This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined (...) over the real unit interval, the rational unit interval, the hyperreals , the strict hyperreals and finite chains, respectively–and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved. (shrink)

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke (...) class='Hi'>semantics. We introduce algebraic Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Gödel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic. (shrink)

Essentialism--roughly, the view that natural kinds have discrete essences, generating truths that are necessary but knowable only _a posteriori_--is an increasingly popular view in the metaphysics of science. At the same time, philosophers of language have been subjecting Kripke’s views about the existence and scope of the necessary _a posteriori_ to rigorous analysis and criticism. Essentialists typically appeal to Kripkean semantics to motivate their radical extension of the realm of the necessary _a posteriori_; but they rarely attempt to provide (...) any semantic arguments for this extension, or engage with the critical work being done by philosophers of language. This collection brings authors on both sides together in one volume, thus helping the reader to see the connections between views in philosophy of language on the one hand and the metaphysics of science on the other. The result is a book that will have a significant impact on the debate about essentialism, encouraging essentialists to engage with debates about the semantic presuppositions that underpin their position, and, encouraging philosophers of language to engage with the metaphysical presuppositions enshrined in Kripkean semantics. (shrink)

In the present paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on Public Announcement Logic without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given current model being replaced with its submodel relativized to the (...) announced proposition. We dually characterize the associated submodel-injection map as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. As is well known, these complex algebras are complete atomic BAOs . The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras . Thanks to this construction, the benefits and the wider scope of applications given by a point-free, intuitionistic theory of epistemic updates are made available. As an application of this dual characterization, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, prove soundness and completeness of IPAL w.r.t. both algebraic and relational models, and show that the well known Muddy Children Puzzle can be formalized in IPAL. (shrink)

We model Correia's hybrid algebraic/structural semantics within a form of the truthmaker semantics in which the propositions of the semantics are taken to to be closed under a well-behaved closure operation.

An algebraic characterisation is given of the Mares-Goldblatt semantics for quantified extensions of relevant and modal logics. Some features of this more general semantic framework are investigated, and the relations to some recent work in algebraic semantics for quantified extensions of non-classical logics are considered.

The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different (...) algebraic semantics. All of these give rise to an algebraic completeness theorem for the deductive system, but their algebraic properties, unlike those of equivalent algebraic semantics, need not reflect the metalogical properties of the deductive system. Many deductive systems that don't have an equivalent algebraic semantics do possess an algebraic semantics; examples of these phenomena are provided. It is shown that all extensions of a deductive system that possesses an algebraic semantics themselves possess an algebraic semantics. Necessary conditions for the existence of an algebraic semantics are given, and an example of a protoalgebraic deductive system that does not have an algebraic semantics is provided. The mono-unary deductive systems possessing an algebraic semantics are characterized. Finally, weak conditions on a deductive system are formulated that guarantee the existence of an algebraic semantics. These conditions are used to show that various classes of non-algebraizable deductive systems of modal logic, relevance logic and linear logic do possess an algebraic semantics. (shrink)

The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different (...) algebraic semantics. All of these give rise to an algebraic completeness theorem for the deductive system, but their algebraic properties, unlike those of equivalent algebraic semantics, need not reflect the metalogical properties of the deductive system. Many deductive systems that don't have an equivalent algebraic semantics do possess an algebraic semantics; examples of these phenomena are provided. It is shown that all extensions of a deductive system that possesses an algebraic semantics themselves possess an algebraic semantics. Necessary conditions for the existence of an algebraic semantics are given, and an example of a protoalgebraic deductive system that does not have an algebraic semantics is provided. The mono-unary deductive systems possessing an algebraic semantics are characterized. Finally, weak conditions on a deductive system are formulated that guarantee the existence of an algebraic semantics. These conditions are used to show that various classes of non-algebraizable deductive systems of modal logic, relevance logic and linear logic do possess an algebraic semantics. (shrink)

Research in dynamic semantics has made strides by studying various aspects of discourse in terms of computational effect systems, for example, monads (Shan, 2002; Charlow, 2014), Barker and 2014), (Maršik, 2016). We provide a system, based on graded monads, that synthesizes insights from these programs by formalizing individual discourse phenomena in terms of separate effects, or grades. Included are effects for introducing and retrieving discourse referents, non-determinism for indefiniteness, and generalized quantifier meanings. We formalize the behavior of individual effects, (...) as well as the interactions between effects, in terms of algebraic laws tailored to the relevant discourse phenomena. The system we propose is thus modular and suggests a novel approach to integrating formal accounts of distinct semantic phenomena. Finally, we give an interpretation of the system into pure \( \lambda \) -calculus that respects the laws. Future work will aim to integrate more discourse phenomena using the same methodology, for example, presupposition and conventional implicature. (shrink)

Alethic pluralism holds that there are many ways of being true. Such a view has been challenged to make sense of the standard account of logical validity as necessary truth preservation. In this paper, the recent solution elaborated by Aaron Cotnoir, based on an algebraic approach, is shown to be untenable. Some reflections about the relation of many-valued logics with truth pluralism are also discussed.

What makes our utterances mean what they do? In this work I formulate and justify a structural constraint on possible answers to this key question in the philosophy of language, and I show that accepting this constraint leads naturally to the adoption of an algebraic formalization of truth-theoretic semantics. I develop such a formalization, and show that applying algebraic methodology to the theory of meaning yields important insights into the nature of language. ;The constraint I propose is, (...) roughly, this: the meaning of an utterance of a natural language sentence is derived from the place of that sentence within a system that includes other sentences, and is structured in virtue of primitive semantic facts about complete sentences. This constraint calls for a formal semantics program where the meaning of natural language sentences is represented by an appropriate assignment of the elements of an algebraic structure to sentences, an assignment that represents facts concerning complete sentences. I call the above constraint 'The Algebraic Thesis', and the corresponding formal semantics program 'Algebraic Semantics.' ;I introduce AT by carving it out of the holistic views of language of Quine and Davidson, and I show that it is independent of other important features of these philosophers' views. I then argue for AT, taking into account recent work by Brandom, Dummett, Fodor and Lepore, Gaifman and Perry. In developing Algebraic Semantics I present structures of two basic types: Boolean algebras and cylindric algebras; these structures are construed in the context of AT in a novel way: they are treated as applying to classes of sentences as numbers apply to physical objects in measurement. Applying algebraic methodology to formal semantics and to the theory of meaning, I develop these results: ; An account of the semantics of necessity and possibility that avoids both Quine's dismissal of these notions and their reduction by David Lewis to quantification over possible worlds; ; The application of the model-theoretic notion of expansion to capture the process of our learning progressively more of the meaning of expressions in our first languages. (shrink)

A structural theorem for Kleene algebras is proved, showing that an element of a Kleene algebra can be looked upon as an ordered pair of sets, and that negation with the Kleene property is describable by the set-theoretic complement. The propositional logic \ of Kleene algebras is shown to be sound and complete with respect to a 3-valued and a rough set semantics. It is also established that Kleene negation can be considered as a modal operator, due to a (...) perp semantics of \. Moreover, another representation of Kleene algebras is obtained in the class of complex algebras of compatibility frames. One concludes with the observation that \ can be imparted semantics from different perspectives. (shrink)

The properties of belief revision operators are known to have an informal semantics which relates them to the axioms of conditional logic. The purpose of this paper is to make this connection precise via the model theory of conditional logic. A semantics for conditional logic is presented, which is expressed in terms of algebraic models constructed ultimately out of revision operators. In addition, it is shown that each algebraic model determines both a revision operator and a (...) logic, that are related by virtue of the stable Ramsey test. (shrink)

We study the variety of equivalential algebras with zero and its subquasivariety that gives the equivalent algebraic semantics for the ‐fragment of intuitionistic propositional logic. We prove that this fragment is hereditarily structurally complete. Moreover, we effectively construct the finitely generated free equivalential algebras with zero.

This paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators—akin to the theory of modal Algebras—RBAs serve as models of modal logics in which truth (...) is relative. In particular, modal RBAs provide semantics for various well-known awareness logics and an alternative view of possibility semantics. (shrink)

A well-known result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix 〈,F〉, whereis an algebra of the appropriate type, andFa subset of the domain of, called the set of designated elements. In particular, every quasi-classical modal logic—a set of modal formulas, containing the smallest classical modal logicE, which is closed under the inference rules of substitution and modus ponens—is characterized by such a matrix, (...) wherenow is a modal algebra, andFis a filter of. If the modal logic is in fact normal, then we can do away with the filter; we can study normal modal logics in the setting of varieties of modal algebras. This point of view was adopted already quite explicitly in McKinsey and Tarski [8]. The observation that the lattice of normal modal logics is dually isomorphic to the lattice of subvarieties of a variety of modal algebras paved the road for an algebraic study of normal modal logics. The algebraic approach made available some general results from Universal Algebra, notably those obtained by Jónsson [6], and thereby was able to contribute new insights in the realm of normal modal logics [2], [3], [4], [10].The requirement that a modal logic be normal is rather a severe one, however, and many of the systems which have been considered in the literature do not meet it. For instance, of the five celebrated modal systems, S1–S5, introduced by Lewis, S4 and S5 are the only normal ones, while only SI fails to be quasi-classical. The purpose of this paper is to generalize the algebraic approach so as to be applicable not just to normal modal logics, but to quasi-classical modal logics in general. (shrink)

For the modal logic |$\textsf {S4}^{C}_{I}$|⁠, we identify the class of completable |$\textsf {S4}^{C}_{I}$|-algebras and prove for them a Stone-type representation theorem. As a consequence, we obtain strong algebraic and topological completeness of the logic |$\textsf {S4}^{C}_{I}$| in the case of local semantic consequence relations. In addition, we consider an extension of the logic |$\textsf {S4}^{C}_{I}$| with certain infinitary derivations and establish the corresponding strong completeness results for the enriched system in the case of global semantic consequence relations.

In the first chapter of From Metaphysics to Ethics, Frank Jackson begins, as he puts it, ‘by explaining how serious metaphysics by its very nature raises the location problem.’ (1998, p. 1) He gives us two examples of location problems. The first concerns semantic properties, such as truth and reference: Some physical structures are true. For example, if I were to utter a token of the type ‘Grass is green’, the structure I would thereby bring into existence would be true (...) ... How are the semantic properties of the sentence related to the non-semantic properties of the sentence? Where, if anywhere, are the semantic properties of truth, content and reference to be found in the non-semantic, physical or naturalistic account of the sentence?’ (1998, p. 2) Jackson notes two possible answers to questions of this kind. The first denies that there are any such semantic properties. (shrink)

The paper provides a preliminary study of algebraic semantics for modal and superintuitionistic non-monotonic logics. The main question answered is: how can non-monotonic inference be understood algebraically?

A systematic approach to the algebraic and Kripke semantics for logics with restricted structural rules, notably for logics on an underlying non-distributive lattice, is developed. We provide a new topological representation theorem for general lattices, using the filter space X. Our representation involves a galois connection on subsets of X, hence a closure operator $\Gamma$, and the image of the representation map is characterized as the collection of $\Gamma$-stable, compact-open subsets of the filter space . The original lattice (...) ${\cal L}$ is thus imbedded in the complete lattice of stable sets. The representation is shown to be functorial and is extended to a Stone-type duality. ;In the presence of additional operators, the extension problem is addressed in terms of what we call adjoint completions of normal operators. An operator f on a lattice L is normal if f: $L\sp{} \times \cdots \times L\sp{} \to L\sp{}$ is a monotone map, where each $L\sp{}$, $i \in n$ + 1, is either L or $L\sp{op}$ and f sends finite joins of $L\sp{}$ to joins in $L\sp{}$. Every familiar operator, such as $\circ$ , + , $\gets, \to, \neg, \square$ O is normal. By a uniform argument we show that normal operators $f\sb{i}$ on L extend to operators $F\sb{i}$ on $\Gamma X$ that distribute over arbitrary joins or meets and that \sb{i\in I})$ is a universal such extension. ;We also raise and resolve the question of representability of operators on stable sets by relations. We give an extensive number of examples of representation of specific operators and provide explicit constructions of canonical Kripke frames, based on our extension arguments, for a number of logical systems, notably for Intuitionistic and Classical Linear Logics, or Non-commutative Linear Logic and of fragments thereof. (shrink)

In this book Ludlow uses the metaphysics of time as a case study and focuses on the dispute between A-theorists and B-theorists about the nature of time.

Lawvere hyperdoctrines give categorical algebraic semantics for intuitionistic predicate logic. Here we extend the hyperdoctrinal semantics to a broad variety of substructural predicate logics over the Typed Full Lambek Calculus, verifying their completeness with respect to the extended hyperdoctrinal semantics. This yields uniform hyperdoctrinal completeness results for numerous logics such as different types of relevant predicate logics and beyond, which are new results on their own; i.e., we give uniform categorical semantics for a broad variety (...) of non-classical predicate logics. And we introduce an analogue of Lawvere–Tierney topology and cotopology in the hyperdoctrinal setting, which gives a unifying perspective on different logical translations, in particular allowing for a uniform treatment of Girard’s exponential translation between linear and intuitionistic logics and of Kolmogorov’s double negation translation between intuitionistic and classical logics. In the hyerdoctrinal conception, type theories are categories, logics over type theories are functors, and logical translations between them, then, are natural transformations, in particular Lawvere–Tierney topologies and cotopologies on hyperdoctrines. The view of logical translations as hyperdoctrinal Lawvere–Tierney topologies and cotopologies has not been elucidated before, and may be seen as a novel contribution of the present work. From a broader perspective, this work may be regarded as taking first steps towards interplay between algebraic and categorical logics; it is, technically, a combination of substructural algebraic logic and hyperdoctrinal categorical logic, as the hyperdoctrinal completeness theorem is shown via the integration of the Lindenbaum–Tarski algebra construction with the syntactic category construction. As such this work lays a foundation for further interactions between algebraic and categorical logics. (shrink)

In the first chapter of From Metaphysics to Ethics, Frank Jackson begins, as he puts it, ‘by explaining how serious metaphysics by its very nature raises the location problem.’ (1998, p. 1) He gives us two examples of location problems. The first concerns semantic properties, such as truth and reference: Some physical structures are true. For example, if I were to utter a token of the type ‘Grass is green’, the structure I would thereby bring into existence would be true (...) ... How are the semantic properties of the sentence related to the non-semantic properties of the sentence? Where, if anywhere, are the semantic properties of truth, content and reference to be found in the non-semantic, physical or naturalistic account of the sentence?’ (1998, p. 2) Jackson notes two possible answers to questions of this kind. The first denies that there are any such semantic properties. (shrink)

(1996). Compatibilist semantics in metaphysics: A case study. Australasian Journal of Philosophy: Vol. 74, No. 1, pp. 117-134. doi: 10.1080/00048409612347101.

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)

In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5Π with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such (...) a proof. This result raises the question: For which normal modal logics. (shrink)

We provide a new semantics and a slightly different formalisation for the propositional logic with superposition introduced and studied in Tzouvaras [. Propositional superposition logic...

I propose a metaphysical position I call 'limited metaphysical realism', and I link it to a position in the philosophy of language I call 'psychologistic semantics'. Limited metaphysical realism asserts that there is a mind-independent, discourse-independent world, but posits a sparse ontology. Psychologistic semantics construes truth not as direct word/world correspondence, and not as warranted assertibility (or Putnam's "ideal" warranted assertibility), but rather as 'correct assertibility'. I argue that virtues of this package deal over each (...) of the two broad positions that have recently dominated metaphysics and philosophy of language--positions I call package deal metaphysical realism, and package deal anti-realism. (shrink)

Three influential forms of realism are distinguished and interrelated: realism about the external world, construed as a metaphysical doctrine; scientific realism about non-observable entities postulated in science; and semantic realism as defined by Dummett. Metaphysical realism about everyday physical objects is contrasted with idealism and phenomenalism, and several potent arguments against these latter views are reviewed. -/- Three forms of scientific realism are then distinguished: (i) scientific theories and their existence postulates should be taken literally; (ii) the existence (...) of unobservable entities posited by our most successful scientific theories is justified scientifically; and (iii) our best current scientific theories are at least approximately true. It is argued that only some form of scientific realism can make proper sense of certain episodes in the history of science. -/- Finally, Dummett’s influential formulation of semantic issues about realism considered. Dummett argued that in some cases, the fundamental issue is not about the existence of entities, but rather about whether statements of some specified class (such as mathematics) have an objective truth value, independently of our means of knowing it. Dummett famously argued against such semantic realism and in favor of anti-realism. The relation of semantic realism to the metaphysical construal of realism and Dummett’s main argument against semantic realism is examined. (shrink)

It is widely assumed in philosophy that there is a tight connection between semantics and metaphysics. Semantic theories about the meanings of natural language terms and phrases are taken to provide evidence for and against various metaphysical theses about the nature of non-linguistic parts of the world. Call this view the widespread thesis. I argue that the widespread thesis is mistaken: semantic theories do not generally have robust metaphysical consequences. I contend that the best arguments for the (...) widespread thesis turn on an interpretation of semantic theories according to which they are metaphysically illuminating: the metaphysical structure of the world is directly mirrored in the structure of the theory. Terms and existential quantification in the theory signal the existence of entities in the world, predicates signal the existence of properties, and the structures of set theoretic objects invoked by the theory directly correspond to structural features of the world. Such an interpretation is not the only one available, however, and it is a substantive philosophical (and linguistic) issue which interpretation is correct. Accordingly, I develop and defend an alternative interpretation of semantic theories according to which the structural features of these theories and of their theorems reflect biological and computational constraints on the architecture of the language faculty and historical contingencies in its evolutionary development. Supposedly metaphysically committing features of the theory—including the appearance of quantification and (ostensibly) referring terms—may thus be arepresentational artifacts of these constraints, rather than representational features that reflect metaphysical reality. To determine the metaphysical consequences of a semantic theory then, we must determine which of its features represent language-independent characteristics of the world and which are arepresentational consequences of the structure of the language faculty. To do this, however, we must have some prior idea what the language-independent world is like—that is, we must engage in prior metaphysical theorizing that is not beholden to the semantic theory itself. (shrink)