We define consciousness as the category of all conscious experiences. This immediately raises the question: What is the essence in which every conscious experience in the category of conscious experiences partakes? We consider various abstract essences of conscious experiences as theories of consciousness. They are: (i) conscious experience is an action of memory on sensation, (ii) conscious experience is experiencing a particular as an exemplar of a general, (iii) conscious experience is an interpretation of sensation, (iv) conscious experience is referring (...) sensation to an object as its cause, and (v) conscious experience is a model of stimulus. Corresponding to each one of these theories we obtain a category of models of conscious experiences: (i) category of actions, (ii) category of idempotents, (iii) category of two sequential maps, (iv) category of brain-generalized figures, and (v) functor categories with intuition as base and conceptual repertoire as exponent, respectively. For each theory of consciousness we also calculate its truth value object and characterize the objective logic intrinsic to the corresponding category of models of consciousness experiences. (shrink)

Logical pluralism is the view that there is more than one correct logic. Most logical pluralists think that logic is normative in the sense that you make a mistake if you accept the premisses of a valid argument but reject its conclusion. Some authors have argued that this combination is self-undermining: Suppose that L1 and L2 are correct logics that coincide except for the argument from Γ to φ, which is valid in L1 but invalid in L2. If (...) you accept all sentences in Γ, then, by normativity, you make a mistake if you reject φ. In order to avoid mistakes, you should accept φ or suspend judgment about φ. Both options are problematic for pluralism. Can pluralists avoid this worry by rejecting the normativity of logic? I argue that they cannot. All else being equal, the argument goes through even if logic is not normative. (shrink)

I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism.

The Fregean tradition promotes a conception of logic as being independent from all psychological acts of the knowing subject. Without questioning logic's status as a paradigm of objectivity the present essay rejects this conception, both on logical grounds and in light of the scholastic theory of intentionality. Finding fault with two key doctrines of the analytic movement, the linguistic turn and anti-psychologism, it reinterprets them to exclude only psychological acts that engender subjective interpretive variability. It then describes (...) class='Hi'>logic's dependence on non-interpretive psychological acts that ensue as a natural result of objects being apprehended by the intellect. These acts, which Hervaeus Natalis and the scholastic tradition thematize as the imposition of second intentional relations of reason, dictate the logically coherent parameters of logical objectivity. (shrink)

A picture of the world as chiefly one of discrete objects, distributed in space and time, has sometimes seemed compelling. It is however one of the main targets of Henry Laycock's book; for it is seriously incomplete. The picture, he argues, leaves no space for "stuff" like air and water. With discrete objects, we may always ask "how many?," but with stuff the question has to be "how much?" Laycock's fascinating exploration also addresses key logical and linguistic questions about the (...) way we categorize the many and the much. (shrink)

Alexius Meinong developed a notion of defective objects in order to account for various logical and psychological paradoxes. The notion is of historical interest, since it presages recent work on the logical paradoxes by Herzberger and Kripke. But it fails to do the job it was designed for. However, a technique implicit in Meinong's investigation is more successful and can be adapted to resolve a similar paradox discovered by Romane Clark in a revised version of Meinong's Theory of Objects due (...) to Rapaport. One family of paradoxes remains, but it is argued that they are unavoidable and relatively harmless. (shrink)

I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that (...) only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument. (shrink)

My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language. -/- Hence I try to explore, what properties objects are presupposed to have, in order to enter the universe of discourse of an interpreted formalized language. -/- First I review Frege′s analysis of the logical structure of truth value definite sentences of scientific colloquial language, to draw suggestions from (...) his saturated vs. unsaturated sentence components paradigm. -/- Next try investigate, in how far reference to non pure math objects might allow for a role as argument of a truth value function. Object kinds to be considered are: common sense objects, technical objects, humanities object kinds (social, psychical, ...), objects of art, ... , be they abstract, concrete, or in this respect mixed objects. -/- Then have a comment on the just referenced label abstract objects. -/- Next try to get an idea wrt the ontological significance of the fact, that pure math objects and functions in some important classical cases can be uniquely defined by means of categorical theories. Here, in the course of my argument, I have a little corollary wrt the standard model of first order Peano arithmetic, reducing the epistemological significance of the existence of the non standard models. -/- Next, wrt a special concept of a formalized empirical theory, I care for whether the impure math objects and mixed objects described here, and complying best with truth value function mapping, are also reliable candidates for the ontological commitment of such theories: and discuss an alternative, which reduces the ontological significance of the universe of discourse of the theories intended models. -/- In the end I sum up what is - from my point of view - the status quo, according to my respective findings. (shrink)

Once upon a time, logical conventionalism was the most popular philosophical theory of logic. It was heavily favored by empiricists, logical positivists, and naturalists. According to logical conventionalism, linguistic conventions explain logical truth, validity, and modality. And conventions themselves are merely syntactic rules of language use, including inference rules. Logical conventionalism promised to eliminate mystery from the philosophy of logic by showing that both the metaphysics and epistemology of logic fit into a scientific picture of reality. For (...) naturalists of all stripes, this was paradise. Alas, paradise was lost. By the end of the twentieth century, logical conventionalism had been almost universally abandoned. But more recently, it has been revitalized. This chapter provides an overview of logical conventionalism and its history, clears away misunderstandings, and briefly responds to both the canonical objections to conventionalism (from Quine, Dummett, Boghossian, and many others) as well as new objections (from Field and Golan). From paradise lost, to paradise regained: logical conventionalism is back. (shrink)

In this article, I re-examine Kant’s many senses of cognition, thought, and object in order to better understand his account of logic, and, in particular, transcendental logic. In laying out the multiple and conflicting ways Kant defines each of those terms, the relation between transcendental logic and pure general logic can be revealed as analogous to, and, indeed, linked to, the isomorphism that exists between the categories of the understanding and the logical forms of judgment, which (...) are equivalent if one ignores the manifold of intuition. This means transcendental logic is a general logic, even as a logic of (some) cognizing. (shrink)

‘Feminist logic’ may sound like an impossible, incoherent, or irrelevant project, but it is none of these. We begin by delineating three categories into which projects in feminist logic might fall: philosophical logic, philosophy of logic, and pedagogy. We then defuse two distinct objections to the very idea of feminist logic: the irrelevance argument and the independence argument. Having done so, we turn to a particular kind of project in feminist philosophy of logic: Valerie (...) Plumwood's feminist argument for a relevance logic. Plumwood's work serves as our primary case study as we turn to the project of considering three different ways we might understand her argument and revisionist arguments like it: as a priori theorizing, as ameliorative conceptual engineering, or as instances of anti-exceptionalist approaches to logic. After arguing that the anti-exceptionalist approach seems to provide the most promising means of understanding the kind of project undertaken in a feminist challenge to classical logic, we briefly address the consequences that this might have for logic instruction. Here, we argue for the perhaps unexpected conclusion that feminist programs ought to offer more, not less, instruction in logic for those who take interest. (shrink)

According to the Bad Company objection, the fact that Frege’s infamous Basic Law V instantiates the general definitional pattern of higher-order abstraction principles is a good reason to doubt the soundness of this sort of definitions. In this paper I argue against this objection by showing that the definitional pattern of abstraction principles – as extrapolated from §64 of Frege’s Grundlagen– includes an additional requirement (which I call the specificity condition) that is not satisfied by the Basic Law V while (...) is satisfied by other higher-order abstractions such as Hume’s Principle. I also show that the failure of this additional requirement in the case of Basic Law V is engendered by an essential feature of Frege’s conception of logic and thus that Frege himself should not have regarded the Basic Law V as a definition by abstraction. (shrink)

Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical (...) interpretation of the connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators. (shrink)

The book is a collection of papers and aims to unify the questions of syntax and semantics of language, which are included in logic, philosophy and ontology of language. The leading motif of the presented selection of works is the differentiation between linguistic tokens (material, concrete objects) and linguistic types (ideal, abstract objects) following two philosophical trends: nominalism (concretism) and Platonizing version of realism. The opening article under the title “The Dual Ontological Nature of Language Signs and the Problem (...) of Their Mutual Relations” provides a broad introduction into the problem area connected with this differentiation, while the logic-formal characteristics of the distinction are framed in the work entitled “On the Type-Token Relationships” (Chapter 1). The basic part of the book deals with issues relating to syntax (Chapters 2-4) and semantics of language (Chapters 5-6), as well as pertaining to syntactic-semantic pragmatic questions (Chapters 7-13). Throughout the book, language, categorial language, is characterized syntactically as generated by classical categorial grammar (Chapter 2) and formalized on two opposing levels: as language of expression-tokens (level of tokens) and language of expression-types (level of types). The author’s considerations contained in Chapters 2 and 4 lead to the important philosophical conclusion that in formal-logical syntactic studies on language the assumption that expression-types constitute the primary language layer while expression-tokens make the secondary one, can be neglected; thus, this speaks in favour of the opposing standpoint—the concretistic one—in the ontology of language syntax. In the works “Meaning and Interpretations”, Parts I and II (Chapters 5 and 6), it is underlined, however, that such semantic concepts as: meaning, denotation and interpretation are defined on the types level, yet their formal definitions require making use of notions of the tokens level. The semantic notions introduced in the above-mentioned articles are also used in the following works of the present selection, under the titles: “Three Principles of Compositionality” and “On Metaknowledge and Truth” (Chapters 7 and 8). They formalize two principles of compositionality that are well known in the literature on the subject, deriving from Frege, i.e. those of meaning and of denotation; they are related to the syntactic principle of compositionality which was introduced by the author. All the three principles are, at the same time, three conditions of homomorphism of categorial language algebra into three kinds of non-standard models of language (one syntactic and two semantic ones: intensional and extensional), which allows introducing three definitions of truthfulness into these models. The next two works in the collection, entitled: “On Language Adequacy” and “What is the Sense in Logic and Philosophy of Language” (Chapters 9 and 10) concern adequacy of categorial language syntax along with its dual semantics: intensional and extensional, and categorial compatibility of any of its syntactic categories with two corresponding semantic categories: intensional and extensional, based on the compatibility the syntactic category of each language expression with the ontological category assigned to its denotatum. The well-known problem of categorial compatibility for first-order quantifiers finds its solution in the paper “Categories of First-Order Quantifiers” (Chapter 11). In the work “Logic and Ontology of Language” (Chapter 12), being in a sense a summary of the considerations presented in the preceding chapters of the book, language is treated as an ontological being, characterized in compliance with the logical conception of language proposed by Ajdukiewicz. Application—like throughout the book—of tools of classical logic and set theory has resulted in emergence of a general formal logical theory of syntax, semantics and pragmatics of language, which takes into account duality in the understanding of linguistic expressions as tokens (concretes) and types (abstract objects). In terms that take into account a functional approach to language itself, there comes out an ontological neutrality of logic with respect to existential assumptions relating to the ontological nature of linguistic expressions and their extra-linguistic ontological counterparts. The issues connected with applying logic while explaining the manner of using linguistic tokens and linguistic types to determine notions of language communication are raised and illustrated in the last chapter of the work, bearing the title “A Logical Conceptualization of Knowledge on the Notion of Language Communication”. (shrink)

Gareth Evans has argued that the existence of vague objects is logically precluded: The assumption that it is indeterminate whether some object a is identical to some object b leads to contradiction. I argue in reply that, although this is true—I thus defend Evans's argument, as he presents it—the existence of vague objects is not thereby precluded. An 'Indefinitist' need only hold that it is not logically required that every identity statement must have a determinate truth-value, not that some such (...) statements might actually fail to have a determinate truth-value. That makes Indefinitism a cousin of mathematical Intuitionism. (shrink)

The purpose of this paper is to explore the question of how truthmaker theorists ought to think about their subject in relation to logic. Regarding logic and truthmaking, I defend the view that considerations drawn from advances in modal logic have little bearing on the legitimacy of truthmaker theory. To do so, I respond to objections Timothy Williamson has lodged against truthmaker theory. As for the logic of truthmaking, I show how the project of understanding the (...) logical features of the truthmaking relation has led to an apparent impasse. I offer a new perspective on the logic of truthmaking that both explains the problem and offers a way out. (shrink)

What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer’s hitherto undiscovered Berlin Lectures, which concern the (...) philosophy of language, logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled ‘rational representationalism’: the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us. (shrink)

This note clarifies an error in the proof of the main theorem of “The Ricean Objection: An Analogue of Rice’s Theorem for First-Order Theories”, Logic Journal of the IGPL, 16(6): 585–590(2008).

We propose a solution to the problem of logical omniscience in what we take to be its fundamental version: as concerning arbitrary agents and the knowledge attitude per se. Our logic of knowledge is a spin-off from a general theory of thick content, whereby the content of a sentence has two components: an intension, taking care of truth conditions; and a topic, taking care of subject matter. We present a list of plausible logical validities and invalidities for the (...) class='Hi'>logic of knowledge per se for arbitrary agents, and isolate three explanatory factors for them: the topic-sensitivity of content; the fragmentation of knowledge states; the defeasibility of knowledge acquisition. We then present a novel dynamic epistemic logic that yields precisely the desired validities and invalidities, for which we provide expressivity and completeness results. We contrast this with related systems and address possible objections. (shrink)

An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize nicely to languages (...) containing the kind of vocabulary that usually motivates non-classical theories—for example, a language containing a naïve truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence, non-classical theorists do not seem entitled to those results. In this paper, we analyze these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity and that, once the ambiguity is removed, there are no reasons to think that non-classical logicians are not entitled to their recapture results. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

The objective Bayesian view of proof (or logical probability, or evidential support) is explained and defended: that the relation of evidence to hypothesis (in legal trials, science etc) is a strictly logical one, comparable to deductive logic. This view is distinguished from the thesis, which had some popularity in law in the 1980s, that legal evidence ought to be evaluated using numerical probabilities and formulas. While numbers are not always useful, a central role is played in uncertain reasoning (...) by the ‘proportional syllogism’, or argument from frequencies, such as ‘nearly all aeroplane flights arrive safely, so my flight is very likely to arrive safely’. Such arguments raise the ‘problem of the reference class’, arising from the fact that an individual case may be a member of many different classes in which frequencies differ. For example, if 15 per cent of swans are black and 60 per cent of fauna in the zoo is black, what should I think about the likelihood of a swan in the zoo being black? The nature of the problem is explained, and legal cases where it arises are given. It is explained how recent work in data mining on the relevance of features for prediction provides a solution to the reference class problem. (shrink)

Two questions are addressed in this paper. First, what is it to see? I argue that it is veridical experience of things outside the perceiver brought about by looking. Second, what is it to see a material object? I argue that it is experience of an occupant of a spatial region that is a logical subject for other visual features, able to move to another spatial region, to change intrinsically, and to interact with other material objects. I show how this (...) theory is different from the idea that object-seeing is merely the visual segregation of a region of the visual field. Finally, I argue that we do not object-see objects reflected in mirrors, surfaces of back-lit objects, and depictions of objects. (shrink)

This book describes how scientists bring their own interests and passions to their work, illustrates the dynamics between researchers and the research community ...

I argue against inferentialism about logic. First, I argue against an analogy between logic and chess, before considering a more basic objection to stipulating inference rules as a way of establishing the meaning of logical constants. The objectionthe Mushroom Omelette Objectionis that stipulative acts are partly constituted by logical notions, and therefore cannot be used to explain logical thought. I then argue that the same problem also attaches to following existing conventional rules, since either those rules have logical (...) contents, or following those conventional rules is done for logical reasons. Lastly, I compare this argument with other arguments found in Quine’s early work, and consider two attempts to reply to Quine. (shrink)

It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about (...) metaphysical necessity, then paradigmatic metaphysical necessities would be necessary in one sense of “necessary”, not necessary in another, and that would be it. The question of whether they were necessary simpliciter would be like the question of whether the Parallel Postulate is true simpliciter – understood as a pure mathematical conjecture, rather than as a hypothesis about physical spacetime. In a sense, the latter question has no objective answer. In this article, I argue that paradigmatic questions of modal metaphysics are like the Parallel Postulate question. I then discuss the deflationary ramifications of this argument. I conclude with an alternative conception of the space of possibility. According to this conception, there is no objective boundary between possibility and impossibility. Along the way, I sketch an analogy between modal metaphysics and set theory. (shrink)

Climate change assessments rely upon scenarios of socioeconomic developments to conceptualize alternative outcomes for global greenhouse gas emissions. These are used in conjunction with climate models to make projections of future climate. Specifically, the estimations of greenhouse gas emissions based on socioeconomic scenarios constrain climate models in their outcomes of temperatures, precipitation, etc. Traditionally, the fundamental logic of the socioeconomic scenarios—that is, the logic that makes them plausible—is developed and prioritized using methods that are very subjective. This introduces (...) a fundamental challenge for climate change assessment: The veracity of projections of future climate currently rests on subjective ground. We elaborate on these subjective aspects of scenarios in climate change research. We then consider an alternative method for developing scenarios, a systems dynamics approach called ‘Cross-Impact Balance’ (CIB) analysis. We discuss notions of ‘objective’ and ‘objectivity’ as criteria for distinguishing appropriate scenario methods for climate change research. We distinguish seven distinct meanings of ‘objective,’ and demonstrate that CIB analysis is more objective than traditional subjective approaches. However, we also consider criticisms concerning which of the seven meanings of ‘objective’ are appropriate for scenario work. Finally, we arrive at conclusions regarding which meanings of ‘objective’ and ‘objectivity’ are relevant for climate change research. Because scientific assessments uncover knowledge relevant to the responses of a real, independently existing climate system, this requires scenario methodologies employed in such studies to also uphold the seven meanings of ‘objective’ and ‘objectivity.’. (shrink)

The logical interpretation of probability, or "objective Bayesianism'' – the theory that (some) probabilities are strictly logical degrees of partial implication – is defended. The main argument against it is that it requires the assignment of prior probabilities, and that any attempt to determine them by symmetry via a "principle of insufficient reason" inevitably leads to paradox. Three replies are advanced: that priors are imprecise or of little weight, so that disagreement about them does not matter, within limits; that (...) it is possible to distinguish reasonable from unreasonable priors on logical grounds; and that in real cases disagreement about priors can usually be explained by differences in the background information. It is argued also that proponents of alternative conceptions of probability, such as frequentists, Bayesians and Popperians, are unable to avoid committing themselves to the basic principles of logical probability. (shrink)

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. -/- There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is (...) countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. -/- Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects. (shrink)

Chapter 3 surveys objectivity in the natural sciences. Thomas Kuhn problematized the logicist understanding of the objectivity or rationality of scientific change, providing a very different picture than that of the cumulative or step-wise progress of theoretical science. Theories often compete, and when consensus builds around one competitor it may be for a variety of reasons other than just the direct logical implications of experimental successes and failures. Kuhn pitted the study of the actual history of science against what Hans (...) Reichenbach referred to as the “logical substitutes” logicists used to reconstruct the rationality of theory change. Such substitutes were promoted to show that theory confirmation is a purely epistemic affair, where purely epistemic implies purely inferential. But where the positivists saw logical compulsion as the reason for theory change, Kuhn saw psychology in values as relevant to how theories are actually confirmed or accepted by the scientific community. (shrink)

Review of Karel Lambert, Meinong and the Principle of Independence: Its Place in Meinong's Theory of Objects and Its Significance in Contemporary Philosophical Logic.

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as (...) the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

Complex logic is a novel logical framework, which formalizes the semantics of the categories of matter, space, and time in a system of logic that operates with complex logical objects. A complex logical object represents a superposition of a logical statement and its logical negation positioning any statement co-relatively to its logical negation. In the system of logical notations, where S is a logical statement and Not S is its logical negation, complex logic includes co-relative logical positions (...) of S and Not S with the probabilities of their truth within the scale 0... 1, excluding the boundary values of 0 and 1, into a single logical superposition. Thus, the logical superposition, summing up the probabilistic positions of S and Not S, has an invariant truth value equal to 1. In this context, complex logic proves to be invariant to reality, offering a unified logical interpretation of reality, spanning from quantum to cosmological scales. By defining the nature of matter and reality as a complex two-format processual essence with corresponding logical consequences, complex logic enhances our concept and understanding of reality. (shrink)

Logic and psychology overlap in judgment, inference and proof. The problems raised by this commonality are notoriously difficult, both from a historical and from a philosophical point of view. Sundholm has for a long time addressed these issues. His beautiful piece of work [A Century of Inference: 1837-1936] begins by summarizing the main difficulty in the usual provocative manner of the author: one can start, he says, by the act of knowledge to go to the object, as the Idealist (...) does; one can also start by the object to go to the act, in the Realist mood; never the two shall meet. He is himself inclined to accept the first perspective as the right one and he has eventually developed an original version of antirealism which starts, not from considerations about the publicity of meaning, in the manner of Dummett, but from an epistemic standpoint, trying to search in a non-Fregean tradition of analysis of judgement and cognate notions a way of founding constructivist semantics. The present paper ploughes the same field. We concentrate on the significance, for Sundholm’s program, of the perspective that has been opened by Twardowski in his important essay on acts and products (1912. (shrink)

In a recent paper in this journal Duncan Pritchard responds to an objection to the modal account of risk pressed by Ebert, Smith and Durbach ( 2020 ). In this paper, I expand upon the objection and argue that it still stands. I go on to consider a more general question raised by this exchange – whether risk is ‘objective’, or whether it is something that varies from one perspective to another.

This paper discusses the Uniqueness Thesis, a core thesis in the epistemology of disagreement. After presenting uniqueness and clarifying relevant terms, a novel counterexample to the thesis will be introduced. This counterexample involves logical disagreement. Several objections to the counterexample are then considered, and it is argued that the best responses to the counterexample all undermine the initial motivation for uniqueness.

We analyze the logical form of the domain knowledge that grounds analogical inferences and generalizations from a single instance. The form of the assumptions which justify analogies is given schematically as the "determination rule", so called because it expresses the relation of one set of variables determining the values of another set. The determination relation is a logical generalization of the different types of dependency relations defined in database theory. Specifically, we define determination as a relation between schemata of first (...) order logic that have two kinds of free variables: (1) object variables and (2) what we call "polar" variables, which hold the place of truth values. Determination rules facilitate sound rule inference and valid conclusions projected by analogy from single instances, without implying what the conclusion should be prior to an inspection of the instance. They also provide a way to specify what information is sufficiently relevant to decide a question, prior to knowledge of the answer to the question. (shrink)

In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl’s ideas of pure grammar, Leśniewski-Ajukiewicz’s theory syntactic/semantic categories and in accordance with Frege’s ontological canons, Bocheński’s famous motto—syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its (...) cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science. (shrink)

The current resurgence of interest in cognition and in the nature of cognitive processing has brought with it also a renewed interest in the early work of Husserl, which contains one of the most sustained attempts to come to grips with the problems of logic from a cognitive point of view. Logic, for Husserl, is a theory of science; but it is a theory which takes seriously the idea that scientific theories are constituted by the mental acts of (...) cognitive subjects. The present essay begins with an exposition of Husserl's act-based conception of what a science is, and goes on to consider his account of the role of linguistic meanings, of the ontology of scientific objects, and of evidence and truth. The essay concentrates almost exclusively on the Logical Investigations of 1900/01. This is not only because this work, which is surely Husserl's single most important masterpiece, has been overshadowed first of all by his Ideas I and then later by the Crisis. It is also because the Investigations contain, in a peculiarly clear and pregnant form, a whole panoply of ideas on logic and cognitive theory which either simply disappeared in Husserl's own later writings or became obfuscated by an admixture of that great mystery which is 'transcendental phenomenology'. (shrink)

There has been much work on ontological dependence in recent literature. However, relatively little of it has been dedicated to the ways in which individual physical objects may depend on other distinct, non-overlapping objects. This paper gives several examples of such object-dependence and distinguishes between different types of it. The paper also introduces and refines the notion of an n-tet. N-tets (typically) occur when there are object-dependence relations between n objects. I claim that the identity (or, rather, what I call (...) the n-dentity) conditions for n-tets are not grounded in the individual identity conditions of each of the n objects, but instead are metaphysically basic. The paper then briefly discusses some ramifications of accepting object-dependence (and n-tets) on the philosophy of biology, ethics, and logic. (shrink)

We think of logic as objective. We also think that we are reliable about logic. These views jointly generate a puzzle: How is it that we are reliable about logic? How is it that our logical beliefs match an objective domain of logical fact? This is an instance of a more general challenge to explain our reliability about a priori domains. In this paper, I argue that the nature of this challenge has not been properly (...) understood. I explicate the challenge both in general and for the particular case of logic. I also argue that two seemingly attractive responses – appealing to a faculty of rational insight or to the nature of concept possession – are incapable of answering the challenge. (shrink)

This is the second part of a two-part series on the logic of hyperlogic, a formal system for regimenting metalogical claims in the object language (even within embedded environments). Part A provided a minimal logic for hyperlogic that is sound and complete over the class of all models. In this part, we extend these completeness results to stronger logics that are sound and complete over restricted classes of models. We also investigate the logic of hyperlogic when the (...) language is enriched with hyperintensional operators such as counterfactual conditionals and belief operators. (shrink)

In this paper I propose and motivate a logic of the interdefined concepts of making true and control, understood as intensional propositional operators to be indexed to an agent. While bearing a resemblance to earlier logics in the tradition, the motivations, semantics, and object language theory differ on crucial points. Applying this logic to widespread formal theories of agency, I use it as a framework to argue against the ubiquitous assumption that the strongest actions or options available to (...) a given agent must always be pairwise incompatible. The conclusion is that this assumption conflicts with failures of higher order control of agents over their degree or precision of control, failures exhibited by such imperfect agents as ourselves. I discuss models in this setting for understanding such imperfectly self-controlling agents. In an appendix, I prove several relevant results about the logic described, including soundness and completeness both for it and for certain natural extensions. (shrink)

Sentences about logic are often used to show that certain embedding expressions are hyperintensional. Yet it is not clear how to regiment “logic talk” in the object language so that it can be compositionally embedded under such expressions. In this paper, I develop a formal system called hyperlogic that is designed to do just that. I provide a hyperintensional semantics for hyperlogic that doesn’t appeal to logically impossible worlds, as traditionally understood, but instead uses a shiftable parameter that (...) determines the interpretation of the logical connectives. I argue this semantics compares favorably to the more common impossible worlds semantics, which faces difficulties interpreting propositionally quantified logic talk. (shrink)

In matters of personal taste, faultless disagreement occurs between people who disagree over what is tasty, fun, etc., in those cases when each of these people seems equally far from the objective truth. Faultless disagreement is often taken as evidence that truth is relative. This article aims to help us avoid the truth-relativist conclusion. The article, however, does not argue directly against relativism; instead, the article defends non-relative truth constructively, aiming to explain faultless disagreement with the resources of semantic (...) contextualism. To this end the article describes and advocates a contextualist solution inspired by supervaluationist truth-value gap approaches. The solution presented here, however, does not require truth value gaps; it preserves both logical bivalence and non-relative truth, even while it acknowledges and explains the possibility of faultless disagreement. The solution is motivated by the correlation between assertions’ being true and their being useful. This correlation, furthermore, is used not only to tell which assertions are true, but also to determine which linguistic intuitions are reliable. (shrink)

The main objective of the paper is to provide a conceptual apparatus of a general logical theory of language communication. The aim of the paper is to outline a formal-logical theory of language in which the concepts of the phenomenon of language communication and language communication in general are defined and some conditions for their adequacy are formulated. The theory explicates the key notions of contemporary syntax, semantics, and pragmatics. The theory is formalized on two levels: token-level and type-level. (...) As such, it takes into account the dual – token and type – ontological character of linguistic entities. The basic notions of the theory: language communication, meaning and interpretation are introduced on the second, type-level of formalization, and their required prior formalization of some of the notions introduced on the first, token-level; among others, the notion of an act of communication. Owing to the theory, it is possible to address the problems of adequacy of both empirical acts of communication and of language communication in general. All the conditions of adequacy of communication discussed in the presented paper, are valid for one-way communication (sender-recipient); nevertheless, they can also apply to the reverse direction of language communication (recipient-sender). Therefore, they concern the problem of two-way understanding in language communication. (shrink)

I discuss Benacerraf's epistemological challenge for realism about areas like mathematics, metalogic, and modality, and describe the pluralist response to it. I explain why normative pluralism is peculiarly unsatisfactory, and use this explanation to formulate a radicalization of Moore's Open Question Argument. According to the argument, the facts -- even the normative facts -- fail to settle the practical questions at the center of our normative lives. One lesson is that the concepts of realism and objectivity, which are widely identified, (...) are actually in tension. (shrink)

Many have claimed that legitimate constitutional democracy is either conceptually or practically impossible, given infinite regress paradoxes deriving from the requirement of simultaneously democratic and constitutional origins for legitimate government. This paper first critically investigates prominent conceptual and practical bootstrapping objections advanced by Barnett and Michelman. It then argues that the real conceptual root of such bootstrapping objections is not any specific substantive account of legitimacy makers, such as consent or democratic endorsement, but a particular conception of the logic (...) of normative standards—the determinate threshold conception—that the critic attributes to the putatively undermined account of legitimacy. The paper further claims that when we abandon threshold conceptions of the logic of legitimacy in favor of regulative-ideal conceptions, then the objections, from bootstrapping paradoxes to the very idea of constitutional democracy, disappear. It concludes with considerations in favor of adopting a more demanding conception of the regulative ideal of constitutional democracy, advanced by Habermas, focusing on potentials for developmental learning. (shrink)

Logic is useful as a neutral formalism for expressing the contents of mental representations. It can be used to extract crisp conclusions regarding the higher-order theory of phenomenal consciousness developed in (McDermott 2001, 20007). A key aspect of conscious perceptions is their connection to the distinction between appearance and reality. Perceptions must often be corrected. To do so requires that the logic of perception be able to represent the logical structure of judgment events, that is, to include the (...) formulas of the logic as objects to be reasoned about. However, there is a limit to how finely humans can examine their own representations. Terms representing primary and secondary qualities seemed to be _locked,_ so that the numbers (or levels of neural activation) that are their essence are not directly accessible. Humans feel a need to invoke ``intrinsic,'' ``nonrelational'' properties of many secondary qualities --- their _qualia_ --- to ``explicate'' how we compare and discriminate among them, although this is not actually how the comparisons are accomplished. This model of qualia explains several things: It accounts for the difference between ``normal'' and ``introspective'' access to a perceptual module in terms of quotation. It dissolves Jackson's knowledge argument by explaining what Mary learns as a fictional but undoubtable belief structure. It makes spectrum inversion logically impossible by providing a degree of freedom between the physical structure of the brain and the representations it contains that redescribes putative cases of spectrum inversion as alternative but equivalent ways of mapping physical states to representational states. (shrink)

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the (...) idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)