What is <Emphasis Type="Italic">the</Emphasis> Logic of Inference?

The topic of this paper is the question whether there is a logic which could be justly called the logic of inference. It may seem that at least since Prawitz, Dummett and others demonstrated the proof-theoretical prominency of intuitionistic logic, the forthcoming answer is that it is this logic that is the obvious choice for the accolade. Though there is little doubt that this choice is correct (provided that inference is construed as inherently single-conclusion and complying with the Gentzenian structural rules), I do not think that the usual justification of it is satisfactory. Therefore, I will first try to clarify what exactly is meant by the question, and then sketch a conceptual framework in which it can be reasonably handled. I will introduce the concept of ‘inferentially native’ logical operators (those which explicate inferential properties) and I will show that the axiomatization of these operators leads to the axiomatic system of intuitionistic logic. Finally, I will discuss what modifications of this answer enter the picture when more general notions of inference are considered. Presented by Hannes Leitgeb     

Complex Systems from the Perspective of Category Theory: II. Covering Systems and Sheaves

Using the concept of adjunction, for the comprehension of the structure of a complex system, developed in Part I, we introduce the notion of covering systems consisting of partially or locally defined adequately understood objects. This notion incorporates the necessary and sufficient conditions for a sheaf theoretical representation of the informational content included in the structure of a complex system in terms of localization systems. Furthermore, it accommodates a formulation of an invariance property of information communication concerning the analysis of a complex system.     

Realism about laws

This paper explores the idea that laws express relationships between properties or universals as defended in Michael Tooley's recent book Causation: A Realist Approach. I suggest that the most plausible version of realism will take a different form than that advocated by Tooley. According to this alternative, laws are grounded in facts about the capacities and powers of particular systems, rather than facts about relations between universals. The notion of lawfulness is linked to the notion of invariance, rather than to the metaphysical notion of a necessary connection.     

Melia and Saatsi on structural realism

Newman’s objection is sometimes taken to be a fatal objection to structural realism (SR). However, ambiguity in the definition of “structure” allows for versions that do not succumb to Newman’s objection. In this paper, I consider some versions of SR that maintain an abstract notion of structure yet avoid Newman’s objection. In particular, I consider versions suggested by Melia and Saatsi. They reject a solution that restricts the domain of the second-order quantifiers, and argue in favor of buttressing the language with intensional operators such as “it is physically necessary that...”. I argue that their favored solution effectively requires the former suggestion that they reject. This argument suggests that a notion of natural properties may be indispensable to SR.     

laws and their stability

Many philosophers have believed that the laws of nature differ from the accidental truths in their invariance under counterfactual perturbations. Roughly speaking, the laws would still have held had q been the case, for any q that is consistent with the laws. (Trivially, no accident would still have held under every such counterfactual supposition.) The main problem with this slogan (even if it is true) is that it uses the laws themselves to delimit qs range. I present a means of distinguishing the laws (and their logical consequences) from the accidents, in terms of their range of invariance under counterfactual antecedents, that does not appeal to physical modalities (or any cognate notion) in delimiting the relevant range of counterfactual perturbations. I then argue that this approach explicates the sense in which the laws possess a kind of necessity.     

Decision framing in judgment aggregation

Judgment aggregation problems are language dependent in that they may be framed in different yet equivalent ways. We formalize this dependence via the notion of translation invariance, adopted from the philosophy of science, and we argue for the normative desirability of translation invariance. We characterize the class of translation invariant aggregation functions in the canonical judgment aggregation model, which requires collective judgments to be complete. Since there are reasonable translation invariant aggregation functions, our result can be viewed as a possibility theorem. At the same time, we show that translation invariance does have certain normatively undesirable consequences (e.g. failure of anonymity). We present a way of circumventing them by moving to a more general model of judgment aggregation, one that allows for incomplete collective judgments.     

Validity and Necessity

In this paper I argue against the commonly received view that Kripke’s formal Possible World Semantics (PWS) reflects the adoption of a metaphysical interpretation of the modal operators. I consider in detail Kripke’s three main innovations vis-à-vis Carnap’s PWS: a new view of the worlds, variable domains of quantification, and the adoption of a notion of universal validity. I argue that all these changes are driven by the natural technical development of the model theory and its related notion of validity: they are dictated by merely formal considerations, not interpretive concerns. I conclude that Kripke’s model theoretic semantics does not induce a metaphysical reading of necessity, and is formally adequate independently of the specific interpretation of the modal operators.     

Synonymy and Intra-Theoretical Pluralism

The starting point of this paper is a version of intra-theoretical (logical) pluralism that was recently proposed by Hjortland [2013]. In a first move, I use synonymy-relations to formulate an intuitively compelling objection against Hjortland's claim that, if one uses a single calculus to characterise the consequence relations of the paraconsistent logic LP and the paracomplete logic K3, one immediately obtains multiple consequence relations for a single language and hence a reply to the Quinean charge of meaning variance. In a second move, I explain how a natural generalisation of the notion of synonymy (adapted to the 3-sided sequent-calculus used by Hjortland) can be used to counter this objection, but I also show how the solution can be turned into an equally devastating ‘one logic after all’ type of objection. Finally, I propose the general diagnosis that these problems could only arise in the presence of conceptual distinctions that are too coarse to accommodate coherent pluralist theses. The latter leads to the general methodological recommendation that the conceptual resources used to think and talk about logic should be kept in line with the formal resources that are used to define and describe a logical theory.     

Minimal Belief Change and the Pareto Principle

This paper analyzes the notion of a minimal belief change that incorporates new information. I apply the fundamental decision-theoretic principle of Pareto-optimality to derive a notion of minimal belief change, for two different representations of belief: First, for beliefs represented by a theory – a deductively closed set of sentences or propositions – and second for beliefs represented by an axiomatic base for a theory. Three postulates exactly characterize Pareto-minimal revisions of theories, yielding a weaker set of constraints than the standard AGM postulates. The Levi identity characterizes Pareto-minimal revisions of belief bases: a change of belief base is Pareto-minimal if and only if the change satisfies the Levi identity (for “maxichoice” contraction operators). Thus for belief bases, Pareto-minimality imposes constraints that the AGM postulates do not. The Ramsey test is a well-known way of establishing connections between belief revision postulates and axioms for conditionals (“if p, then q”). Pareto-minimal theory change corresponds exactly to three characteristic axioms of counterfactual systems: a theory revision operator that satisfies the Ramsey test validates these axioms if and only if the revision operator is Pareto-minimal. This revised version was published online in June 2006 with corrections to the Cover Date.     

Construction of the World: Ontology in the Tractatus

Why does Wittgenstein say in the Tractatus Logico-Philosophicus that the world has as its members only facts, not things? Focusing on this question, I start with the problem, in its general form, “How is it possible to determine something as X?” and establish the excluding-allowing model for determination. From this model, I derive an argument for Wittgenstein’s aforementioned statement. The argument shows that a whole cannot be determined as consisting of components that are determined separately in a strong sense, whereas in a weak sense it can be. This thus demonstrates why the context principle holds. The recommended interpretation places suitable weight on the Tractarian notion of possibility. It provides new insights into Wittgenstein’s conception of logic, and his atomism about facts and states of affairs.     

The Place of Self‐Respect in a Theory of Justice

This essay provides a critical examination of Rawls' (and Rawlsians') conception of self‐respect, the social bases of self‐respect, and the normative justification of equality in the social bases of self‐respect. I defend a rival account of these notions and the normative ideals at stake in political liberalism and a theory of social justice.

I make the following arguments: (1) I argue that it is unreasonable to take self‐respect to be a primary social good, as Rawls and his interpreters characterize it; (2) secondly, drawing on a distinction made by Darwall, I argue that recognition respect provides a far more suitable notion of respect for a theory of justice than Rawls' notion of appraisal respect; (3) thirdly, I argue that Rawls' treatment of self‐respect and the social bases of self‐respect as empirical conceptions should be rejected in favor of normative notions of a reasonable or justified self‐respect and equality in reasonable social bases of self‐respect; (4) I argue that Rawls' notions of political liberalism and public reason provide a way of grounding a notion of the reasonable social bases of self‐respect in political ideals of the person implicit in modern economic institutions, and family relations, ignored by Rawlsians—but as central to reasonable social bases of self‐respect and justice, as Rawlsians' ideal of persons as free and equal citizens.     

Innateness in cognitive science

Innateness is one of the central concepts of cognitive science; but it is also a source of considerable confusion. In this article, I survey recent attempts to understand the notion of innateness as it figures in cognitive science and indicate which is likely to prove most fruitful. One approach draws directly on our "commonsense" views about innate traits. Another aims to characterize innateness in terms of concepts drawn from biology, such as genetic determination. Yet neither strategy has met with much success. This could indicate that a satisfactory account of innateness needs to make use of the conceptual resources of cognitive science itself. A proposal that takes this suggestion seriously is outlined, and an appeal is made for a more systematic assessment of the role and significance of the notion of innateness to cognitive science.     

Realism,Positivism and Reference

Summary  Depending on the realist or instrumentalist twist that is given to positivism, interesting arguments can be made for both causal and classical theories of reference with regard to the use of scientific terms in the language of theory. But my claim is that the rigid foundationalism that supports the theoretical terms via the correspondence rules of the Received View undercuts the notion that it is possible to argue coherently for a causal theory of reference as allied to a positivistic view.     

Categorical invariance and structural complexity in human concept learning

An alternative account of human concept learning based on an invariance measure of the categorical stimulus is proposed. The categorical invariance model (CIM) characterizes the degree of structural complexity of a Boolean category as a function of its inherent degree of invariance and its cardinality or size. To do this we introduce a mathematical framework based on the notion of a Boolean differential operator on Boolean categories that generates the degrees of invariance (i.e., logical manifold) of the category in respect to its dimensions. Using this framework, we propose that the structural complexity of a Boolean category is indirectly proportional to its degree of categorical invariance and directly proportional to its cardinality or size. Consequently, complexity and invariance notions are formally unified to account for concept learning difficulty. Beyond developing the above unifying mathematical framework, the CIM is significant in that: (1) it precisely predicts the key learning difficulty ordering of the SHJ [Shepard, R. N., Hovland, C. L., & Jenkins, H. M. (1961). Learning and memorization of classifications. Psychological Monographs: General and Applied, 75(13), 1-42] Boolean category types consisting of three binary dimensions and four positive examples; (2) it is, in general, a good quantitative predictor of the degree of learning difficulty of a large class of categories (in particular, the 41 category types studied by Feldman [Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature, 407, 630-633]); (3) it is, in general, a good quantitative predictor of parity effects for this large class of categories; (4) it does all of the above without free parameters; and (5) it is cognitively plausible (e.g., cognitively tractable).     

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer??s calculus of ??conditional density operators??. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.     

Aboutness in imagination

I present a formal theory of the logic and aboutness of imagination. Aboutness is understood as the relation between meaningful items and what they concern, as per Yablo and Fine’s works on the notion. Imagination is understood as per Chalmers’ positive conceivability: the intentional state of a subject who conceives that p by imagining a situation—a configuration of objects and properties—verifying p. So far aboutness theory has been developed mainly for linguistic representation, but it is natural to extend it to intentional states. The proposed framework combines a modal semantics with a mereology of contents: imagination operators are understood as variably strict quantifiers over worlds with a content-preservation constraint.     

Supervaluations and the Propositional Attitude Constraint

For the sentences of languages that contain operators that express the concepts of definiteness and indefiniteness, there is an unavoidable tension between a truth-theoretic semantics that delivers truth conditions for those sentences that capture their propositional contents and any model-theoretic semantics that has a story to tell about how indetifiniteness in a constituent affects the semantic value of sentences which imbed it. But semantic theories of both kinds play essential roles, so the tension needs to be resolved. I argue that it is the truth theory which correctly characterises the notion of truth, per se. When we take into account the considerations required to bring model theory into harmony with truth theory, those considerations undermine the arguments standardly used to motivate supervaluational model theories designed to validate classical logic. But those considerations also show that celebration would be premature for advocates of the most frequently encountered rival approach – many-valued model theory.     

An Intensional Theory of Truth: An Informal Report

Saul Kripke’s theory of truth suffers from expressive limitations – in particular, there are no extensional operators within that framework that allow one to characterize those sentences that fail to receive a truth value within the framework. Especially worrisome is the fact that there is no operator that outputs true on exactly the paradoxical sentences. In this paper I extend Kripke’s approach via the addition of extensional operators, which allows us to characterize many (but not all) such sentences, including the paradoxical ones.     

Facing the mirror: A relativist account of immune nonconceptual self-representations

There is a consensus among philosophers that some “I”-thoughts are immune to error through misidentification. In some recent papers, this property has been formulated in the following deflationist way: an “I”-thought is immune to error through misidentification when it can misrepresent the mental or bodily property self-ascribed but cannot misrepresent the subject (if any) possessing that property. However, it has been put forward that the range of mental and bodily states that are immune in that limited sense cannot include nonconceptual forms of self-representation. In this paper, I claim the opposite. I argue in favor of a theoretical framework inspired by semantic relativism that solves the problem of immune nonconceptual self-representations. In order to do so, I refute an argument against the relativist account which is based on the existence of shared representations. This argument, I contend, rests on a confusion between two conditions to which a relativist may appeal when considering whether a certain mental content is relative to the self: a strong invariance condition and a weak invariance condition. I then argue that even if we acknowledge the existence of shared representations, the weak invariance condition is still satisfied, and consequently the relativist framework can make sense of INSRs. I argue that this weak invariance condition is satisfied by a representational function that self-relativizes certain representations. I then provide an empirical instance of such a function by discussing some of the recent literature on motor representations and the sense of agency. In the last part of the paper, I answer several potential objections. These potential objections lead me to distinguish two fundamental kinds of error relative to the self: error through misidentification and error through misapplication. This distinction allows me to answer a fundamental question raised by the very idea of de facto immunity to error through misidentification.     

THE ROLE OF CONSCIOUSNESS AS MEANING MAKER IN SCIENCE, CULTURE, AND RELIGION

Two hundred years ago, Friedrich Schleiermacher took critical issue with Immanuel Kant's intellectual notion of intuition as applied to human nature (Wellmon 2006). He found it necessary to modify—"hermeneutically," as he said—Kant's notion of anthropology by enabling it to include as human the new and strange human tribes Captain Cook found in the Pacific South Seas. A similar hermeneutic move is necessary if physics is to include the local contextual empirical syntheses of relativity and quantum physics. In this hermeneutical revision the synthesis is formed around the notion of a Hilbert Vector Space as the universal grammar of physics, adding to it the dynamic of the Schrödinger equation, and representing empirical "observables" by projection operators that map the subspaces of definite measurable values. Among the set of observable projection operators, some pairs share the same subspace, commute with one another, and share a common laboratory setting. Other pairs do not share this property and are described as being mutually complementary. Complementary symmetries introduce into the discursive language of physics the commonsense notion of contextuality. The new synthesis, proposed by Eugene Wigner, John von Neumann, and (in his own way) Paul Dirac, brought physics into the community of common language and established it as a work of general human achievement. ¹