Coinductive models and normal forms for modal logics (or how we learned to stop worrying and love coinduction)

We present a coinductive definition of models for modal logics and show that it provides a homogeneous framework in which it is possible to include different modal languages ranging from classical modalities to operators from hybrid and memory logics. Moreover, results that had to be proved separately for each different language—but whose proofs were known to be mere routine—now can be proved in a general way. We show, for example, that we can have a unique definition of bisimulation for all these languages, and prove a single invariance-under-bisimulation theorem.We then use the new framework to investigate normal forms for modal logics. The normal form we introduce may have a smaller modal depth than the original formula, and it is inspired by global modalities like the universal modality and the satisfiability operator from hybrid logics. These modalities can be extracted from under the scope of other operators. We provide a general definition of extractable modalities and show how to compute extracted normal forms. As it is the case with other classical normal forms—e.g., the conjunctive normal form of propositional logic—the extracted normal form of a formula can be exponentially bigger than the original formula, if we require the two formulas to be equivalent. If we only require equi-satisfiability, then every modal formula has an extracted normal form which is only polynomially bigger than the original formula, and it can be computed in polynomial time.     

Liar-type Paradoxes and the Incompleteness Phenomena

We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s Paradox in this framework. Furthermore, we define explicit and implicit self-reference in paradoxes in the incompleteness phenomena.     

What Is an Inconsistent Truth Table?

Do truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears in the foreground. Rather than any one logic being privileged, then, on this count partisans across the logical spectrum are in relatively similar dialectical positions.     

Two Ways to Smoke a Cigarette

In the early part of the paper, I attempt to explain a dispute between two parties who endorse the compositionality of language but disagree about its implications: Paul Horwich, and Jerry Fodor and Ernest Lepore. In the remainder of the paper, I challenge the thesis on which they are agreed, that compositionality can be taken for granted. I suggest that it is not clear what compositionality involves nor whether it obtains. I consider some kinds of apparent counterexamples, and compositionalist responses to them in terms of covert indexicality and unspecific meanings. I argue that the last option is the best for most of the cases I consider. I conclude by stressing, as against Horwich and Fodor and Lepore, that the appropriate question concerns the extent to which compositionality obtains in a natural language, rather than whether it obtains or not, so that the answer is essentially messy, requiring detailed consideration of a wide range of examples.     

Abstraction and Individual Style in Syllogistic and Heterogeneous Reasoning

Syllogistic reasoning can be modelled with mental models. Heterogeneous reasoning brings together propositional and visual information, and in the Hyperproof logic-teaching system, the visual can act as a model for the propositional. Both mental models notation and Hyperproof’s visual representations allow abstractions of a limited kind. Individual differences in reasoning performance have been found in both domains. We suggest that at least some of the differences can be attributed to students’ differing ability to manipulate abstractions.     

The motivations for Walter Burley’s theory of the proposition

Walter Burley (born c. 1275; died. c. 1344) claims throughout his career that the mind can make a statement (propositio) out of things. Since things include entities that exist outside of the mind, Burley appears to be claiming that the mind can form a statement out of things that exist outside of it. Most scholars of Burley offer a deflationary reading of this claim, arguing that it confuses two distinct but closely related philosophical issues: the nature of propositional content, on the one hand, and the role of facts in a compelling account of truth, on the other. But I argue that Burley means exactly what he says: that the mind can, quite literally, form statements out of things that exist outside of it. In Burley’s account, statements of this sort function as the propositional contents of our thoughts and written or uttered sentences. This account of propositional content is motivated by three more fundamental theses to which Burley is committed: referentialism, compositionality, and a claim about truth-conditionality I call intellectualism.     

Perceptual experience and seeing that p

I open my eyes and see that the lemon before me is yellow. States like this—states of seeing that $p$ —appear to be visual perceptual states, in some sense. They also appear to be propositional attitudes (and so states with propositional representational contents). It might seem, then, like a view of perceptual experience on which experiences have propositional representational contents—a Propositional View—has to be the correct sort of view for states of seeing that $p$ . And thus we can’t sustain fully general non-Propositional but Representational, or Relational Views of experience. But despite what we might initially be inclined to think when reflecting upon the apparent features of states of seeing that $p$ , a non-propositional view of seeing that $p$ is, I argue, perfectly intelligible.     

Frege's Content-Principle and Relevant Deducibility

Given the harmony principle for logical operators, compositionality ought to ensure that harmony should obtain at the level of whole contents. That is, the role of a content qua premise ought to be balanced exactly by its role as a conclusion. Frege's contextual definition of propositional content happens to exploit this balance, and one appeals to the Cut rule to show that the definition is adequate.We show here that Frege's definition remains adequate even when one relevantizes logic by abandoning an unrestricted Cut rule. The proof exploits the fact that in the relevantized logic, which abandons the unrestricted rule of Cut, any failure of the transitivity of deduction is offset by the epistemic gain involved in learning that a stronger-than-expected result holds.     

Supervaluation-Style Truth Without Supervaluations

Kripke’s theory of truth is arguably the most influential approach to self-referential truth and the semantic paradoxes. The use of a partial evaluation scheme is crucial to the theory and the most prominent schemes that are adopted are the strong Kleene and the supervaluation scheme. The strong Kleene scheme is attractive because it ensures the compositionality of the notion of truth. But under the strong Kleene scheme classical tautologies do not, in general, turn out to be true and, as a consequence, classical reasoning is no longer admissible once the notion of truth is involved. The supervaluation scheme adheres to classical reasoning but violates compositionality. Moreover, it turns Kripke’s theory into a rather complicated affair: to check whether a sentence is true we have to look at all admissible precisification of the interpretation of the truth predicate we are presented with. One consequence of this complicated evaluation condition is that under the supervaluation scheme a more proof-theoretic characterization of Kripke’s theory becomes inherently difficult, if not impossible. In this paper we explore the middle ground between the strong Kleene and the supervaluation scheme and provide an evaluation scheme that adheres to classical reasoning but retains many of the attractive features of the strong Kleene scheme. We supplement our semantic investigation with a novel axiomatic theory of truth that matches the semantic theory we have put forth.     

Self-knowledge and commitments

In this paper I provide an outline of a new kind of constitutive account of self-knowledge. It is argued that in order for the model properly to explain transparency, a further category of propositional attitudes—called “commitments”—has to be countenanced. It is also maintained that constitutive theories can’t remain neutral on the issue of the possession of psychological concepts, and a proposal about the possession of the concept of belief is sketched. Finally, it is claimed that in order for a constitutive account properly to explain authority, it has to take a rather dramatic constructivist turn, which makes it suitable as an explanation of self-knowledge only for a limited class of mental states.     

The Contents of Perception and the Contents of Emotion

Several philosophers think there are important analogies between emotions and perceptual states. Furthermore, considerations about the rational assessibility of emotions have led philosophers—in some cases, the very same philosophers—to think that the content of emotions must be propositional content. If one finds it plausible that perceptual states have propositional contents, then there is no obvious tension between these views. However, this view of perception has recently been attacked by philosophers who hold that the content of perception is object‐like. I shall argue for a view about the content of emotions and perceptual states which will enable us to hold both that emotional content is analogous to perceptual content and that both emotions and perceptual states can have propositional contents. This will involve arguing for a pluralist view of perceptual content, on which perceptual states can have both contents which are proposition‐like and contents which are object‐like. I shall also address two significant objections to the claim that emotions can have proposition‐like contents. Meeting one of these objections will involve taking on a further commitment: the pluralist account of perceptual content will have to be one on which the contents of perception can be non‐conceptual.     

Piktorielle Repräsentationen als unterbestimmte räumliche Modelle

Central for the discussion about the pros and cons of pictorial representations is the indeterminacy problem. Whereas propositional representations can be underdetermined, it is a widespread opinion that pictorial representations are completely determined and committed to details. In contrast to this view, which is held — for example by Dennett, Pylyshyn and Iorger — in the present paper a variable system of commitments is proposed: Users of pictorial representations have alternatives with respect to interpretations and commitments since there are implicit annotations to these representations. The annotations refer to axiomatized geometric systems, which — from a cognitive point of view — can be seen as systems of commitments.     

The dialectics of infinitism and coherentism: inferential justification versus holism and coherence

This paper formally explores the common ground between mild versions of epistemological coherentism and infinitism; it proposes—and argues for—a hybrid, coherentist–infinitist account of epistemic justification. First, the epistemological regress argument and its relation to the classical taxonomy regarding epistemic justification—of foundationalism, infinitism and coherentism—is reviewed. We then recall recent results proving that an influential argument against infinite regresses of justification, which alleges their incoherence on account of probabilistic inconsistency, cannot be maintained. Furthermore, we prove that the Principle of Inferential Justification has rather unwelcome consequences—formally resembling the Sorites paradox—as soon as it is iterated and combined with a natural Bayesian perspective on probabilistic inferences. We conclude that strong versions of foundationalism and infinitism should be abandoned. Positively, we provide a rough sketch for a graded formal coherence notion, according to which infinite regresses of epistemic justification will often have more than a minimal degree of coherence.     

Non-classical Metatheory for Non-classical Logics

A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete.     

Visuelle und räumliche Interferenzen beim Schlußfolgern

The present paper provides evidence for a differential involvement of spatial and visual mental resources in propositional and spatial reasoning. Two experiments consider the load on visuospatial working memory during reasoning. Subjects solve propositional and spatial reasoning tasks either alone or in combination with visual tracking. In the first experiment, subjects resource allocation strategy is manipulated. Independently of the allocation of mental resources to the reasoning tasks or the tracking task, tracking is found to interfere much more with spatial than with propositional reasoning. In the second experiment, propositional reasoning after a comprehension training is disrupted by a simultaneous secondary spatial task, but not by the visual tracking task.     

Graphical language games: interactional constraints on representational form

The emergence of shared symbol systems is considered to be a pivotal moment in human evolution and human development. These changes are normally explained by reference to changes in people's internal cognitive processes. We present 2 experiments which provide evidence that changes in the external, collaborative processes that people use to communicate can also affect the structure and organization of symbol systems independently of cognitive change. We propose that mutual-modifiability-opportunities for people to edit or manipulate each other's contributions-is a key constraint on the emergence of complex symbol systems. We discuss the implications for models of language development and the origins of compositionality.     

The Cultural Evolution of Structured Languages in an Open‐Ended,Continuous World

Language maps signals onto meanings through the use of two distinct types of structure. First, the space of meanings is discretized into categories that are shared by all users of the language. Second, the signals employed by the language are compositional: The meaning of the whole is a function of its parts and the way in which those parts are combined. In three iterated learning experiments using a vast, continuous, open‐ended meaning space, we explore the conditions under which both structured categories and structured signals emerge ex nihilo. While previous experiments have been limited to either categorical structure in meanings or compositional structure in signals, these experiments demonstrate that when the meaning space lacks clear preexisting boundaries, more subtle morphological structure that lacks straightforward compositionality—as found in natural languages—may evolve as a solution to joint pressures from learning and communication.     

Imaginative Attitudes

The point of this paper is to reveal a dogma in the ordinary conception of sensory imagination, and to suggest another way forward. The dogma springs from two main sources: a too close comparison of mental imagery to perceptual experience, and a too strong division between mental imagery and the traditional propositional attitudes (such as belief and desire). The result is an unworkable conception of the correctness conditions of sensory imaginings—one lacking any link between the conditions under which an imagining aids human action and inference and the conditions under which it is veridical. The proposed solution is, first, to posit a variety of imaginative attitudes—akin to the traditional propositional attitudes—which have different associated correctness (or satisfaction) conditions. The second part of the solution is to allow for imaginings with “hybrid” contents, in the sense that both mental images and representations with language‐like constituent structure contribute to the content of imaginings.     

Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic

We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using filtrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.