Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `Syntactic aspects of modal incompleteness theorems,' and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

Hybrid type theory: a quartet in four movements

This paper sings a song -a song created by bringing together the work of four great names in the history of logic: Hans Reichenbach, Arthur Prior, Richard Montague, and Leon Henkin. Although the work of the first three of these authors have previously been combined, adding the ideas of Leon Henkin is the addition required to make the combination work at the logical level. But the present paper does not focus on the underlying technicalities (these can be found in Areces, Blackburn, Huertas, and Manzano [to appear]) rather it focusses on the underlying instruments, and the way they work together. We hope the reader will be tempted to sing along

The master argument and branching time

It is argued that reconstructions of the so-called ‘Master Argument’ of Dideros Cronos to the effect that possibility should be understood as present or future truth, essentially relies on two axioms: i) that every true proposition concerning the past is necessary, and ii) that it follows necessarily from a proposition being true that it always has been the case that it would be true. It is furthermore argued that these two axioms are inconsistent in the sense that any tense/modal semantics which incorporates both collapses either modally (fails to distinguish between truth simpliciter and modalised truth) or temporally (fails to offer a plausible semantical account for propositions about the future). This finding is, furthermore, taken as indicator for the more generel claim that there are principled difficulties involved in construing semantics for combined tense/modal logical systems

Philosophical lessons of entanglement

The quantum-mechanical description of the world, including human observers, makes substantial use of entanglement. In order to understand this, we need to adopt concepts of truth, probability and time which are unfamiliar in modern scientific thought. There are two kinds of statements about the world: those made from inside the world, and those from outside. The conflict between contradictory statements which both appear to be true can be resolved by recognising that they are made in different perspectives. Probability, in an objective sense, belongs in the internal perspective, and to statements in the future tense. Such statements obey a many-valued logic, in which the truth values are identified as probabilities.Comment: Talk given at 75 Years of Quantum Entanglement, Kolkata, India, 10 January 201

Moderate Relativism

In modal logic, propositions are evaluated relative to possible worlds. A proposition may be true relative to a world w, and false relative to another world w'. Relativism is the view that the relativization idea extends beyond possible worlds and modalities. Thus, in tense logic, propositions are evaluated relative to times. A proposition (e.g. the proposition that Socrates is sitting) may be true relative to a time t, and false relative to another time t'. In this paper I discuss, and attempt to rebut, two classical objections to Relativism. The first objection, due to Frege, is the objection from incompleteness. I distinguish two possible relativist responses to that objection, one of which corresponds to the view I actually defend : Moderate Relativism. The second objection is due to Mark Richard, who argued that the objects of belief cannot be relativistic. I show that that objection can be met within the Moderate Relativist framework. In the last section, I deal with special forms of disagreement that have loomed large in recent discussions of Relativism

The categories of grammar in resource-books of English for advanced learners

The courses of morphology and syntax of the English language as linguistic disciplines studied at university level elaborate on a systematic description of the language, logically categorising the individual items of the language system. In doing so, the traditional structural terminology is employed, describing the phenomena such as complex verb phrases which may comprise modal exponents, i.e. epistemic and deontic modality, progressive and perfective aspects or the category of active vs. passive voice. As in any scientific field, the language used should provide clarity and precision and ambiguities should be avoided. In the field of applied linguistics, however, the requirement of precision in the description of the language categories seems impossible to comply with. This paper discusses the ways the categories of English grammar are defined in standard structural linguistic resources and compares them with publications in the field of applied linguistics at advanced levels. It also contrasts these against the lexical approach introduced by Lewis. The aim of the paper is to highlight the presumed discrepancies between the aforementioned types of resources and help to facilitate the ELT in the field of presenting English grammar