We prove that the finite-model version of arithmetic with the divisibility relation is undecidable . Additionally we prove FM-representability theorem for this class of finite models. This means that a relation R on natural numbers can be described correctly on each input on almost all finite divisibility models if and only if R is of degree ≤0′. We obtain these results by interpreting addition and multiplication on initial segments of finite models with divisibility only.

Krynicki, M. and M. Mostowski, Decidability problems in languages with Henkin quantifiers, Annals of Pure and Applied Logic 58 149–172.We consider the language L with all Henkin quantifiers Hn defined as follows: Hnx1…xny1…yn φ iff f1…fnx1. ..xn φ, ...,fn). We show that the theory of equality in L is undecidable. The proof of this result goes by interpretation of the word problem for semigroups.Henkin quantifiers are strictly related to the function quantifiers Fn defined as follows: Fnx1…xny1…yn φ iff fx1…xn (...) φ,...,f). In contrast with the first result we show that the theory of equality with all quantifiers Fn is decidable.We also consider decidability problems for other theories in languages L and L. (shrink)

The problem of computational complexity of semantics for some natural language constructions – considered in [M. Mostowski, D. Wojtyniak 2004] – motivates an interest in complexity of Ramsey quantifiers in finite models. In general a sentence with a Ramsey quantifier R of the following form Rx, yH(x, y) is interpreted as ∃A(A is big relatively to the universe ∧A2 ⊆ H). In the paper cited the problem of the complexity of the Hintikka sentence is reduced to the problem of (...) computational complexity of the Ramsey quantifier for which the phrase “A is big relatively to the universe” is interpreted as containing at least one representative of each equivalence class, for some given equvalence relation. In this work we consider quantifiers Rf, for which “A is big relatively to the universe” means “card(A) > f (n), where n is the size of the universe”. Following [Blass, Gurevich 1986] we call R mighty if Rx, yH(x, y) defines N P – complete class of finite models. Similarly we say that Rf is N P –hard if the corresponding class is N P –hard. We prove the following theorems. (shrink)

We investigate some logics with Henkin quantifiers. For a given logic L, we consider questions of the form: what is the degree of the set of L–tautologies in a poor vocabulary (monadic or empty)? We prove that the set of tautologies of the logic with all Henkin quantifiers in empty vocabulary L*∅ is of degree 0’. We show that the same holds also for some weaker logics like L ∅(Hω) and L ∅(Eω). We show that each logic of the form (...) L ∅ (k)(Q), with the number of variables restricted to k, is decidable. Nevertheless – following the argument of M. Mostowski from [Mos89] – for each reasonable set theory no concrete algorithm can provably decide L (k) (Q), for some (Q). We improve also some results related to undecidability and expressibility for logics L(H4) and L(F2) of Krynicki and M. Mostowski from [KM92]. (shrink)

Writing from a scientifically and philosophically informed perspective, the authors provide a critical overview of the conceptual difficulties encountered in many current neuroscientific and psychological theories.

A long-standing theme in discussion of perception and thought has been that our primary cognitive contact with individual objects and events in the world derives from our perceptual contact with them. When I look at a duck in front of me, I am not merely presented with the fact that there is at least one duck in the area, rather I seem to be presented withthisthing (as one might put it from my perspective) in front of me, which looks to (...) me to be a duck. Furthermore, such a perception would seem to put me in a position not merely to make the existential judgment that there is some duck or other present, but rather to make a singular, demonstrative judgment, that that is a duck. My grounds for an existential judgment in this case derives from my apprehension of the demonstrative thought and not vice versa. (shrink)

This book defends antitheodicism, arguing that theodicies, seeking to excuse God for evil and suffering in the world, fail to ethically acknowledge the victims of suffering. The authors argue for this view using literary and philosophical resources, commencing with Immanuel Kant's 1791 "Theodicy Essay" and its reading of the Book of Job. Three important twentieth century antitheodicist positions are explored, including "Jewish" post-Holocaust ethical antitheodicism, Wittgensteinian antitheodicism exemplified by D.Z. Phillips and pragmatist antitheodicism defended by William James. The authors argue (...) that these approaches to evil and suffering are fundamentally Kantian. Literary works such as Franz Kafka'sThe Trial, Samuel Beckett's Waiting for Godot, and George Orwell's Nineteen Eighty-Four, are examined in order to crucially advance the philosophical case for antitheodicism. (shrink)

In his last book, David Armstrong sets out his metaphysical system in a set of concise and lively chapters each dealing with one aspect of the world. He begins with the assumption that all that exists is the physical world of space-time. On this foundation he constructs a coherent metaphysical scheme that gives plausible answers to many of the great problems of metaphysics. He gives accounts of properties, relations, and particulars; laws of nature; modality; abstract objects such as numbers; and (...) time and mind. (shrink)