Being Sceptical about Kripkean A Posteriori Necessities and Natural Kinds

The article discusses Saul Kripke’s influential theories of a posteriori necessary truths and natural kinds. With respect to the statements of identity involving proper names, it is argued that although their truth is a posteriori and necessary in the specific sense of counterfactual invariance, this is of no significance for substantial philosophical issues beyond the philosophy of language, because this counterfactual invariance is a trivial consequence of the use of proper names as rigid designators. The case is made that the expansion of the realm of necessary a posteriori truths to the statements of theoretical identity that involve “natural kind terms”, as well as the Kripkean essentialist theory of natural kinds, have no weighty argumentative support and fit badly both with science and language practice. This sets the stage for the development of an appropriately sophisticated “descriptivist” account of meaning and reference that would be better suited for a widened range of Kripke-Putnam style thought experiments. The general outlines of such a descriptivist account are provided.

Epistemic and Doxastic Contents

The case for making belief states the primary focus of our analysis and for including impossible worlds in that analysis is outlined in this chapter. This allows the reader to deny various closure principles, although this won’t help defeat worries about external-world scepticism. The issue that concerns the authors most is the problem of bounded rationality: belief states seem to be closed under ‘easy’ trivial consequence, but not under full logical consequence, and yet the former implies the latter. The solution presented here is that some trivial closure principle must fail on a given belief state, yet it is indeterminate just where this occurs. Formal models of belief states along these lines are given and it is shown that they respect the indeterminacy-of-closure intuition. Finally, the chapter discusses how we might square this approach with the fact that some people seem to believe contradictions.

Defining Cognitive Logics by Non-Classical Tableau Rules

In the paper we propose a new approach to formalization of cognitive logics. By cognitive logics we understand supraclassical, but non-trivial consequence operations, defined in a propositional language. We extend some paradigm of tableau methods, in which classical consequence Cn is defined, to stronger logics - monotonic, as well as non-monotonic ones - by specific use of non-classical tableau rules. So far, in that context tableaus have been treated as a way of formalizing other approaches to supraclassical logics, but we use them autonomically to generate various consequence operations. It requires a description of the hierarchy of non-classical tableau rules that result in different supraclassical consequence operations, so we give it.

On the Modal μ-Calculus Over Finite Symmetric Graphs

In this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.

ALGEBRAIC GEOMETRIC INVARIANTS OF PARAFREE GROUPS

Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL2ℂ inherits the structure of an algebraic variety known as the representation variety of G in SL2ℂ. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that are not free share a host of properties with free groups. In this paper algebraic geometric invariants involving the number of maximal irreducible components (mirc) of R(G), and the dimension of R(G) for certain classes of parafree groups are computed. It is shown that in an infinite number of cases these invariants successfully discriminate between ismorphism types within the class of parafree groups of the same rank. This is quite surprising, since an n generated group G is free of rank n if and only if Dim (R(G)) = 3n. In fact, a trivial consequence of Theorem 1.8 in this paper is that given an arbitrary positive integer k, and any integer r ≥ 2, there exist infinitely many non-isomorphic fg parafree groups of rank r and deviation 1 with representation varieties of dimension 3r, having more than k mirc of dimension 3r. This paper also introduces many novel and surprising dimension formulas for the representation varieties of certain one-relator groups.

Kinematic models cannot provide insight into motor control

In Plamondon & Alimi's target article, a bell-shaped velocity profile typically observed in fast movements is used as a basis for the “kinematic theory” of motor control. In our opinion, kinematics is a necessary but insufficient ground for a theory of motor control. Relationships between different kinematic characteristics are an emergent property of the system dynamics controlled by the brain in a specific way. In particular, bell-shaped velocity profiles with or without additional waves are a trivial consequence of shifts in the equilibrium state of the system as suggested, for example, in the λ-model of motor control.