Second-order propositional modal logic: expressiveness and completeness results

In this paper we advance the state-of-the-art on the application of second-order propositional modal logic (SOPML) in the representation of individual and group knowledge, as well as temporal and spatial reasoning. The main theoretical contributions of the paper can be summarised as follows. Firstly, we introduce the language of (multi-modal) SOPML and interpret it on a variety of different classes of Kripke frames according to the features of the accessibility relations and of the algebraic structure of the quantification domain of propositions. We provide axiomatisations for some of these classes, and show that SOPML is unaxiomatisable on the remaining classes. Secondly, we introduce novel notions of (bi)simulations and prove that they indeed preserve the interpretation of formulas in (the universal fragment of) SOPML. Then, we apply this formal machinery to study the expressiveness of Second-order Propositional Epistemic Logic (SOPEL) in representing higher-order knowledge, i.e., the knowledge agents have about other agents’ knowledge, as well as graph-theoretic notions (e.g., 3-colorability, Hamiltonian paths, etc.). The final outcome is a rich formalism to represent and reason about relevant concepts in artificial intelligence, while still having a model checking problem that is no more computationally expensive than that of the less expressive quantified boolean logic

Tarski’s Practice and Philosophy: Between Formalism and Pragmatism: What has Become of Them?

Collection : Synthese Library, n°341International audienceConsidering works by Tarski, the author claims that Tarski’s fundamental aim was to establish formal semantics as a new branch of mathematics

Two-sorted Frege Arithmetic is not Conservative

Neo-Fregean logicists claim that Hume's Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A longstanding problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck's Two-sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn't. In fact, 2FA is not conservative over $n$-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic

Frege's Theory of Real Numbers: A consistent Rendering

Frege's definition of the real numbers, as envisaged in the second volume of \textit{Grundgesetze der Arithmetik}, is fatally flawed by the inconsistency of Frege's ill-fated \textit{Basic Law V}. We restate Frege's definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege's own indications is possible at all

Axiomatic set theory as a basis for the construction of mathematics

It is widely known that one of the major tasks of 'Foundations' is to construct a formal system which can he said to contain the whole of mathematics. For various reasons axiomatic set theory is a very suitable choice for such a system and it is one which has proved acceptable to both logicians and mathematicians. The particular demands of mathematicians and logicians, however, are not the same. As a result there exist at the moment two different formulations of set theory which can be roughly said to cater for logicians and mathematicians respectively. It is these systems which are the subject of this dissertation. The system of set theory constructed for logicians is by P. Bernays. This will be discussed in chapter II. For mathematicians No Bourbaki has constructed a system of set theory within which he has already embedded a large part of mathematics. This system will be discussed in chapter III. Chapter I is historical and contains some of Cantor's original ideas. The relationship between Bernays' system and (essentially) Bourbaki's system is commented upon in chapter IV. <p