We present infinite extensive strategy profiles with perfect information and we show that replacing finite by infinite changes the notions and the reasoning tools. The presentation uses a formalism recently developed by logicians and computer science theoreticians, called coinduction. This builds a bridge between economic game theory and the most recent advance in theoretical computer science a...

In this paper, we propose a unified framework for the syntactico-semantic learning of natural languages. This framework integrates recent results on grammatical inference with positive structural examples and an algebraic characterization of the Principle of Compositionality, introduced by linguists and logicians. The purpose is to give account of how children learn to understand their native l...

Logicians frequently use axiom schemata to encode (potentially infinite) sets of sentences with particular syntactic form. In this paper we examine a first-order language in which it is possible to write expressions that both describe sentences and assert the truth of the sentences so described. The effect of adding such expressions to a knowledge base is the same as directly including the set ...

A formalization of Gödel’s incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows Świerczkowski (2003), who gave a detailed proof using hereditarily finite set theory. The adoption of this theory is generally beneficial, but it poses certain technical issues that...

In this paper, we show that different semantics of ability give rise to different validity sets. The issue is important for several reasons. First, many logicians identify a logic with its set of true sentences. As a consequence, we prove that different notions of ability induce different logics in the traditional sense. Secondly, the study can be seen as the first systematic step towards satis...

Many contemporary logicians acknowledge a plurality of logical theories and accept that theory choice is in part motivated by evidence. However, just as there no agreement on theories, also consensus what constitutes In this paper, I outline Jean Piaget’s psychological reasoning show how he used it to diagnose solve one the paradoxes material implication. assess use psychology source evidence f...

In a definition (x)((xr)D[x]) of the set r, the definiens D[x]must not depend on the definiendum r. This implies that all quantifiers in D[x] are independent of r and of (x). This cannot be implemented in the traditional first-order logic, but can be expressed in IF logic. Violations of such independence requirements are what created the typical paradoxes of set theory. Poincaré's Vicious Circl...

A semi-batch crystallization system is used to determine growth kinetics and morphology of barium carbonate crystals, under a constant pH value and constant temperature in the various relative supersaturations. Results show that crystal growth rate satisfies a first-order kinetic expression at high relative supersaturation, while at relative supersaturation values lower than 3,  the k...

Two things are striking about this medieval definition. The first is that it is very close to the modern conception of formal consequence one finds in Bolzano and Tarski3. The second is that although Buridan and the other fourteenth-century logicians state these distinctions with a great deal of precision, they say almost nothing about the point of the distinctions. Why are these distinctions d...