Francois Lepage

This paper is composed of two independent parts. The first is concerned with Russell’s early philosophy of mathematics and his quarrel with Poincaré about the nature of their opposition. I argue that the main divergence between the two philosophers was about the nature of definitions. In the second part, I briefly present Le!niewski’s Ontology and suggest that Le!niewski’s original treatment of definitions in the foundations of mathematics is the natural solution to the problem that divided Russell and Poincaré

The aim of this paper is to present a strongly complete first order functional predicate calculus generalized to models containing not only ordinary classical total functions but also arbitrary partial functions. The completeness proof follows Henkin’s approach, but instead of using maximally consistent sets, we define saturated deductively closed consistent sets . This provides not only a completeness theorem but a representation theorem: any SDCCS defines a canonical model which determine a unique partial value for every predicate symbol and any function symbol. Any SDCCS can thus be interpreted as an epistemic state

This paper has four parts. In the first part, I present Leniewski's protothetics and the complete system provided for that logic by Henkin. The second part presents a generalized notion of partial functions in propositional type theory. In the third part, these partial functions are used to define partial interpretations for protothetics. Finally, I present in the fourth part a complete system for partial protothetics. Completeness is proved by Henkin's method [4] using saturated sets instead of maximally saturated sets. This technique provides a canonical representation of a partial semantic space and it is suggested that this space can be interpreted as an epistemic state of a non-omniscient agent.