Investigations into Entailment and Knowledge

Abstract

The concept of semantic consequence, or briefly entailment, is of basic importance to logic. Chapter 1 is devoted to a generalization of the concept. A semantic relation between a family of sets of propositions and a set of propositions, dubbed generalized entailment, is defined and examined. The underlying intuition is: if each set in the family contains a true proposition, then the entailed set contains a true proposition. The above condition gains intuitive plausibility when one thinks of sets of propositions as representing search spaces rather than knowledge bases. Single- and multiple-conclusion entailments are definable in terms of generalized entailment. Chapter 2 is devoted to some special case of generalized entailment, dubbed constructive generalized entailment.

Chapter 3 presents a logical calculus which offers a uniform proof-mechanism for proofs of valid formulas, refutations of inconsistent formulas, and refutations of contingent formulas.

Chapter 4 takes under scrutiny two basic approaches to the nature of propositional knowledge, namely the knowledge as true belief “plus something else” account and the knowledge as true conviction account. Three, so far unnoticed, problems/paradoxes concerning knowledge conceived these ways are pointed out and discussed. Chapter 5, in turn, proposes a change in perspective. The concept of being epistemically permitted is introduced. In general, being epistemically permitted is different from being epistemically possible and from being known; the latter concepts can be defined in terms of the former, however. A relation akin to entailment, dubbed transmission of epistemic permittance, is then defined and analysed.

In Chapters 6 and 7 two concepts of single- and multiple-conclusion entailment, based on the idea of minimality, are introduced and studied. The analysed entailments, dubbed “strong,” have some intuitively plausible properties which their standard counterparts lack. Chapter 8 is devoted to the issue of emplacement of strong entailments among alternative proposals, including classical, relevant and connexive stances. Chapter 9 presents an application of the concept of strong single-conclusion entailment in the area of belief revision. Some proof-theoretic accounts of strong entailments are offered in Chapter 10. Chapter 11 briefly indicates how the formal apparatus introduced in the previous chapter can be employed in analysing further issues.