The foundations of logic should be obvious and compelling. They should at least explain why we may not deny the tautologies of this or that logical system, if we are to have a rational systen of beliefs. I believe that standard semantics helps to explain why the tautologies of the sentential calculus and lower predicate calculus should not be rejected but does little to explain why those of higher order predicate calculus and of modal sentential and predicate calculi may not be denied. I do not propose to argue here that this is so. I wish only to show how a foundation of a different kind can succeed in doing simply and elegantly what I claim standard semantics fails to do. By standard semantics I mean the kind of semantics which can be found in almost any textbook on logic in which the logical connectives and operators of a formally (syntactically) defined language-structure are defined in terms of truth and falsity conditions. I shall call any such foundation for a logical system a truth semantics. The need for a truth semantics derives from the standard concept of validity of argument according to which an argument is valid iff there is no interpretation of its non-logical terms, (or alternatively, no possible world) in which its premises are true and its conclusion false. For it follows from this definition that to understand any sentence sufficiently for all of the purposes of logic we need to know under what interpretations (or in what possible worlds) it would be true or false. I proceed from a different concept of validity, which leads to a different programme of analysis, and hence to a different kind of foundation for logical systems. I consider an argument in a given language to be valid iff there is no rational belief system on that language in which its premises are accepted and its conclusion rejected. Thus, for me, validity is an epistemic notion, and my problem is to define a rational belief system on a language, and to specify acceptance and rejection conditions for the sentences of that language. I begin in the usual way be defining a language-structure syntactically, so that the sentences, operators, connectives, etc. of the language-structure can