The logician recognizes the following propositional forms as necessary and sufficient for the expression of any truth.

It is shown that a formula of modal propositional logic has precisely the same models as a sentence of the first-order language of a single dyadic predicate iff its class of models is closed under ultraproducts. as a corollary, any modal formula definable by a set of first-order conditions is always definable by a single such condition. these results are then used to show that the formula (lmp 'validates' mlp) is not first-order definable.

It is proposed in this paper to develop a method by which the most general problem of the algebra of propositions is solved. This problem is to construct all propositions whose truth is independent of the form of the variables. As might be expected this method will enable us to determine without the use of matrices the consistency and independence of propositions, except in the case of those fundamental properties, which taken together define consistency itself. In the discussion which follows (...) no limitations are placed on the meaning of the terms and the word proposition is used simply in the sense of a variable. If we were to say, for example, that our variables must be restricted to zero or one values, we would be introducing a limitation which would reduce the generality of our science. A logic may be false in the sense that its postulates lead to contradictory theorems, or it may be false in the sense that it lacks generality, and the situation is not saved by calling it a two-valued, or a three-valued logic, or a logic of “propositions,” or by designating it by some other euphuism. (shrink)