what can be said and what can only be shown. It is here, for instance, that he first distinguishes between material and grammatical propositions, noting: A philosophical treatise attempts to say something where nothing can properly be said. It is predicated upon the idea that philosophy should be pursued in a way analogous to the natural sciences; that philosophers are looking to construct true theories. This sense of philosophy does not coincide with Wittgenstein's conception of philosophy. Wittgenstein is to be credited with the popularization of truth tables (4.31) and truth conditions (4.431) which now constitute the standard semantic analysis of first-order sentential logic. The philosophical significance of such a method for Wittgenstein was that it alleviated a confusion, namely the idea that logical inferences are justified by rules. If an argument form is valid, the conjunction of the premises will be logically equivalent to the conclusion and this can be clearly seen in a truth table; it is displayed. The concept of tautology is thus central to Wittgenstein's Tractarian account of logical consequence, which is strictly deductive. Proposition 6.N At the beginning of Proposition 6, Wittgenstein postulates the essential form of all sentences. He uses the notation , where stands for all atomic propositions, stands for any subset of propositions, and stands for the negation of all propositions making up . Proposition 6 says that any logical sentence can be derived from a series of NOR operations on the totality of atomic propositions. Wittgenstein drew from Henry M. Sheffer's logical theorem making that statement in the context of the propositional calculus. Wittgenstein's N-operator is a broader infinitary analogue of the Sheffer stroke, which applied to a set of propositions produces a proposition that is equivalent to the denial of every member of that set. Wittgenstein shows that this operator can cope with the whole of predicate logic with identity, defining the quantifiers at 5.52, and showing how identity would then be handled at 5.53–5.532. The subsidiaries of 6. contain more philosophical reflections on logic, connecting to ideas of knowledge, thought, and the a priori and transcendental. The final passages argue that logic and mathematics express only tautologies and are transcendental, i.e. they lie outside of the metaphysical subject's world. In turn, a logically "ideal" language cannot supply meaning, it can only reflect the world, and so, sentences in a logical language cannot remain meaningful if they are not merely