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[QUOTE]blah blah blah[/QUOTE] to reply to BobNOMAAMRooney nli.
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[QUOTE="BobNOMAAMRooney%20nli:686071"]One sometimes says that Tarski's definition of satisfaction is compositional, meaning that the class of assignments which satisfy a compound formula F is determined solely by (1) the syntactic rule used to construct F from its immediate constituents and (2) the classes of assignments that satisfy these immediate constituents. (This is sometimes phrased loosely as: satisfaction is defined recursively. But this formulation misses the central point, that (1) and (2) don't contain any syntactic information about the immediate constituents.) Compositionality explains why Tarski switched from truth to satisfaction. You can't define whether ‘For all x, G’ is true in terms of whether G is true, because in general G has a free variable x and so it isn't either true or false. The name ‘compositionality’ is from a paper of Katz and Fodor in 1963 on natural language semantics. In talking about compositionality, we have moved to thinking of Tarski's definition as a semantics, i.e. a way of assigning ‘meanings’ to formulas. (Here we take the meaning of a sentence to be its truth value.) Compositionality means essentially that the meanings assigned to formulas give at least enough information to determine the truth values of sentences containing them. One can ask conversely whether Tarski's semantics provides only as much information as we need about each formula, in order to reach the truth values of sentences. If the answer is yes, we say that the semantics is fully abstract (for truth). One can show fairly easily, for any of the standard languages of logic, that Tarski's definition of satisfaction is in fact fully abstract. As it stands, Tarski's definition of satisfaction is not an explicit definition, because satisfaction for one formula is defined in terms of satisfaction for other formulas. So to show that it is formally correct, we need a way of converting it to an explicit definition. One way to do this is as follows, using either higher order logic or set theory. Suppose we write S for a binary relation between assignments and formulas. We say that S is a satisfaction relation if for every formula G, S meets the conditions put for satisfaction of G by Tarski's definition. For example, if G is ‘G1 and G2’, S should satisfy the following condition for every assignment a: S(a,G) if and only if S(a,G1) and S(a,G2). We can define ‘satisfaction relation’ formally, using the recursive clauses and the conditions for atomic formulas in Tarski's recursive definition. Now we prove, by induction on the complexity of formulas, that there is exactly one satisfaction relation S. (There are some technical subtleties, but it can be done.) Finally we define a satisfies F if and only if: there is a satisfaction relation S such that S(a,F). It is then a technical exercise to show that this definition of satisfaction is materially adequate. Actually one must first write out the counterpart of Convention T for satisfaction of formulas, but I'm moving with my auntie and uncle in Bel-Air. I whistled for a cab, and when it came near, the license plate said "fresh" and there was dice in the mirror. If anything I could say that this cab was rare, but I thought "naw forget it, yo homes to Bel-Air!" I pulled up to the house about seven or eight and I yelled to the cabbie "yo homes smell ya later!" Looked at my kingdom, I was finally there. To sit on my throne as the prince of bel-air.[/QUOTE]
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