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Download Decidable Theories II by J.R. Büchi, D. Siefkes, D. Siefkes, G.H. Müller PDF

By J.R. Büchi, D. Siefkes, D. Siefkes, G.H. Müller

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Extra info for Decidable Theories II

Sample text

It is alphabetical order, thus, that helps us pair up the variables across conjunctions. The difference bet\veen 'x conquered y' and 'y conquered z' is after all not gratuitous, when you think how differently they fare in conjunction with some further clause such as 'z killed x'. We arrived at a compact statement of the satisfaction conditions of negations and conjunctions, relative t~ their constituents. The negation is satisfied by just the sequences that its constituent is not satisfied by, and the conjunction is satisfied by just the sequences that its constituents are both satisfied by.

Logical grammar Grammar 23 The rest of the gran1n1ar consists of further gran1n1a tical con~ structions. One of these is predication of a one-place predicate. It consists in joining such a predicate, perhaps the verb 'walks', and a variable to forn1 a sentence: 'x walks'. The result is an atol11ic sentence, in the sense of containing no subordinate sentence. Also it is an open sentence, because of the variable. It is true for certain values of the variable, namely, those that walk, and false for other values, but of itself it is neither true nor false; such is the way of an open sentence.

Predicates are wanted in all sentences regardless of whether there are attributes to refer to, and the dummy predicate 'F' is wanted generally for expository purposes without thought of its being a quantifiable variable taking attributes as values. If we are also going to quantify over attributes and refer to them, then clarity is served by using recognizable variables and distinctive names for the purpose and not mixing these up with the predicates. The distinction between lexicon and particles is not, I have twice said, a distinction between names and other words.