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By John N. Crossley (ed.)

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Extra resources for Sets, Models and Recursion Theory (Logic Colloquium'65)

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For example, if we did a truth table we would not get a tautology. FOR PRECISELY THIS REASON, I REINTRODUCED ARISTOTLE’S DISTINCTION BETWEEN SUBJECT AND PREDICATE – OBJECTS AND THE THINGS WE SAY ABOUT THEM – INTO MY LOGIC. This might be seen as making logic sensitive to the grammar of the sentences in an argument. NOT THE ACTUAL WORDS, BUT THE STRUCTURE OF THE SENTENCES BECOMES MIRRORED IN THE LOGICAL SYMBOLS. 48 Predicate Calculus In Russell’s Predicate Calculus, lowercase letters stand for objects: a, b, c … stand for specific-named objects and x, y, z stand for as yet unspecified objects.

I’M JUST GOING OUT TO THE SHOPS, CAN I GET YOU ANYTHING? OH, CAN YOU GET ME SOME GRAPES … … AND SOME SCOURING PADS … This continued re-application is called recursion and is vital for the construction of models. It allows us to construct an infinite number of sentences from a few simple rules and a finite vocabulary. 51 … AND SOME CORN FLAKES … … AND SOME BLEACH. 9 14 +1 Hilbert had a view of mathematics that he called formalism. The idea is that the things mathematics talks about are nothing but symbols.

The second rule shows how to build a new well-formed formula out of a sentence that already exists and in addition the sequence “which”, predicate, name. For example, “Homo erectus evolved into Homo sapiens which evolved into Homo habilis”. Using this model we can construct an infinite number of sentences via the recursive application of Rule Two. Of course only a few of these sentences will be true, but it should now be clear that this familiar diagram is yet another application of logic. 1917) has suggested that we can apply this idea to English and every other natural language, filling in the gaps with a semantic model.

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