Informal natural deduction 3. Response to your second question given in an edit. He has published a monograph on lamda-trees, which are generalisations of ordinary trees. Rather too much of a good thing? This blog has now moved Go to logicmatters.
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First, propositional logic. Then we get the quantifier-free part of first-order logic, dealing with properties and relations, functions, and identity.
So at this second stage we get the idea of an interpretation, of truth-in-a-structure, and we get added natural deduction rules for identity and the handling of the substitution of terms. At both these first two stages we get a Hintikka-style completeness proof for the given natural deduction rules. Only at the third stage do quantifiers get added to the logic and satisfaction-by-a-sequence to the semantic apparatus.
This does make for a great gain in accessibility. The really cute touch is to introduce the idea of polynomials and diophantine equations early — in fact, while discussing quantifier-free arithmetic — and to state without proof! This is all done with elegance and a light touch — not to mention photos of major logicians and some nice asides — making an admirably attractive introduction to the material. Rather too much of a good thing? Still, you can easily skim and skip. After Ch.
The presentation of the formal natural deduction system is not exactly my favourite in its way of graphically representing discharge of assumptions I fear that some readers might be puzzled about vacuous discharge and balk at Ex.
After a short interlude, Ch. The treatment of the semantics without quantifiers in the mix to cause trouble is very nice and natural; likewise at the syntactic level, treatment of substitution goes nicely in this simple context.
Again we get a soundness and Hintikka-style completeness proof for an appropriate natural deduction system. Adding natural deduction rules on the syntactic side and a treatment of satisfaction-by-finite-n-tuples on the semantic side all now comes very smoothly after the preparatory work in Ch.
The Hintikka-style completeness proof for the new logic builds very nicely on the two earlier such proofs: this is about as accessible as it gets in the literature, I think. But the core key sections on soundness and completeness proofs and associated metalogical results are second to none for their clarity and accessibility.
CHISWELL HODGES MATHEMATICAL LOGIC PDF