Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid to which properties of theories result in the presence of which rules of inference, and to restrictions on the sets of formulas to which the rules may be employed, restrictions (...) determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stålmarck. (shrink)

Studying some familiar truth-conditions, I shall detail the role they play in elementary logic and inquire into our grounds for holding them true. I shall discharge the first of these assignments with a good deal of assurance, but the second with far less; to my mind, below mirrors our use of ‘and’, our use of ‘every’, our use of ‘necessarily’, etc. pretty accurately, but little evidence to that effect has ever been supplied, and — disturbingly enough — many take exception (...) to the conditions under study. (shrink)