By Sara Negri

This e-book keeps from the place the authors' past booklet, Structural evidence conception, ended. It offers an extension of the tools of study of proofs in natural good judgment to undemanding axiomatic structures and to what's referred to as philosophical common sense. A self-contained short advent to the facts concept of natural good judgment is incorporated that serves either the mathematically and philosophically orientated reader. the tactic is outfitted up steadily, with examples drawn from theories of order, lattice conception and uncomplicated geometry. the purpose is, in all the examples, to aid the reader seize the combinatorial behaviour of an axiom procedure, which usually results in decidability effects. The final half provides, as an software and extension of all that precedes it, a proof-theoretical method of the Kripke semantics of modal and similar logics, with a good number of new effects, supplying crucial interpreting for mathematical and philosophical logicians.

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C⊃D C⊃D C⊃D E. . ∨E .. C [D. l ] .. E ⊃E After the permutation conversion the part is .. m ] .. C⊃D E l .. [D.. ] .. C E ⊃E E. . n ] .. C⊃D E l .. [D.. ] .. C E ⊃E ∨E Finally, we have permutation convertibilities in which the conversion formula is ⊥ derived by ⊥E . 3. A simplification convertibility in a derivation is an instance of an E-rule with no discharged assumptions, or an instance of ∨E with no discharges of at least one disjunct. As with permutation conversions, simplification conversions also apply to all E-rules when general elimination rules are used.

B A detour convertibility on disjunction is quite similar. m ] .. B A ⊃B ⊃I C. . .. n ] .. C ⊃E .. m× ... .. B .. A .. m× ... .. n× .. .. C. . .. A There is no I-rule for ⊥, so no detour convertibility either. 2. An E-rule with a major premiss derived by an E-rule is a permutation convertibility. The novelty of general elimination rules is that permutation conversions apply to all cases in which a major premiss of an E-rule has been derived. With six E-rules, this gives 36 convertibilities of which we show a couple: 23 24 Proof systems based on natural deduction A permutation convertibility on major premiss C&D derived by &E on A&B and its conversion are ..

The number of rules drops down to four instead of six (plus the two of partial order). Moreover, the subterm property has an almost immediate proof. We consider also a formulation of strict order with eigenvariable rules, which permits the introduction (in a literal sense) of a relation of equality. A normal form for derivations and some of its consequences such as the conservativity of strict order with equality over the strict partial order fragment and the subterm property are shown. 1 Order relations (a) Partial order.

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