By Heinrich Rickert

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CONTENTS
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Preface
CHAPTER ONE. fundamentals FROM ALGEBRA AND TOPOLOGY
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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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