Still, it would certainly be a nice bonus if some modified version of the strong conditional could serve as one. Finally, I suspect the strong conditional will be of more use for logic rather than the philosophy of language, and I will make no claim that the strong conditional is a good model for any particular use of the indicative conditional in English or other natural languages. ) with more than one conditional, and it may be that no single conditional will satisfy all of our intuitions about how a conditional should behave. In any case, one can always augment one’s language (. My conditional has some counterintuitive consequences of its own, but I think its pros outweigh its cons. It is stronger than the material conditional and is not subject to the standard paradoxes of material implication, nor is it subject to some of the standard paradoxes of C. In this article I define a strong conditional for classical sentential logic, and then extend it to three non-classical sentential logics. Since not every derivation of a sequent can be permuted into every other derivation of that sequent, proof terms provide a non-trivial account of the identity of proofs, independent of the syntactic representation of those proofs. Further- more, proof terms are invariants for sequent derivations in a strong sense-two derivations δ1 and δ2 have the same proof term if and only if some permutation of derivation steps sends δ1 to δ2 (given a relatively natural class of permutations of derivations in the sequent calculus). However, unlike cut elimination in the sequent calculus, each proof term has a unique normal form (from which all cuts have been eliminated) and it is straightforward to show that term reduction is strongly normalising-every reduction process terminates in that unique normal form. - Proof terms can be simplified in a process corresponding to the elimination of cut inferences in sequent derivations. There may be different ways to connect those premises and conclusions. However, not all proof terms for a sequent Σ≻Δ (. This encoding is many–to–one in the sense that different derivations can have the same proof term, since different derivations may be different ways of representing the same underlying connection between premises and conclusions. The term for a derivation δ of a sequent Σ≻Δ encodes how the premises Σ and conclusions Δ are related in δ. I give an account of proof terms for derivations in a sequent calculus for classical propositional logic. In selecting a book for classroom use, I recommend checking one thing: how much meta-theory is included, so that the book is neither below nor above the level students can handle. Instructors will have their own favorites. Meta-theoretical results for propositional logic are also generally classified as "proof theory," "model theory," "mathematical logic," etc.īecause of the age of propositional logic there are literally hundreds of introductions to logic which cover this subject reasonably well. a proposition is either true or false, not neither, and not both. Also appropriate here are modest extensions of propositional logic, provided that Boole's three laws of thought are not violated, viz. As such, non-standard propositional logics are not normally classified in this category-unless a comparison between classical logic and another logic is being drawn or one is reduced to the other-although restrictions of propositional logic in which nothing not a theorem in ordinary propositional logic is a theorem in the restriction do fit here. This leaf node is a sub-category of classical logic. The principle by which the meaning or truth conditions of compound propositions can be recovered by this "building up" process is known as compositionality. In classical propositional logic, molecular or compound propositions are built up from atomic propositions by means of the connectives, whose meaning is given by their truth tables. It ignores entirely the structure within propositions. Propositional logic is the simpler of the two modern classical logics.