Neil Tennant

The system of natural deduction that originated with Gentzen , and for which Prawitz proved a normalization theorem, is re-cast so that all elimination rules are in parallel form. This enables one to prove a very exigent normalization theorem. The normal forms that it provides have all disjunction-eliminations as low as possible, and have no major premisses for eliminations standing as conclusions of any rules. Normal natural deductions are isomorphic to cut-free, weakening-free sequent proofs. This form of normalization theorem renders unnecessary Gentzen's resort to sequent calculi in order to establish the desired metalogical properties of his logical system.Ultimate normal forms are well-adapted to the needs of the computational logician, affording valuable constraints on proof-search. They also provide an analysis of deductive relevance. There is a deep isomorphism between natural deductions and sequent proofs in the relevantized system

Peacocke argues for a ‘generalized rationalism’, holding that ‘all entitlement has a fundamentally a priori component.’ (2) But his rationalism ‘differs from those of Frege and Gödel, just as theirs differ from that of Leibniz.’ He requires both substantive theories of intentional content and of understanding, and systematic formal theories of referential semantics and truth. We need an externalist theory of content: ‘Only mental states with externally individuated contents can make judgements about the external, mind-independent world rational.’ (123) Purely evidential conceptions of meaning and content are inadequate. (34-49) They cannot account for the following: a thinker often has to work out what would be evidence for a content; contents cannot depend, for their identity, on all of the infinitely ramifying evidential connections among them; and thinkers conceive, however tacitly, of (at least some) observed properties as categorical. By contrast with an evidential theory, a truthconditional theory of content can account for all these problematic facts. Peacocke states, develops and defends three principles of rationalism which collectively ‘relate entitlement to truth, to the identity of states and their intentional contents, and to the a priori.’ (3-4) He does not thoroughly explain his central notion of entitlement, but this much is clear: any thinker is entitled to various transitions in, or into, thought. An example of a transition into thought would be that from one’s perceptual experience to an observational judgment. An example of a transition in thought would be a logical inference from certain premises to a conclusion. A transition is rational just in case the thinker is entitled to it. (Note that this aims to explain rationality in terms of entitlement, not the other way round.) It is clear from 28 that Peacocke needs an abstract ontology of entitlements (such as proofs, in the case of mathematics). Yet he does not endorse ‘Gödel’s obscure quasi-perceptual and quasi-causal epistemology of mathematics and the abstract sciences.’ (54..

This study continues the anti-realist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the anti-realist’s knowability principle ‘ϕ, therefore 3Kϕ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of ϕ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before applying the proposed restriction. A similar restriction is proposed for the converse inference, the so-called Rule of Factiveness ‘3Kϕ therefore ϕ’. The proposed restriction appears to block another Fitch-style derivation that uses the KK -thesis in order to get around..