By Ruy J G B De Queiroz, Anjolina G De Oliveira, Dov M Gabbay

This complete booklet offers an sufficient framework to set up numerous calculi of logical inference. Being an 'enriched' process of normal deduction, it is helping to formulate logical calculi in an operational demeanour. through uncovering a definite concord among a sensible calculus at the labels and a logical calculus at the formulation, it permits mathematical foundations for platforms of common sense presentation designed to address meta-level good points on the object-level through a labelling mechanism, comparable to the D Gabbay's Labelled Deductive structures. The ebook actually demonstrates that introducing 'labels' comes in handy to appreciate the proof-calculus itself, and in addition to explain its connections with model-theoretic interpretations.

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E. ) will be dealt with separately from (yet harmoniously with) the logical calculus on the formulas, by a calculus of functional expressions. Whereas the variables for individuals are handled ‘naturally’ in the interpretation of first-order logic with our labelled natural deduction, the introduction of variables to denote contexts, or possible-worlds (structured collection of labelled formulas), as in [de Queiroz and Gabbay (1997)], is meant to account for Montague’s senses. 2 Reassessing Frege’s two-dimensional calculus In our attempt to reassess the benefits of having those two sides working together, we would like to insist on the two sides being treated separately.

Here the labels (or proof constructions) take over the main r^ole of measuring instrument. e. those connectives whose elimination rules may violate the subformula property of a deduction, such as ∨, ∃, . =), but it also seems to retake Gentzen’s analysis of deduction in its more general sense. That is to say, the emphasis is put back into the deductive properties of the logical connectives, rather than on the truth of the constituent formulas. Remark. Notice that the mere condition of ‘elimination followed by introduction’ is not sufficient to allow us to perform an η-conversion.

In particular, when the natural deduction presentation system is based on a ‘labelling’ mechanism the binding of free variables in the labels corresponds to the discharge of respective assumptions. f (x)’) to bind free-variables and discharge the assumption labelled by the corresponding variable. e. the deduction rests on no assumption. In other words, a formula is a theorem if it admits a categorical proof to be constructed. 1 Dividing the tasks: A functional calculus on the labels, a logical calculus on the formula We have seen that the origins of variable binding mechanisms, both on the formulas of logic (the propositions) and on the expressions of the functional calculus (the terms), go back at least as far as Frege’s early investigations on a ‘language of concept writing’.

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