Gentzen’s and Jaśkowski’s formulations of natural deduction are logically equivalent in the normal sense of those words. However, Gentzen’s formulation more straightforwardly lends itself both to a normalization theorem and to a theory of “meaning” for connectives . The present paper investigates cases where Jaskowski’s formulation seems better suited. These cases range from the phenomenology and epistemology of proof construction to the ways to incorporate novel logical connectives into the language. We close with a demonstration of this latter aspect by (...) considering a Sheffer function for intuitionistic logic. (shrink)

Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Jaśkowski in 1934. This article traces the development of natural deduction from the view that these founders embraced to the widespread acceptance of the method in the 1960s. I focus especially on the different choices made by writers of elementary textbooks—the standard conduits of (...) the method to a generation of philosophers—with an eye to determining what the ‘essential characteristics’ of natural deduction are. (shrink)

There is widespread belief that connectionist networks are dramatically different from classical or symbolic models. However, connectionists rarely test this belief by interpreting the internal structure of their nets. A new approach to interpreting networks was recently introduced by Berkeley et al. (1995). The current paper examines two implications of applying this method: (1) that the internal structure of a connectionist network can have a very classical appearance, and (2) that this interpretation can provide a cognitive theory that cannot be (...) dismissed as a mere implementation. (shrink)

The argument from design is one of the most widely debated arguments for the existence of God. There has been much written in support of and in criticism of the argument's basic structure and conclusion. I shall attempt to clarify these positions, and to argue that the theistic account provides a more rationally justified explanation of human life on earth than the atheistic account. Many philosophers think that any proof for the existence of God is mere "metaphysical speculation." Many times (...) these philosophers use the criteria of scientific empiricism as the standard for an "acceptable" scientific theory, regardless of the subject matter. I shall formulate the argument from design as an empirical scientific theory according to the school of scientific falsificationism as stated by Karl Popper. The principal literature to be investigated in this study will be the most recent articles and texts which discuss relevant scientific data, and their philosophical implications for the argument from design. The results of this investigation demonstrate that it is possible to formulate an argument for the existence of God which meets all of the criteria of the falsificationists school of scientific empiricism. (shrink)

The power of Wittgenstein's N operator described in the Tractatus is that every proposition which can be expressed in the Russellian variant of the predicate calculus familiar to him has an equivalent proposition in an extended variant of his N operator notation. This remains true if the bound variables are understood in the usual inclusive sense or in Wittgenstein's restrictive exclusive sense. The problematic limit of Wittgenstein's N operator comes from his claim that symbols alone reveal the logical status of (...) propositions. This would require knowledge of the size of the Tractarian domain of unchangeable, simple objects, and this, Wittgenstein maintained, cannot be known. (shrink)

We present the design goals and metatheory of the Monadic Hybrid Calculus, a new formal system that has the same power as the Monadic Predicate Calculus. MHC allows quantification, including relative quantification, in a straightforward way without the use of bound variables, using a simple adaptation of modal logic notation. Thus “all Greeks are mortal” can be written as [G]M. MHC is also ‘hybrid’ in that it has individual constants, which allow us to formulate statements about particular individuals. Thus “Socrates (...) is Athenian and mortal” can be formalised as s.For our proof system, we use a simple adaptation of Beth-style tableaus. The availability of individual constants eliminates the need for labelled deduction.We discuss first the pragmatic and pedagogical advantages of MHC. Then we present the metatheory: formal syntax, semantics, proof rules, soundness, completeness and expressive equivalence with the Monadic Predicate Calculus. (shrink)