These long-awaited, well-edited volumes complete the projected ten-volume edition, six volumes of which appeared in 1931-5. Volume VII contains, among other things, the important but previously unpublished "On the Logic of drawing History from Ancient Documents," "The Association of Ideas," and "Habit," as well as a sharp criticism of telepathy. Volume VIII reprints Peirce's major reviews of such works as Frazer's Berkeley, Royce's The World and the Individual, and Pearson's Grammar of Science, and contains some of his correspondence with pivotal (...) figures such as Carus, Dewey, James, and Lady Welby. The bibliography is superb.--P. W. (shrink)
The psychologist-philosopher B.F. Skinner and the physicist-philosopher P.W. Bridgman, both dedicated empiricists, initially entered into an intellectual relationship that seemed destined to be warm and fruitful. Yet, it ended up unfulfilled. Since I am now perhaps one of the few who knew both men as colleagues for many years, I might be able to throw some unique light on their interaction, and on what I consider to be one of the missed opportunities in the history of ideas.
The syntactic structure of the system of pure implicational relevant logic P - W is investigated. This system is defined by the axioms B = (b → c) → (a → b) → a → c, B' = (a → b) → (b → c) → a → c, I = a → a, and the rules of substitution and modus ponens. A class of λ-terms, the closed hereditary right-maximal linear λ-terms, and a translation of such λ-terms M to BB'I-combinators (...) M + is introduced. It is shown that a formula α is provable in P - W if and only if α is a type of some λ-term in this class. Hence these λ-terms represent proof figures in the Natural Deduction version of P - W. Errol Martin (1982) proved that no formula with form α → α is provable in P - W without using the axiom I. We show that a β-normal form λ-term M in the class is η reducible to λ x.x if the translated BB'I-combinator M + contains I. Using this theorem and Martin's result, we prove that a λ-term in the class is βη-reducible to λ x.x if the λ-term has a type α → α. Hence the structure of proofs of α → α in P - W is determined. (shrink)
The logical system P-W is an implicational non-commutative intuitionistic logic defined by axiom schemes B = (b → c) → (a → b) → a → c, B' = (a → b) → (b → c) → a → c, I = a → a with the rules of modus ponens and substitution. The P-W problem is a problem asking whether α = β holds if α → β and β → α are both provable in P-W. The answer is (...) affirmative. The first to prove this was E. P. Martin by a semantical method. In this paper, we give the first proof of Martin's theorem based on the theory of simply typed λ-calculus. This proof is obtained as a corollary to the main theorem of this paper, shown without using Martin's Theorem, that any closed hereditary right-maximal linear (HRML) λ-term of type α → α is βη-reducible to λ x.x. Here the HRML λ-terms correspond, via the Curry-Howard isomorphism, to the P-W proofs in natural deduction style. (shrink)
The syntactic structure of the system of pure implicational relevant logic $P - W$ is investigated. This system is defined by the axioms $B = (b \rightarrow c) \rightarrow (a \rightarrow b) \rightarrow a \rightarrow c, B' = (a \rightarrow b) \rightarrow (b \rightarrow c) \rightarrow a \rightarrow c, I = a \rightarrow a$, and the rules of substitution and modus ponens. A class of $\lambda$-terms, the closed hereditary right-maximal linear $\lambda$-terms, and a translation of such $\lambda$-terms $M$ to $BB'I$-combinators (...) $M^+$ is introduced. It is shown that a formula $\alpha$ is provable in $P - W$ if and only if $\alpha$ is a type of some $\lambda$-term in this class. Hence these $\lambda$-terms represent proof figures in the Natural Deduction version of $P - W$. Errol Martin (1982) proved that no formula with form $\alpha \rightarrow \alpha$ is provable in $P - W$ without using the axiom $I$. We show that a $\beta$-normal form $\lambda$-term $M$ in the class is $\eta$ reducible to $\lambda x.x$ if the translated $BB'I$-combinator $M^+$ contains $I$. Using this theorem and Martin's result, we prove that a $\lambda$-term in the class is $\beta\eta$-reducible to $\lambda x.x$ if the $\lambda$-term has a type $\alpha \rightarrow \alpha$. Hence the structure of proofs of $\alpha \rightarrow \alpha$ in $P - W$ is determined. (shrink)
This is a commissioned review of Copan, P. and Craig, W. The Kalām Cosmological Argument Volume Two: Scientific Evidence for the Beginning of the Universe New York: Bloomsbury, US$172.50, ISBN 978-1-50-133587-7.
We describe new results in parametrized complexity theory. In particular, we prove a number of concrete hardness results for W[P], the top level of the hardness hierarchy introduced by Downey and Fellows in a series of earlier papers. We also study the parametrized complexity of analogues of PSPACE via certain natural problems concerning k-move games. Finally, we examine several aspects of the structural complexity of W [P] and related classes. For instance, we show that W[P] can be characterized in terms (...) of the DTIME ) and NP. (shrink)
In “Imagination and Judgment” W.P. Ker argues, contrary to the “ordinary teaching” of the moralists of his day, that we have good reason to consider imagination as “the highest form of practical wisdom or prudence” (475). Modes of imaginative thought that direct human passion towards morally valuable ends are best understood as a form of reason or an intellectual virtue, as opposed to a dangerous distraction from reality and threat to good judgment. Ker’s piece remains of interest partly because it (...) anticipates some of the most important contributions to moral theory made by philosophers, most notably Iris Murdoch and Martha Nussbaum, who have developed conceptions of ‘moral imagination’ in more recent decades. More significantly, reflecting on Ker’s catalogue of the positive and direct roles played by imagination in moral reasoning reveals that there is further work to be done in clarifying the concept of imagination in relation to practical reason. (shrink)