On interpreting Gödel's Second Theorem

Summary In this paper I have considered various attempts to attribute significance to G2.²⁵ Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are literally false. Two others (BCR and Resnik's Interpretation), I have argued, are groundless.I would like to thank Dale Gottlieb, Stephen Barker, Tim McCarthy, Philip Kitcher, Michael Resnik and Richmond Thomason for extensive and helpful discussion of this work.     

The simple substitution property of Gödel's intermediate propositional logics S n 's

The simple substitution property provides a systematic and easy method for proving a theorem from the additional axioms of intermediate prepositional logics. There have been known only four intermediate logics that have the additional axioms with the property. In this paper, we reformulate the many valued logics S' _n defined in Gödel [3] and prove the simple substitution property for them. In our former paper [9], we proved that the sets of axioms composed of one prepositional variable do not have the property except two of them. Here we provide another proof for this theorem.     

Wittgenstein's picture‐theory of Language 1

I argue that the current view (as held by, eg., Warnock, Anscombe and Stenius) of Wittgenstein's theory of language in the Tractates is mistaken. This view maintains that Wittgenstein's theory is one of ‘isomorphism'; that, roughly, a sentence has meaning in virtue of its being a facsimile of a fact or possible fact. But a detailed study of significant passages in the Tractattis shows that Wittgenstein held no such view. His use of important terms, such as Salz, Bild, Sachverhalt, Tatsache, etc., has been crucially misunderstood. There is no isomorphism. The theory is not about sentences at all, but about talking sense.     

Wittgenstein's Temple: Or How Cool Is Philosophy?

How should we understand Wittgenstein's comment in 1929 that his ‘ideal’ was ‘a certain coolness’? Does it have the implication for the practice of philosophy that is suggested by the late Dewi Phillips? Wittgenstein's use of the metaphor of a temple in relation to the passions is curiously reminiscent in its structure of Rilke's first sonnet to Orpheus. In Zettel a similar preoccupation seems to be manifested in the long and unexpected passage that Wittgenstein copies out from Plato, a passage which is juxtaposed to the famous remark that the philosopher is the citizen of no community of ideas.     

Wittgenstein's picture‐theory

In a paper published in this journal, Vol. 5, No. 1, 1962, pp. 46–64, Mr. H. R. G. Schwyzer has argued that ‘the current view (as held by, eg., Wamock, Anscombe and Stemus) of Wittgenstein's theory of language in the Tractatus is mistaken’. The editor of the journal has asked me for a reply. My reply concerns only my own book, and it amounts to the statement that Mr. Schwyzer's attack on the book has very little to do with what is said in it.     

Is Wittgenstein's Resort to Ordinary Language an Appeal to Empirical facts?

There are two widely held views in the literature as regards Wittgenstein's philosophy. One says that Wittgenstein in his later work appeals to ordinary language in his effort to show how the philosophical problems can be dissolved, and the other says that his investigation is a grammatical one. This paper undertakes to examine what is meant by a grammatical investigation, especially in view of the fact that this investigation relies on empirical facts that have to do with linguistic usage. The examination is carried out by concentrating on what Wittgenstein has to say on the issue of knowledge – in particular, how the way we use the word contributes to the dismissal of Moore's answer to the challenge of scepticism. The conclusion is that Wittgenstein's resort to ordinary language is not typically empirical. The examples of ordinary usage that he cites may be contingent, but they could not have been different given the language games they are part of. The correct use of words Wittgenstein appeals to is not fixed by some kind of essence, but neither is it decided by a majority rule. It gets entrenched in a complex nexus of practices. Wittgenstein's reference to "use" instead of 'usage"and to "linguistic facts" instead of "sociological facts" lends support more to a logical than to an empirical investigation.     

Some Applications of Lawvere’s Fixpoint Theorem

The famous diagonal argument plays a prominent role in set theory as well as in the proof of undecidability results in computability theory and incompleteness results in metamathematics. Lawvere (1969) brings to light the common schema among them through a pretty neat fixpoint theorem which generalizes the diagonal argument behind Cantor’s theorem and characterizes self-reference explicitly in category theory. Not until Yanofsky (2003) rephrases Lawvere’s fixpoint theorem using sets and functions, Lawvere’s work has been overlooked by logicians. This paper will continue Yanofsky’s work, and show more applications of Lawvere’s fixpoint theorem to demonstrate the ubiquity of the theorem. For example, this paper will use it to construct uncomputable real number, unnameable real number, partial recursive but not potentially recursive function, Berry paradox, and fast growing Busy Beaver function. Many interesting lambda fixpoint combinators can also be fitted into this schema. Both Curry’s Y combinator and Turing’s Θ combinator follow from Lawvere’s theorem, as well as their call-by-value versions. At last, it can be shown that the lambda calculus version of the fixpoint lemma also fits Lawvere’s schema.     

An Infinitary Extension of Jankov’s Theorem

It is known that for any subdirectly irreducible finite Heyting algebra A and any Heyting algebra B, A is embeddable into a quotient algebra of B, if and only if Jankov’s formula χ _A for A is refuted in B. In this paper, we present an infinitary extension of the above theorem given by Jankov. More precisely, for any cardinal number κ, we present Jankov’s theorem for homomorphisms preserving infinite meets and joins, a class of subdirectly irreducible complete κ-Heyting algebras and κ-infinitary logic, where a κ-Heyting algebra is a Heyting algebra A with # ≥  κ and κ-infinitary logic is the infinitary logic such that for any set Θ of formulas with # Θ ≥  κ, ∨Θ and ∧Θ are well defined formulas.     

A Simplified Proof of the Church–Rosser Theorem

Takahashi translation * is a translation which means reducing all of the redexes in a λ-term simultaneously. In [4] and [5], Takahashi gave a simple proof of the Church–Rosser confluence theorem by using the notion of parallel reduction and Takahashi translation. Our aim of this paper is to give a simpler proof of Church–Rosser theorem using only the notion of Takahashi translation.     

A New Proof of the McKinsey–Tarski Theorem

It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure (and hence modal box as interior), then \(\mathsf S4\) is the logic of any dense-in-itself metrizable space. The McKinsey–Tarski Theorem relies heavily on a metric that gives rise to the topology. We give a new and more topological proof of the theorem, utilizing Bing’s Metrization Theorem.     

Historians and Philosophers of Logic: Are They Compatible? The Bolzano-Weierstrass Theorem as a Case Study

This paper combines personal reminiscences of the philosopher John Corcoran with a discussion of certain conflicts between historians of logic and philosophers of logic. Some mistaken claims about the history of the Bolzano-Weierstrass Theorem are analyzed in detail and corrected.