Logicism

CORCORAN RECOMMENDS ANDREWS ON TYPE THEORY

1988. Andrews, P. An Introduction to Mathematical Logic and Type Theory. Computer Science and Applied Mathematics. Academic Press.1986.
ISBN 0-12-058535-9; 0-12-058536-7. Mathematical Reviews MR0859866.

The heuristic, pedagogic, historical, and philosophical importance of type theory (or higher-order logic) is well established, even if Skolem, Tarski, Quine and others prove to be correct in questioning its status as “genuine logic” and in impugning the foundational significance of results which presuppose it. Much of the literature of mathematical logic prior to 1950 is virtually inaccessible to persons not versed in type theory. A mathematically rigorous introduction to the subject, such as that presented in this book, is therefore a welcome addition to the literature, especially in view of the fact that in recent years such treatments have been rare.[…] This book has been written with clarity, reasoned with rigor, proofread with care, and produced with taste and imagination. It is a credit to its author and to its publisher.

Relational statements are not regulation propositions with an existential purpose, the statement of things as things in terms of their character, action, or description as such: “the house is white”, “those birds fly”, “that child is very intelligent”. Relational statements are expressions, in one way or another, of things across things: “The Empire State is taller than Nelson’s Column”; “I am conscious of a bird flying overhead”; “1 + 1 = 2”. Logicians have traditionally failed to categorically distinguish these two forms of statement, with some dire consequences: numbers have been treated as objects; consciousness is thought to be something the mind does; relations are said to be qualities. I call relational statements forms of specification and seek in this essay to iron out some of the problems (for “ontology”, for “metaphysics”) that such forms raise. “A is taller than B” is not an “existential” thing to say; it is, nonetheless, a real thing to say. How is that the case?

Frege proposed his doctrine of unsaturatedness as a solution to the problems of the unity of the proposition and the unity of the sentence. I show that Frege’s theory is mystical, ad hoc, ineffective, paradoxical and entails that singular terms cannot be predicates. I explain the traditional solution to the problem of the unity of the sentence, as expounded by Mill, which invokes a syncategorematic sign of predication and the connotation and denotation of terms. I streamline this solution, bring it up to date and contrast the resulting conventionalist account with Frege’s unsaturatedness account. I argue that the conventionalist account provides a clear and intelligible solution to the problem of the unity of the sentence which is free of the defects of Frege’s account. I suggest that the problem of the unity of the proposition is spurious. I recommend that the notion of unsaturatedness be extruded from serious debate.

The hypothesis that language has a cognitive role in verbal minds has found serious opposition in mainstream Cognitive Science. Cognitivism, either in the form of Fodor’s rational nativism, or in the form of the massive modularity of Evolutionary Psychology, cannot accommodate such a role. The assumption that cognition is a matter of logical inference (computation) over logical formulae of an innate language of thought makes language a peripheral to the mind, decoupleable from the rest of the cognitive system. In order to properly assess the empirical evidence, then, such a basic set of assumptions has to be questioned in the first place. We will call into question the arguments for the language of thought, reply to the arguments from this standpoint against the very possibility for language to play a cognitive role; and will show that this perspective fails to provide a proper understanding of human development, which is the period when language acquisition takes place.

According to what was the standard view (Poincaré; Wang, etc.), although Frege endorses, and Kant denies, the claim that arithmetic is reducible to logic, there is not a substantive disagreement between them because their conceptions of logic are too different. In his “Frege, Kant, and the logic in logicism,” John MacFarlane aims to establish that Frege and Kant do share enough of a conception of logic for this to be a substantive, adjudicable dispute. MacFarlane maintains that for both Frege and Kant, the fundamental defining characteristic of logic is “that it provides norms for thought as such (MacFarlane, 2002, p.57). I defend the standard view. I show that MacFarlane's argument rests on conflating the way that pure general logic is normative as a canon and as a propaedeutic, and that once these are distinguished the argument is blocked.

In the paper we consider the classical logicism program restricted to first-order logic. The main result of this paper is the proof of the theorem, which contains the necessary and sufficient conditions for a mathematical theory to be reducible to logic. Those and only those theories, which don't impose restrictions on the size of their domains, can be reduced to pure logic. Among such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), the elementary theory of topoi and many others. It is interesting to note that the initial formulation of the problem of reduction of mathematics to logic is principally insoluble. As we know all theorems of logic are true in the models with any number of elements. At the same time, many mathematical theories impose restrictions on size of their models. For example, all models of arithmetic have an infinite number of elements. If arithmetic was reducible to logic, it would had finite models, including an one-element model. But this is impossible in view of the axiom 0 ̸ = x ′ .

Russell decided that the only way to avoid the paradox that bears his name was to reject certain combinations of words that were apparently about classes as being in fact nonsensical.  In doing this he was not accusing some rival school of philosophers of talking nonsense; only his own previous self.  This gave his position an air of non-polemical neutrality.  So long as no alternative was known to his treatment of the matter, it looked as though he had DISCOVERED that seemingly meaningful, well-formed sentences could be in reality nonsensical.  In spite of the fact that Russell made only a limited use of this idea in his own philosophy, I argue that he gave a decisive impetus to nonsensicalism by providing what seemed to be a museum exhibit of philosophical nonsense.

Neo-Fregeans such as Bob Hale and Crispin Wright seek a foundation of mathematics based on abstraction principles. These are sentences involving a relation called the abstraction relation. It is usually assumed that abstraction relations must be equivalence relations, so reflexive, symmetric and transitive. In this paper I argue that abstraction relations need not be reflexive. I furthermore give an application of non-reflexive abstraction relations to restricted abstraction principles.

Ce travail propose une reevaluation du schisme phenomenologico-analytique a la lumiere des textes de Husserl et de Carnap qui en constituent l’un des fondements et qui cependant emergent d’un contexte philosophique et scientifique similaire. L’idee carnapienne de constitution comme « reconstruction rationnelle » et arbitraire du monde peut en effet paraitre s’opposer terme a terme au «se-constituer» des choses que deploie la phenomenologie husserlienne, mais l’emploi par Carnap du vocabulaire de la constitution nous impose d’interroger le lien que l’entreprise de l’Aufbau entretient avec la constitution idealiste transcendantale. La these de ce travail revient a affirmer que l’opposition Husserl/Carnap ne peut etre interpretee dans les termes d’une opposition entre phenomenologie et analyse logique, ni non plus sur la base des concepts d’idealisme transcendantal, de logicisme ou de phenomenalisme. Comprendre l’opposition entre les deux auteurs (et donc plus lointainement entre les d...

A second short quiz: this time checking whether the students had understood what logicism was, and how much they had retained about some common fallacies, etc. Haack also makes a little fun of herself, asking students to explain the two meanings of "there is a typo in every PowerPoint presentation."

Neo-Fregeans need their stipulation of Hume’s Principle – the number of Fs = the number of Gs if and only if F and G are equinumerous – to do two things. First, it must implicitly define the term-forming operator ‘the number of’, and second it must guarantee that Hume’s Principle as a whole is true. I distinguish two senses in which the neo-Fregeans might ‘stipulate’ Hume’s Principle, and argue that while one sort of stipulation fixes a meaning for ‘the number of' and the other guarantees the truth of Hume’s Principle, neither does both.