The relation between Wittgenstein's philosophy of mathematics and mathematical Intuitionism has raised a considerable debate. My attempt is to analyse if there is a commitment in Wittgenstein to themes characteristic of the intuitionist movement in Mathematics and if that commitment is one important strain that runs through his Remarks on the foundations of mathematics. The intuitionistic themes to analyse in his philosophy of mathematics are: firstly, his attacks on the unrestricted use of the Law of Excluded Middle; secondly, his distrust (...) of non-constructive proofs; and thirdly, his impatience with the idea that mathematics stands in need of a foundation. These elements are Fogelin's starting point for the systematic reconstruction of Wittgenstein's conception of mathematics. (shrink)

Identity, we're told, is the binary relation that every object bears to itself, and to itself only. But how can a relation be binary if it never relates two objects? This puzzled Russell and led Wittgenstein to declare that identity is not a relation between objects. The now standard view is that Wittgenstein's position is untenable, and that worries regarding the relational status of identity are the result of confusion. I argue that the rejection of identity as a binary relation (...) is perfectly tenable. To this end, I outline and defend a logical framework that is not committed to an objectual identity relation but is nevertheless expressively equivalent to first-order logic with identity. After it has thus been shown that there is no indispensability argument for objectual identity, I argue that we have good reasons for doubting the existence of such a relation, and rebut a number of attempts at discrediting these reasons. (shrink)

Abstract This paper offers an appraisal of Phillip Pettit's approach to the problem how a merely finite set of examples can serve to represent a determinate rule, given that indefinitely many rules can be extrapolated from any such set. I argue that Pettit's so-called ethnocentric theory of rule-following fails to deliver the solution to this problem he sets out to provide. More constructively, I consider what further provisions are needed in order to advance Pettit's general approach to the problem. I (...) conclude that what is needed is an account that, whilst it affirms the view that agents' responses are constitutively involved in the exemplification of rules, does not allow such responses the pride of place they have in Pettit's theory. (shrink)

In the Tractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, (...) we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of the Tractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in the Tractatus when taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality. As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression. 5.503. (shrink)

Proposition 5.122 of Wittgenstein’s Tractatus has been the source of much puzzlement among interpreters, so much so that no fully satisfactory account is yet available. This is unfortunate, if only because the containment account of logical consequence has a venerable tradition behind it. Pasquale Frascolla’s interpretation of proposition 5.122 is based on a valid argument and one true premise. However, the argument explains sense containment only in an indirect way, leaving some crucial questions unanswered. Besides, Frascolla does not address the (...) issue of how to make sense containment notationally perspicuous, an essential theme in Wittgenstein’s reflections. In this paper, we elaborate on Frascolla’s account by looking at the issue through the Tractarian notion of logical space. Our analysis shows that, for containment to be fully appreciated, one should adopt a negative perspective on the notion of sense, in line with the exclusionary theory of conceptual content, as labelled by Ian Rumfitt. Besides this, we introduce and discuss two methods—one envisaged by Wittgenstein himself—for making sense containment notationally perspicuous. (shrink)

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. -/- There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably (...) infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. -/- Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects. (shrink)

In this paper, I first review previous interpretations of Wittgenstein's remarks on private language, revealing their inadequacies, and then present my own interpretation. Basing mainly on Wittgenstein's notes for lectures on private sensations, I establish the following points: ‘remembering the connection right’ means ‘reidentifying sensation-types’; the reason for ‘no criterion of correctness’ is that nothing, especially no inner mechanisms nor external devices, can be utilised by the private speaker to tell whether some sensations are of one type or different types; (...) and private names are not really names, private language is not really a language, therefore, private language is a grammatical illusion. My interpretation has the advantage of being able to reconcile Wittgenstein's conception of philosophy, which is to dissolve philosophical problems by rearranging grammatical facts, with his actual philosophical practice, at least in the case of private language. (shrink)

Wittgenstein's insistence in his later philosophy that explanation comes to an end in the explication of what it is to follow a rule provides a locus for the awakening of wonder, analogous to the mystical awe referred to in the "Tractatus". While Wittgenstein did not explore this analogy, it provides a point of entry into the examination of the relevance of his work to religious concerns. Every regular practice is built on capacities of reaction, uptake, and response which are the (...) endpoints of explanation, whose sheer givenness in their relation to our physical and social environment can bring on amazement. This wonder at the end of explanation has been explored -- from quite a different perspective -- by Stephen Gould, whose conclusions complement Wittgenstein's own. (shrink)

ABSTRACTIn 1931 Wittgenstein wrote: ‘the limit of language manifests itself in the impossibility of describing the fact that corresponds to a sentence without simply repeating the sentence’. Here, Wittgenstein claims, ‘we are involved … with the Kantian solution of the problem of philosophy’. This paper shows how this remark fits with Wittgenstein's early account of the substance of the world, his account of logic, and ultimately his view of philosophy. By contrast to the currently influential resolute reading of the Tractatus, (...) the paper argues that the early Wittgenstein did not aim at destroying the idea of a limit of language, but that the notion lies at the very heart of Wittgenstein's early view. In doing so, the paper employs and defends the Kantian interpretation of Wittgenstein's early philosophy. (shrink)