Truth, Proof and Gödelian Arguments : A Defence of Tarskian Truth in Mathematics

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion.

Tyhjyyden reunalla, helvetin kannella : Yhtenäisen subjektin hajoaminen Saul Bellowin Herzogissa

The study shows that the reading paradigm derived from codes Roland Barthes presented in his S/Z exposes the postmodernic nature of Saul Bellow s Herzog (1964), and embodies in which way the novel is organized as a radical rewriting of modernism and its ideology. I explore how solid, compact and modernistic subject breaks down at every level of Herzog s narration. I actually argue that the heteroglossic novel is representative of an early American postmodern movement in literaure, and it should by no means be dissected narrowly as a realistic or naturalistic novel at all. The intertextual code verifies that the interpretation of Herzog remains inadequate if one doesn t take account of the novel s significant intertextual references to other texts. In fact, even the mind of Moses Herzog, the protagonist of the novel, is a mosaic of citations. It emerges from the dissertation that the figure of Don Quixote follows Moses Herzog as an ambiguous shadow while the professor of history struggles with his anxiety and anguish, and travels in a mentally confused state around the U.S. for five days in the early 1960s, encountering the impending atmosphere of transition as the country is on the threshold of a significant cultural and social change. There is a strong necessity for updated interpretation of Herzog partly because its centrality to Bellow s own career as a writer but mainly because it has been previously read trough modernistic lenses. I shall try to proof in my study that American Jewish Saul Bellow s (1915 2005) Herzog escapes any kind of simple, elemental or essentialistic construction or reading and in real terms it doesn t offer any comprehensive, total or coherent solution or system for those philosophical doctrines it criticizes and makes fun of. The philosophical, conceptual and cultural substance of Herzog does not constitute an independent or autonomous theoretical tract which would have a life of its own. Altough the novel reflects the continuity of Bellow s writing it is clearly some kind of conscious or unconscious experiment during his long career as a writer. He hasn t been so radical before or since the publication of Herzog. It is unarguably his most postmodern novel.The reading paradigm based on specific codes demonstrates how deep into the basic questions of his personal life and existence itself he must dive in oder to find his many ways towards authentic or primordial self in fragmented and shattered world which is constanly rewritten and which makes human being a tourist of his own life. In that ongoing process the protagonist has to accept the ultimate plurality of his mind and self. He must confront that the modernistic definition of identity as a solid, monolithic and stable entity has broken down into different, inconsistent and even contradictory possibilities of identification. Moses Herzog embodies obviously Stuart Hall s description of the postmodern self his identity has turn into a movable feast.

On the Modal Content of A Posteriori Necessities

This paper challenges the Kripkean interpretation of a posteriori necessities. It will be demonstrated, by an analysis of classic examples, that the modal content of supposed a posteriori necessities is more complicated than the Kripkean line suggests. We will see that further research is needed concerning the a priori principles underlying all a posteriori necessities. In the course of this analysis it will emerge that the modal content of a posteriori necessities can be best described in terms of a Finean conception of modality – by giving essences priority over modality. The upshot of this is that we might be able to establish the necessity of certain supposed a posteriori necessities by a priori means.

A New Definition of A Priori Knowledge: In Search of a Modal Basis

In this paper I will offer a novel understanding of a priori knowledge. My claim is that the sharp distinction that is usually made between a priori and a posteriori knowledge is groundless. It will be argued that a plausible understanding of a priori and a posteriori knowledge has to acknowledge that they are in a constant bootstrapping relationship. It is also crucial that we distinguish between a priori propositions that hold in the actual world and merely possible, non-actual a priori propositions, as we will see when considering cases like Euclidean geometry. Furthermore, contrary to what Kripke seems to suggest, a priori knowledge is intimately connected with metaphysical modality, indeed, grounded in it. The task of a priori reasoning, according to this account, is to delimit the space of metaphysically possible worlds in order for us to be able to determine what is actual.