Undecidability In Wittgenstein's Picture Theory Of Meaning

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Sherman, Joseph
James Hamilton
Richard Lambert
Alfred Murray
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Undecidability In Wittgenstein's Picture Theory Of Meaning
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In early 1931 Kurt Goedel published a paper with the ominous title "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I," which, though tech nically accessible to only a limited number of individuals, was recognized to be revolutionary in its impact on formal logic. It put a virtual end to the attempts of formalists to give absolute proofs of the internal consistency of logical systems large enough to comprise the whole of arithmetic Concurrently, philosophers of language were attempting to characterize language, or at least a substantial part of it, via structures referred to as "ideal languages." It was hoped that a large number of conceptual incongruities would disappear once the language they were framed in became clear and unambiguous. Ludwig Wittgenstein's Tractatus LogicoPhilosophicus was commonly interpreted to be one such attempt. For as Wittgenstein states in Remark 2.0201, Every statement about complexes can be resolved t into a statement about their constituents and into the propositions that describe the complexes completely. 1 According to the Tractatus, propositions "picture" facts because their constituents represent the objects in the world, and their structure "mirrors" the arrangement of objects in the world. It is extremely important for this picture theory of meaning that propositions he given precise and unambiguous meaning in order for the system to be consistent. I maintain that Wittgenstein's picture theory fails to sufficiently work out this precision, and I intend to show this insufficiency from within the logical structure of the system itself. The most important consideration of which I make use is the Goedel paper. Wittgenstein claims that to know the sense of a proposition is to understand what state of affairs obtains if the proposition is true. Given the results of the Goedel paper, it can be argued that Wittgenstein has not properly determined the truth conditions of propositions: the truth conditions of atomic propositions do not account for those of the remaining statements in the system. According to Goedel, since the system of the Tractatus is not finitistic, some propositions must be true if and only if they are undecidable. Therefore, this system lacks a decision procedure for all propositions, and thus it fails to sufficiently characterize what it means in general for a proposition to be true. First of all, I intend to examine the picture theory of meaning established in the Tractatus, in order to demonstrate its requirement that it be an atomistic theory, and to show that this requirement is not constructed in toto within the Tractatus itself. Secondly, I will offer a correction of the picture theory which is explained in P. F. Strawson's essay "On Referring," which does take, into account a general relation between propositions and truth and falsity; this correction still satisfies Wittgenstein's Tractatus intentions. Finally, I will maintain that Goedel's theorem may he applied directly to the Tractatus picture theory to show its insufficiency, and indirectly to the Strawson model as an indication of the way in which the philosophy of language might proceed. A reading of this thesis should indicate to those aware of recent philosophical trends that philosophy has actually proceeded in this direction. The importance of this thesis resides in the fact that these considerations are taken from within the correspondence theory of meaning itself; only in the closing remarks shall I appeal to any extraneous conception of the linguistic task. I feel that the points raised in this paper are all the stronger for it.
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