3/24/2023 0 Comments A false notion definition![]() ![]() This is, roughly put, how Curry’s Paradox allows us to conclude the consequent of γ thus, in this particular case, it allows us to conclude that snow is white. From these, applying modus ponens again, we conclude that snow is white. At this point, then, we have the following two claims: first, that if \(T\ulcorner \gamma \urcorner \), then snow is white and, second, that \(T\ulcorner \gamma \urcorner \). So we just proved γ itself and, hence, \(T\ulcorner \gamma \urcorner \). Since we have concluded that snow is white under the supposition that \(T\ulcorner \gamma \urcorner \), we can now conclude that if \(T\ulcorner \gamma \urcorner \), then snow is white. We can apply next modus ponens and conclude, under the assumption that \(T\ulcorner \gamma \urcorner \), that snow is white. Then, under this supposition, what γ says is the case to wit, if \(T\ulcorner \gamma \urcorner \) then snow is white. Suppose, now, that \(T\ulcorner \gamma \urcorner \). $$ (\gamma) T\ulcorner\gamma\urcorner\to \text $$ ![]() Footnote 6 Suppose, furthermore, that we have a Curry sentence γ, which is a sentence that asserts that if itself is true then snow is white: Suppose that T is the truth predicate and that it obeys the so called ‘T-schema’, according to which, for any sentence ϕ, \(T\ulcorner \phi \urcorner \leftrightarrow \phi \). Footnote 5 It can be presented concisely as follows. The first counterexample to Definition 1 I want to consider is Curry’s paradox. I will offer two arguments that I claim can reasonably be considered paradoxes but that do not satisfy Definition 1. I want to show next that Definition 1 is too narrow. So the apparent validity of the argument, the apparent truth of the premises, and the apparent falsity of the conclusion are forced by the tension that arises from the concepts involved in the paradox that is, we would be willing to state that the argument is valid, that the premises are true and that the conclusion is false but since this is, in principle, impossible, we are forced to declare these properties as apparent. That is why solving a paradox must involve giving up some of these core intuitions, all of which are, typically, equally cherished, and as a result, there is no agreement whatsoever as to which of them has to be abandoned. ![]() In this sense, then, a paradox points to a tension in the basic intuitions governing one or more of our concepts. Footnote 3 Accordingly, the sense of ‘paradox’ that I am trying to help elucidate in this paper is the technical or semi-technical sense that is usually used in the literature in philosophy of logic and language (what Quine 1966 calls ‘antinomy’). ![]() Thus, the premises of a paradox are apparently true in the sense that their being not true would violate some of our core intuitions with respect to some of the concepts-either explicitly or implicitly-involved in them (and the same can be said about the apparent falsehood of the conclusion). And the same occurs with the other conditions of the traditional characterization. To wit, a paradox is not an apparently valid argument in the sense that it merely seems valid, but in the sense that, declaring it invalid, implies giving up strong intuitions about logic. It is important to note that the sense in which ‘apparently valid’ (‘true’, ‘false’) is used in this definition is quite strong-although not as strong as declaring it valid (true, false), of course. Let us state again the traditional definition of the notion of paradox, henceforth ‘Definition 1’: Definition 1Ī paradox is an apparently valid argument with apparently true premises and an apparently false conclusion. ![]()
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