We begin with a prepositional language Lp containing conjunction (Λ), a class of sentence names {Sα}αϵA, and a falsity predicate F. We (only) allow unrestricted infinite conjunctions, i.e., given any non-empty class of sentence names {Sβ}βϵB,is a well-formed formula (we will use WFF to denote the set of well-formed formulae).The language, as it stands, is unproblematic. Whether various paradoxes are produced depends on which names are assigned to which sentences. What is needed is a denotation function:For example, the LP sentence “F(S1)” (i.e., Λ{F(S1)}), combined with a denotation function δ such that δ(S1)“F(S1)”, provides the (or, in this context, a) Liar Paradox.To give a more interesting example, Yablo's Paradox [4] can be reconstructed within this framework. Yablo's Paradox consists of an ω-sequence of sentences {Sk}kϵω where, for each n ϵ ω:Within LP an equivalent construction can be obtained using infinite conjunction in place of universal quantification - the sentence names are {Si}iϵω and the denotation function is given by:We can express this in more familiar terms as:etc.
Logical pluralism is the view that there is more than one correct logic. In this article, I explore what logical pluralism is, and what it entails, by: (i) distinguishing clearly between relativism about a particular domain and pluralism about that domain; (ii) distinguishing between a number of forms logical pluralism might take; (iii) attempting to distinguish between those versions of pluralism that are clearly true and those that are might be controversial; and (iv) surveying three prominent attempts to argue for logical pluralism and evaluating them along the criteria provided by (ii) and (iii).
Relativism Versus Pluralism 1One is a relativist about a particular phenomenon X if and only if one thinks that the correct account of X is a function of some distinct set of facts Y. Thus, relativism about X amounts to acceptance of the following schema: Philosophy Compass 5/6 (2010):For our purposes, we can define a formal logic (hereafter, simply 'logic') to be any pair
Truth values are, properly understood, merely proxies for the various relations that can hold between language and the world. Once truth values are understood in this way, consideration of the Liar paradox and the revenge problem shows that our language is indefinitely extensible, as is the class of truth values that statements of our language can take -in short, there is a proper class of such truth values. As a result, important and unexpected connections emerge between the semantic paradoxes and the set-theoretic paradoxes.
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