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By pooling together exhaustive analyses of certain philosophical paradoxes, we can prove a series of fascinating results regarding philosophical progress, agreement on substantive philosophical claims, knockdown arguments in philosophy, the wisdom of philosophical belief (quite rare, because the knockdown arguments show that we philosophers have been wildly wrong about language, logic, truth, or ordinary empirical matters), the epistemic status of metaphysics, and the power of philosophy to refute common sense. As examples, this Element examines the Sorites Paradox, the Liar Paradox, and the Problem of the Many – although many other paradoxes can do the trick too.
Roy T Cook examines the Yablo paradox—a paradoxical, infinite sequence of sentences, each of which entails the falsity of all others later than it in the sequence—with special attention paid to the idea that this paradox provides us with a semantic paradox that involves no circularity. The three main chapters of the book focus, respectively, on three questions that can be (and have been) asked about the Yablo construction. First we have the Characterization Problem, which asks what patterns of sentential reference (circular or not) generate semantic paradoxes. Addressing this problem requires an interesting and fruitful detour through the theory of directed graphs, allowing us to draw interesting connections between philosophical problems and purely mathematical ones. Next is the Circularity Question, which addresses whether or not the Yablo paradox is genuinely non-circular. Answering this question is complicated: although the original formulation of the Yablo paradox is circular, it turns out that it is not circular in any sense that can bear the blame for the paradox. Further, formulations of the paradox using infinitary conjunction provide genuinely non-circular constructions. Finally, Cook turns his attention to the Generalizability Question: can the Yabloesque pattern be used to generate genuinely non-circular variants of other paradoxes, such as epistemic and set-theoretic paradoxes? Cook argues that although there are general constructions-unwindings—that transform circular constructions into Yablo-like sequences, it turns out that these sorts of constructions are not 'well-behaved' when transferred from semantic puzzles to puzzles of other sorts. He concludes with a short discussion of the connections between the Yablo paradox and the Curry paradox.
What role, if any, does formal logic play in characterizing epistemically rational belief? Traditionally, belief is seen in a binary way - either one believes a proposition, or one doesn't. Given this picture, it is attractive to impose certain deductive constraints on rational belief: that one's beliefs be logically consistent, and that one believe the logical consequences of one's beliefs. A less popular picture sees belief as a graded phenomenon. This picture (explored more bydecision-theorists and philosophers of science thatn by mainstream epistemologists) invites the use of probabilistic coherence to constrain rational belief. But this latter project has often involved defining graded beliefs in terms of preferences, which may seem to change the subject away fromepistemic rationality.Putting Logic in its Place explores the relations between these two ways of seeing beliefs. It argues that the binary conception, although it fits nicely with much of our commonsense thought and talk about belief, cannot in the end support the traditional deductive constraints on rational belief. Binary beliefs that obeyed these constraints could not answer to anything like our intuitive notion of epistemic rationality, and would end up having to be divorced from central aspects of ourcognitive, practical, and emotional lives.But this does not mean that logic plays no role in rationality. Probabilistic coherence should be viewed as using standard logic to constrain rational graded belief. This probabilistic constraint helps explain the appeal of the traditional deductive constraints, and even underlies the force of rationally persuasive deductive arguments. Graded belief cannot be defined in terms of preferences. But probabilistic coherence may be defended without positing definitional connections between beliefsand preferences. Like the traditional deductive constraints, coherence is a logical ideal that humans cannot fully attain. Nevertheless, it furnishes a compelling way of understanding a key dimension of epistemic rationality.
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Offers a systematic introduction and discussion of all the main solutions to the sorites paradox and its areas of influence.
Infinity is paradoxical in many ways. Some paradoxes involve deterministic supertasks, such as Thomson's Lamp, where a switch is toggled an infinite number of times over a finite period of time, or the Grim Reaper, where it seems that infinitely many reapers can produce a result without doing anything. Others involve infinite lotteries. If you get two tickets from an infinite fair lottery where tickets are numbered from 1, no matter what number you saw on the first ticket, it is almost certain that the other ticket has a bigger number on it. And others center on paradoxical results in decision theory, such as the surprising observation that if you perform a sequence of fair coin flips that goes infinitely far back into the past but only finitely into the future, you can leverage information about past coin flips to predict future ones with only finitely many mistakes. Alexander R. Pruss examines this seemingly large family of paradoxes in Infinity, Causation and Paradox. He establishes that these paradoxes and numerous others all have a common structure: their most natural embodiment involves an infinite number of items causally impinging on a single output. These paradoxes, he argues, can all be resolved by embracing 'causal finitism', the view that it is impossible for a single output to have an infinite causal history. Throughout the book, Pruss exposits such paradoxes, defends causal finitism at length, and considers connections with the philosophy of physics (where causal finitism favors but does not require discretist theories of space and time) and the philosophy of religion (with a cosmological argument for a first cause).
Foundationalism is a view about the structure of knowledge and justification. The heart of the thesis is the claim that if there is any knowledge or justified belief at all, then there is a kind of knowledge and justified belief that does not require inference from something else known or justifiably believed. This Element begins by exploring abstract arguments for foundationalism and against proposed alternatives. It then explores disagreements among foundationalists about how to understand foundational knowledge and justified belief, what is plausibly included in the foundations, and what is required for legitimate inference from foundations to the rest of what we believe. The author argues for the conclusion that one can combine insights captured by different versions of foundationalism by making a distinction between ideal justification and justification that falls short of that ideal.
The book provides a thorough exploration of the epistemic dimensions of ignorance: what is ignorance and what are its varieties?
In 1945 Alonzo Church issued a pair of referee reports in which he anonymously conveyed to Frederic Fitch a surprising proof showing that wherever there is (empirical) ignorance there is also logically unknowable truth. Fitch published this and a generalization of the result in 1963. Ever since, philosophers have been attempting to understand the significance and address the counter-intuitiveness of this, the so-called paradox of knowability. This collection assembles Church's referee reports, Fitch's 1963 paper, and nineteen new papers on the knowability paradox. The contributors include logicians and philosophers from three continents, many of whom have already made important contributions to the discussion of the problem. The volume contains a general introduction to the paradox and the background literature, and is divided into seven sections that roughly mark the central points of debate. The sections include the history of the paradox, Michael Dummett's constructivism, issues of paraconsistency, developments of modal and temporal logics, Cartesian restricted theories of truth, modal and mathematical fictionalism, and reconsiderations about how, and whether, we ought to construe an anti-realist theory of truth.