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Forms of thought are involved whenever we name, describe, or identify things, and whenever we distinguish between what is, might be, or must be the case. It appears to be a distinctive feature of human thought that we can have modal thoughts, about what might be possible or necessary, and conditional thoughts, about what would or might be the case if something else were the case. Even the simplest thoughts are structured like sentences, containing referential and predicative elements, and studying these structures is the main task of philosophical logic. This clear and accessible book investigates the forms of thought, drawing out and focusing on the central logical notions of reference, predication, identity, modality and conditionality. It will be useful to students and other interested readers in epistemology and metaphysics, philosophy of mind and language, and philosophical logic.
In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book - extensively rewritten for its second edition - will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.