Key Ideas in the theory of proofs #1: The Duality of Proofs and Counterexamples

Abstract: Suppose we have a language involving non-denoting singular terms. (The language of everyday mathematics provides one example. Terms like \(\frac{n}{m}\) and \(\lim_{x\to\infty} f(x)\) do not denote, for appropriate choices of \(m\) and of \(f\).) It is not too difficult to define inference rules for an appropriately free logic that...

Abstract: Michael Dummett, in a presentation to the Aristotelian Society 65 years ago (in 1959) inaugurated a long-running debate over semantic realism and anti-realism, and the issue of settling an appropriate logic as a necessary prolegomenon to fruitful discussion metaphysics. Dummett argued that the principles of intuitionistic logic are...

Generic claims, such as Birds fly, Men are violent, and Mosquitos carry Ross River Fever, seem pervasive across human thought and talk. We use generic claims to express our understanding of the world around us and our place in it. These generic claims are useful even though they admit exceptions. We can agree that birds fly, even though emus...

Abstract: When I use a calculator to tell me that 245 × 46 = 11,270, I learn something that I didn’t know before, even though calculators don’t have any beliefs or knowledge. Even small children know how to count things, and it is through our own capacity to enumerate and count things that we learn basic arithmetic. Calculators do not count...

py2010: Intermediate Logic is a University of St Andrews undergraduate subject in which we cover important results in logic to philosophy students. It’s taught by Greg Restall, together with a committed crew of postgraduate students. The subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that...