Calendar

Round Squares are No Contradictions
Jean-Yves Beziau (Federal University of Rio de Janeiro and UC San Diego)
4:10pm, Friday, April 24, 716 Philosophy Hall, Columbia University

Abstract. When talking about contradictions many people think of a round square as a typical example. We will explain in this talk that this is the result of a confusion between two notions of oppositions: contradiction and contrariety. The distinction goes back to Aristotle but it seems that up to now it has not been firmly implemented in the mind of many rational animals nor in their languages.According to the square of opposition, two propositions are contradictory iff they cannot be true and cannot be false together and they are contrary iff they cannot be true together but can be false together. The propositions “X is a square” and “X is a circle” cannot be true together according to the standard definitions of these geometrical objects, but they can be false together: X can be a triangle, something which is neither a square, nor a circle. A round square is a contrariety, not a contradiction. Aristotle insisted that there were two different kinds of oppositions, from this distinction grew a theory of oppositions that was later on shaped in a diagram by Apuleius and Boethius.It is easy to find examples of contrarieties, but not so of contradictions. Many pairs of famous oppositions are rather contraries: black and white (think of the rainbow), right and left (think of the center), day and night (think of dawn or twilight, happy and sad (think of insensibility), noise and silence (think of music), etc. Examples of “real” contradictions are generally from mathematics: odd and even, curved and straight, one and many, finite and infinite. We can indeed wonder if there are any contradictions in (non-mathematical) reality or if it is just an abstraction of our mind expressed through classical negation according to which p and ¬p is a contradiction.

UNIVERSITY SEMINAR ON LOGIC, PROBABILITY, AND GAMES
Gödel on Russell: Truth, Perception, and an Infinitary Version of the Multiple Relation Theory of Judgment
Juliet Floyd (Boston University)
4:10 pm, May 8, 2014
Faculty House, Columbia University

Abstract. Many philosophers think that mathematics is about ‘structure’. Many philosophers would also explicate this notion of ‘structure’ via model theory. But the Compactness and Löwenheim–Skolem theorems lead to some famously hard questions for this view. They threaten to leave us unable to talk about any particular ‘structure’.

In this talk, I outline how we might explicate ‘structure’ without appealing to model theory, and indeed without invoking any kind of semantic ascent. The approach involves making use of internal categoricity. I will outline the idea of internal categoricity, state some results, and use these results to make sense of Putnam’s beautiful but cryptic claim: “Models are not lost noumenal waifs looking for someone to name them; they are constructions within our theory itself, and they have names from birth.”