András Máté Department of Logic, ELTE

Where and when does this (hi)story begin? Greek classical antiquity What is logic? The science and/or art of valid inference + a lot of connected topics Interpretation of propositions Connections between concepts Heuristics … Method of limiting the area in this course: I follow mainly the borderlines of modern, 20th-21st century logic But on a loose, non-exclusive way Restricted to the European cultural sphere

Logical theory: 1.Rules or criteria of validity for a broad group of inferences. 2.The rules themselves are organized to a logically connected theory. Plato’s dialogues: many loose occurrences of type 1. claims. The first example of 2.: the theory of categorical syllogisms in Aristotle’s Prior Analytics. Logical practice: Systematic use of (logical) inference in arguments to convince other people. Distinction between reference to facts, experience, tradition, (intuition?), etc., and reference to inferences (or even to logical truths). Areas of logical practice: philosophical, mathematical, legal argumentation. First testimonies of logical practice: 5th century B.C, all the three areas. Logical theory: (critical) reflection to logical practice (sophistry).

Traces of logical practice in Greek philosophy and mathematics in the 5th century B.C. – the question of priority. Árpád Szabó’s thesis: priority belongs to philosophy, namely, Eleatic philosophy. Parmenides and Zeno wanted to refute the relevance of experience and intuition. Their main achievement: discovery of indirect argumentation. In an indirect argumentation you don’t set out from facts. The starting point is a supposed counterfactual situation.

Deductive turn in Greek mathematics: an answer to the Elean challenge. Egyptian mathematics: some arithmetical calculation, area calculations (with obvious practical aims). Babylonian mathematics: astronomical calculations, systematic solutions to excercises up to equation soulutions. Greek mathematics at the border of archaic and classical era: intuitive mathematics with general theorems and some intuitive support for their validity. A glance to it later. Deductive and counter-intuitive turn in the 5th century. Older explanation: the discovery of irrationality. Szabó: conversely, this discovery is now a consequence, an achievement of the turn.

An amusing testimony Epicharmus (comedy writer, first half of the 5th century B. C.) Fragment 1. A But the gods were always there, of course: they never were lacking; and these things (probably, 'that which is divine') always exist in a similar form and through the same causes. B But still, it is said that Chaos was the first of the gods to be created. A How can that be? It is impossible for a 'first' thing to come from something and into something. B Then was there no first thing that came? A Certainly not! Nor a second either, at any rate of these things (the divine?) of which we are now speaking thus; but they were always there. A How can that be? It is impossible for a 'first' thing to come from something and into something. The Parmenidean argument for the eternity of the Being! Philosophical (cosmogonical) argumentation.

Fr. 2 A Suppose to an odd number, or to an even if you like, one chooses to add a pebble or else to take one from those already there: do you think that the number remains the same? B No, of course not. A Nor, furthermore, if one chooses to add to a cubit another measure of length, or to cut off a length from what was there before: does the former measure still remain? B No. A Now look at human beings in this way: one grows another wastes away, and all are in process of change all the time. But that which changes its nature and never remains in the same state, must also be different by now from that which has changed. So both you and I were yesterday other men, and we are other men now, and again (we shall be) other men (in the future), and never the same, according to the same Law (Logos). Context: The first person wants to convince the other that he (A) is not bound to give back the money he loaned from B (because he is not the same person any more).

„Legal” argument. Uses mathematical example. Parmenides argues (roughly): if the Being changed its properties, it wouldn’t be the same (i.e. Being) any more, Therefore it would be not-Being, But that is impossible. A’s argument is just the same in counterposed form. Core problem: identity during change. Personal identity.