Computational Aristotelian Term Logic: Notation

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Remarks on notation

Terms and "quantors"

More than 2000 years of Aristotelian logic lead to a lot of different notational systems. One of Aristotle's great achievements in formal logic was the introduction of Greek capital letters for terms. Later, in medieval times, the abbreviations A, E, I, and O for the categorical propositions came into use, and the "small" version a, e, i, and o as well. Fortunately, this did not change - although, from time to time, modern authors (like Corcoran, using A, N, S, and $ instead), propose a different notation.

Having decided to use the classical AEIO- notation (and thus not aeio), we still have to choose a notation for terms. Greek letters still present problems in some computer contexts, so we follow the habit of most medieval and modern logicians, denoting terms by Latin letters.

Having made the decision for the AEIO - notation, it seems to be only consequent to use small capital letters for terms. This is what we will do in our texts.

Let us resume:

  1. We utilize capital Latin letters A, E, I, and O for the "Aristotelian quantors" (who knows a better name ...) "ALL ...", "NO ...", "SOME ...", "SOME ... NOT ...".
  2. We use small Latin letters t, u, ..., x, y, z as symbols for terms

How to denote a categorical proposition?

A categorical proposition will be written as a string consisting of two terms x and y together with one of the capital letters A, E, I, or O. Thus, there are three different possibilities for a notational system (we take U as variable for AEIO):

  1. U x y
  2. x U y
  3. y x U

All these notational systems have been used or are still in use today. This does not seem to present a real problem - but there is another difficulty hidden behind the seamingly simple combinatorical problem of placing the "quantor" with respect to the terms.

The problem is that A and O are no "symmetric operators"; i.e. A x y and A y x denote different propositions. Therefore, having choosen one of the three ways of writing propositions, we still have to decide whether, f.i., A x y stands for "All x are y" or "All y are x". While the second variant looks unreasonable for modern people who automatically regard x and y as sets of individuals, it did not for logicians from the time of Aristotle up to, let us say, Leibniz.

For Aristotle, terms do NOT denote sets of individuals! This is not the place to discuss this topic in depth - let us just remark that the Aristotelian terms should be better considered as standing for concepts (like man, animal, living being, gold, etc.). Thus the "relation" A between terms denotes a relation between concepts in an intensional manner. Aristotle says " y is predicated of all x " instead of " All x are y".
So what shall we choose to be denoted by A u v ? "All u are v" (extensional) or "u is predicated of all v" (intensional, Aristotelian)?

We decided to follow Leibniz who was full aware of the significant difference between an intensional and an extensional interpretation of Aristotle's logic. Leibniz favoured the intensional way of looking at the world, but, nevertheless, used the notion A(B,C) for "All B are C" or, speaking Aristotelian, "C is predicated of all B" in a way most people do it today.

So shall we: Our propositions have the form A(x,y) (or Axy in a shorter version) to denote "All x are y" or, "y is predicated of all x":

A(x,y) or Axy All x are y y is predicated of all x
E(x,y) or Exy No x are y y is predicated of no x
I(x,y) or Ixy Some x are y y is predicated of some x
O(x,y) or Oxy Some x are not y y is not predicated of all x

This type of notation will usually be found in books on formal logic. Authors coming from the philological side more often use the other notation, where A(x,y) denotes "x is predicated of all y". For example, in the notation is GaF, which is, in our notation, A(F,G) (NOT A(G,F)!).

Thus, if you once notice a proposition written, f.i., as aMN or MaN, you CANNOT be sure whether this has to be translated into our system as A(M,N) or A(N,M)!

Let me close with the remark that the different notational systems are a cause of permanent problems and also errors. I would appreciate comments on this question, and I am not totally sure that I will stay with my decision for the present notation.

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