## 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
Having decided to use the classical 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: - We utilize capital Latin letters
**A, E, I, and O**for the "Aristotelian quantors" (who knows a better name ...) "ALL ...", "NO ...", "SOME ...", "SOME ... NOT ...". - 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): - U x y
- x U y
- y x U
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
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 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":
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 faculty.washington.edu/smcohen/433/Syllogistic.pdf 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 |