Axioms or circularity
When I was at primary school I had a special dictionary for
schools and I wanted to find the meaning of a word A. Well the dictionary told me it meant B. But what did B mean? Well the dictionary said that B meant A. So I asked my parents and they were most
irritated and clearly felt I should show more initiative. Deadlock.
In an earlier Issue I argued that the purpose of life is
love but that the purpose of love is life, so that anyone seeking the purpose
of life would simply go round in circles:
Well, this won’t do for argumentation, and we have heard the
accusation that certain arguments are circular.
Thus some people will say that abortion is murder. So how do you define murder? Oh, to include abortion. We are in fact on a better path if we say
that life is sacred, since as long as you can define life - i.e. what is the
meaning of ‘life’ as opposed to what is the meaning of life – then you are on
firmer ground, but nobody can argue with you.
So in logic, one builds on axioms, things we take as
given. In fact I myself spent 20 years
of my life working as a statistician.
And within the various argumentations of statistics, one thing is left
undefined, and that is probability itself.
Or at least that was the case nearly 50 years ago when I first studied
statistics. Such undefined items are known as ‘primitives’.
We may note that relatable to axioms are theorems:
propositions that may be proved from axioms (and/or existing theorems) . Now if I have a formula containing
expressions such as (a+b), ab, (a Ù b), etc, we have a principle of uniform substitutability. Because all of these operators are
commutative, we can replace them with their ‘back-to-front’, i.e. commute
them. We may do this in a system which
has an axiom or theorem of the form
(a.b) = (b.a), but we must do so consistently.
Another rule is that if we know (i) that A is always true,
and (ii) A implies B, then B must always be true. This is the rule of Detachment (or in
Latin Modus Ponens). Some logicians do without this rule.
Now all logicians agree that when proving things, one must
build up axioms from ‘primitives’ and definitions. One usually has two primitives, such as ‘not’
and ‘or’. This was the starting point of
Bertrand Russell and P.H. Whitehead in their Principia Mathematica. Mordechaj Wajsberg on the other hand,
produced several systems including one in which implication and falsity were
taken as primitive.
But let us tabulate all this:
 |
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Tarski, Bernays, and Wajsberg, ‘basis 1’, 1937
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Note that axioms may be built up depending on how many
variables or separate propositions they involve: if an axiom involves one
variable, generally you cannot say anything about a proposition with two
variables, etc.
But maybe we have a circularity here as well: for logic you
have to turn to philosophy for its axioms and ‘primitives’. Look at that primitive falsehood! But philosophy cannot proceed without being
logical!
By Martin Prior
References
Hackstaff, L.H. (1966)
Systems of Formal Logic. D.
Reidel Publishing, 1966.
Wajsberg, Mordechaj (1937-9) “Metalogische Beiträge” In: Wiadomości
Matematyczne Volumes 43 (1937) and 47 (1939), translated by McCall and
(partly) P. Woodruff in Contributions to Metalogic (1967).
Whitehead, Alfred North and Bertrand Russell (1910) Principia
Mathematica. Cambridge: Cambridge
University Press.
The Philosophy Takeaway 'Open Topic' Issue 33
