Aristotle has propounded three basic laws of thought:
(i)
Law of Identity,
(ii)
Law of Non-Contradiction,
(iii)
Law of Excluded middle.
To my mind these are in fact one law, when expressed in
terms of formal logic. But firstly
let me return to my contribution to the previous issue (‘Axioms or
Circularity’), where I argued that we needed axioms to avoid arguments which
went ‘round and round in circles’.
I stated that we need at least one new axiom if we add a new
concept. Let us informally try
this out:
- Negation. Two negatives make an affirmative.
- One
proposition. p=
p, p implies itself, p and ~ p (not p) cannot be true
at the same time.
- Two
propositions. If p implies q, then if q is untrue,
then p must also be untrue.
- Three
propositions. If p implies q, and q implies r, then p implies r.
(transitivity)
My main concern here is with item1 – one proposition – but I
would to look a little at items 2 and 3 first. If we look at the question of two propositions, an
important concept is that a&b is equivalent to (and indeed implies)
b&a. This is known as
commutativity, and of course goes on the tree of logic. But this is a bit of a red herring:
we know that meeting the love of your life and getting married is not the same thing as getting married and meeting the love of your
life. But logicians can show
that this is saying the same thing as “If p
implies q, then if q is untrue, then p must also be untrue.”
And this is the important thing for reasoning.
Likewise transitivity is important for reasoning: “If p implies q, and q implies r, then p implies r.” But associativity is also important for logicians, and
also for mathematicians (along with commutativity). But associativity is likewise a red herring for the layman
– and perhaps the non-logician philosopher – since there’s always degrees of
association! Which of course is
important for ethics in matters that are not black and white.
So both associativity, along with double negation, go on the
Tree of Logic. … :] Well let us now get back to the question of item 1, one
proposition, where a proposition is equivalent to itself, implies itself, and
cannot be true at the same time as its opposite. Let us look more closely at Aristotle’s three basic laws of
thought:
(i)
Law of Identity,
(ii)
Law of Non-Contradiction,
(iii)
Law of Excluded Middle.
The Law of Identity simply states that a=a, where to say p=q
means that to say anything about p will always be equally true of q. Aristotle argues in Metaphysics, Book VII, Part 17, that “the fact that a thing is
itself is the single reason and the single cause to be given in answer to all
such questions as why the man is man, or the musician musical…”
The Law of Non-Contradiction states, according to Aristotle,
that “one cannot say of something that it is and that it is not in the same
respect and at the same time”, and relates to what I stated above, namely that
p and ~ p (not p) cannot be true at the same time. In formal logic this is represented by
~ (p & ~p)
But Aristotle is talking about what things are, so that we
consider object x, and property A, we can never have a proposition of
the form
~ (Ax & ~Ax)
i.e. x is both A and
not A.
Now the Law of the Excluded Middle deals with exactly the
same issue as above, but makes clear that if a reference is ambiguous, a
proposition may appear to be both true and false at the same time, but there
can be no contradiction in addressing the facts themselves.
But in all of this, the axiom ~ (p & ~p) is ever
present, either in this form, or in the form (p É p),
p implies p. For logicians (p É
q) is by definition saying the same thing as ~ (p & ~p). This is not so clear in natural
language, since if I consider whether p
implies q, then if p is in fact false, our immediate
reaction is that we cannot tell.
And “p implies p” is the same thing as “p is equivalent to
p” - since equivalence here means that p implies p and vice versa!
As I said earlier, if we look at formalisms, it is useful to
see how they apply to robust argumentation, and here the idea that p implies p
can be expanded to the following three principles:
(i)
p is equal to and equivalent to itself,
(ii)
if we say that x is A, x cannot also be not A at
the same time,
(iii)
but just to be precise, if we say that x is A, it cannot also be not A at the same
time.
Note the very subtle difference between (ii) and (iii).
(iota being the
affirmative operator ! )
Martin Prior
The Philosophy Takeaway 'Open Topic' Issue 34
