A while back I blogged about Leibniz's Law of Identity and how all other logical truths descend from this one necessary formula. I faced some criticism from a few people claiming that axioms such as Modus Ponens and the Law of the Excluded Middle were not evident within the Law of Identity.
Leibniz's law can be stated thus: A=A. It is evident how this principle is a necessary precursor to any subsequent logic, as if A is not identical to itself, then no other true statement can be made. If A is not A then 1+1 no longer equals 2, as the definition of 1 is fluid and not static. Staticity of terms (ie, self-identity) is necessary for any inference from the nature of those terms. New inferences cannot be developed without this principle.
But how is A=A a sufficient basis for all other self-evident logical truth?
Modus Ponens is the basic law of inference in logic. It can be stated thus:
1.P -> Q.
It states that if we accept "P entails Q" (for example, being upper class entails voting conservative), and if P happens to be true, then Q must also be true. In effect, it states the principle that if we accept one thing leading to another thing, we must accept that when the
one thing happens the second does too. Common sense.
It can be restated as an equation:
if (P->Q) then (if(P)->(Q))
P->Q = P->Q.
Given this form, it is clear to see how it is merely an affirmation of tautology. It bears precisely the same form A=A. It holds no additional content, and therefore is not a seperate truth, but simply a derivation from the Law of Identity. Leibniz's law is both the necessary and sufficient condition for ensuring the validity of Modus Ponens.
The Law of the Excluded Middle is similarly obvious:
A statement must be either true, or if not true, then false. There is no inbetween, a statement cannot be both true and false, and cannot be neither true nor false. It must be one and only one of these. Without wishing to go into Logical Positivist-type delineations of some statements being meaningless and therefore outside the realms of truth or falsehood because they contain no sensible assertion, it should be clear how the LEM is true.
We can restate the LEM as A=not-not-A, ie A cannot equal the negation of A. Or, A=A.
What Leibniz believed was that all analytic (self-evident) truth was explicitly derived from A=A. This, he saw as the emanation of truth from (and within) the mind of God. Human intellect could trace this path through the use of logic. Synthetic (contingent) truth was similarly derived solely from A=A but the process by which this happens is not accessible to the human mind. The manifestation of reality and the circumstances which surround us are necessarily derived from God's own nature and thought - therefore this is necessarily the best of all possible worlds (as it must be, if derived from the source of goodness itself).
A=A is itself a statement (albeit an obscure one) about God's nature: God is identity. I am that I am (Exodus 3:14)