Necessity and Logic

1. The Principle of Contradiction.  The principle says, basically, that if something is contradictory it is false, and hence if the negation of something is contradictory, then the thing is true.

o The principle contains a number of different sub-principles.  For instance, at one point Leibniz breaks it up into:

o Non-contradiction (strictly speaking): p is not both true and false

o Bivalence: either p is true or p is false.

o Leibniz doesn’t give us a proof of the proposition.  He takes it to be crucial and self-evident.  It’s interesting to note that some logicians deny bivalence—and manage to do a lot of things without it.

o There are other basic rules Leibniz gives.  For instance, not-not-p being p, etc.  It seems that Leibniz uses “Principle of Contradiction” as a catch-all title for all purely logical rules, other than purely self-evident assertions such as the concept of a horse is not the concept of a chair.

o Leibniz rightly sees the principle as at the heart of the method of proof known as reductio ad absurdum.  Suppose we want to prove something.  We can prove it by showing that its negation implies a self-contradiction.  Any proof can be put in this way.  Here’s a modern example.  Some numbers are rational, in that they can be written in the form n/m where n and m are integers.  Suppose we want to show: Sometimes an irrational number raised to an irrational power is rational.  We can prove it by reductio ad absurdum.  Suppose this is false.  Then, whenever an irrational number is raised to an irrational power, the answer is irrational.  Thus, Ö2^Ö2 is irrational.  Hence, (Ö2^Ö2)^Ö2 is also irrational.  But (Ö2^Ö2)^Ö2=Ö2^Ö2×Ö2=Ö2²=2.  Hence it is false that whenever an irrational number is raised to an irrational power and it is true (by assumption).  Hence, we have a contradiction, and so the proof is complete: we have shown that assuming the negation of the claim we are to prove leads to a contradiction.

o Anything that can be proved by means of the principle of contradiction is to be called a logically necessary truth.  A proposition that can’t thus be proved or disproved is logically contingent.  A contingent truth is a truth that can’t be proved by means of the principle of contradiction.

2. The second principle is the Principle of Sufficient Reason (PSR).

• As used by Leibniz, the PSR says that for every true proposition, (a) there is a reason why it is true rather than being false, and (b) this reason is sufficient in the sense that it could be used to derive the true proposition.  Some will drop the latter condition, but it is essential to L’s thought.  Thus, Leibniz will not allow one to say that the reason Smith was murdered was because Jones made a free choice whether to murder Smith or not.  For from the fact that Jones made a free choice whether to murder Smith or not it does not follow that Smith was murdered.
• Let’s concretize.  Leibniz will say—and we will soon see him saying it—that the reason for p is to be found in God’s nature.  It is a part of God’s nature to be perfectly good, and hence to create things in the best possible way.  So r is something like: There is a perfectly good God who always does the very best that can without contradiction be done.  Then r explains p and hence explains all contingent facts according to L.  But this doesn’t mean that all contingent facts stop being contingent.  Consider some contingent fact: Alex Pruss has an even number of hairs on his head now.  The explanation then says that this is better, and hence God made it be so.  But that it is better would require an infinite number of steps to verify.  One would have to compare infinitely many possible universes where I have an odd number of hairs on my head now, and make sure that none of those is better than or as good as this universe.  And we can’t even do the comparison of two universes very well.  Take two different societies: How often can we see conclusively that one is better than another?  To do so would require much more analysis than we’re ordinarily capable of, except in extreme cases (clearly Stalinist society was rather worse than most!)

3. Free will.

• Actions do come from will. Freedom consists in doing what you willed, rather than acting against your will.
• “The wise mind always acts according to principles”. The principles are general and not exceptions.

4. “What is independent of sense and matter”

• Sensible qualities—i.e., what is sensed—are actually occult (i.e., hidden).
• See text.

5. Copernicus and “hypotheses”.