[EM] Logical Implication

[Anthony Simmons]

Repl to: MIKE OSSIPOFF <nkklrp at hotmail.com>
Subject: Re: [EM] Richard's frontrunners example

Forgive me for butting in; I've been skulking in the corners
until now.

>> Now I accept that it does seem that (not A) implies (if A,
>> then B). When A isn't true, I've heard people say things
>> like "If A, then I'm a monkey's uncle!". I had no idea
>> that it's literally true.

Absolutely.  It's a theorem of logic.  If you think about it,
that's how we reason when we prove that something is not true
by assuming that it's true and showing that it leads to a
contradiction.

>> Adopting and believing a definition that says that
>> something is true that (as I believed at the time) is
>> actually false, seems like more than just using
>> mathematical language. In that case, I'd rather make an
>> exception and not use that particular piece of
>> mathematical language, a definition of "if" that implies
>> something that is (as I then believed) false.

>> I still haven't heard anyone actually say what the
>> mathematical definiton of "if" is. Will it be in a
>> definition of mathematics? Won't someone tell me what it
>> is?

Sentential logic (in which the fundamental units are complete
sentences) is a kind of algebra.  In ordinary algebra,
addition is just a function.  Put a pair of numbers in, get a
sum out.  Logical implication (if ... then ...) works the
same way, and is also a function, except that the possible
values are "true" and "false".  Put a pair of logical values
in, get a logical value out.

Each logical function can be defined by a truth table, which
works just like a multiplication table.  For implication:

         A -> B

            B
           T   F
          ---|---
       T | T | F |
    A    |---|---|
       F | T | T |
          ---|---

There are sixteen possible binary operators, corresponding to
the sixteen possible truth tables with two variables.

The algebra of logic is really just a kind of number system,
and in fact if the symbols are interpreted properly, it is
just the arithmetic of the integers modulo two (in which
there are only two numbers, and 1+1=0).  Or, as someone just
did, we can interpret the algebra as being basic set theory
(e.g. Venn diagrams).

The meaning of "if ... then ..." (which is functionally a
single symbol) is nothing more than what's contained in that
table.

A caveat for purists only:  Functions are defined withing the
context of set theory, which is defined in terms of logic,
which has just been defined in terms of functions.  That kind
of circular definition causes loss of bladder control in
logicians, so it isn't really a valid mathematical
definition.  The rigorously correct way to define logical
implication is simply to say that it's one of the symbols in
a system that obeys certain rules, and is essentially
undefined, like the primitive (basic, undefined) concepts in
high school geometry.  Then the truth table is merely a
description rather than a definition.

Since this is my first message, I would also like to say that
I appreciate the effort people have put into sharing their
knowledge and thoughts.

    Tony