## Tuesday, May 8, 2012

### Math & Logic

When it comes down to it, logic (AKA: propositional calculus) is an attempt to turn language into mathematics. We were so impressed, and properly so, with mathematics that we try to systematize most, if not all, human understanding in terms of mathematics, and language is no different. Ever hear that inevitable kid in math class moan, "Why should we learn math anyway?" Well, this is the answer. If you understand mathematics, you understand how we have tried to understand and depict just about every field of human study - which gives you a huge head start in learning any more specialized field.

Mathematics is also a logical/deductive system and works by means of well-established operators, inferences, and well-formed formulas. It's axiomatic structure has been detailed with meticulous rigour, so whether mathematics is in the domain of logic, or logic attempts to emulate mathematics is a kind of chicken/egg issue, but my purpose in linking the two together here is to discuss something aout logic that most, it seems, do not realize.

### The Dumbing Down of Logic

When I was going to university an interesting transition was underway. Logic wasn't terribly popular, because it was difficult, but logic was still the big money-maker for what was otherwise a department in chaos. Many were saying it was important to make logic more accessible, in the form of what was called "critical thinking." Critical thinking focused less on the math and more on fallacies. I was one of the lucky ones; I managed to steal my logic education before the shift was enacted. I assume this kind of shift in orientation was not uncommon among philosophy departments because nowadays we see plenty of people who can recite fallacy names, but do not understand a very important thing...

### What Logic Is ... and Isn't

Logic is not what most think it is; it is not a path to truth. Logic is a test of "internal consistency." Does the argument agree with itself? Despite that logic works with what are called "truth values," truth values do not necessarily map onto reality-truth. This is the difference between validity and soundness. Validity is a assessment of the argument form - the equation, as it were - whereas soundness refers to whether the content of the argument maps onto reality. A valid argument is one in which if the premises are true, then the conclusion must be true. A sound argument is one in which the argument is both valid and the premises are true. It is quite possible for an argument to be valid without being sound. Let me express this in a simple math example.

2+2=4. Now we know that this is true, courtesy of the definitions of 2, +, = and 4. We also know that if there are two groups of two oranges on the table in front of us, that we have 4 oranges total. Yes, this skeptic just said "know." What we do not know is whether we have 2 groups of two oranges on the table before us or not. The equation doesn't tell us that. To determine that, we actually have to look at the table and check. Thinking of a logical argument as an if-then conditional might be one of the best ways of understanding what logic is. If the premises are true, then the conclusion must be true. 2+2 may equal 4, but that is not particularly helpful if we are not referring to two actual groups of two. If there are two groups of three, then the 2 in 2+2 is garbage in, and the 4 is garbage out.

Now let me express this in the form of a categorical syllogism:

All humans are fish. (If A then B)
I am a human. (C is A)
Therefore, I am a fish (C is B)

This argument is valid, but unsound, for obvious reasons (all humans are not fish, indeed no humans are). Therefore the conclusion that I am a fish cannot be relied upon. "All humans are fish" is garbage in, and "I am a fish" is garbage out.

And this points at the issue with logic. The propositions actually have content. It may be mistaken content, but it is content nonetheless. We can plug any content into a valid argument form and crank the equation to get a conclusion, but if the conten t is mistaken, then there is a very good probability the conclusion will be as well. Of course, matters are a little more complicated than that, but it serves well enough for our purposes here.

### Truth: Analytic vs Synthetic

Moving via wff (well-formed formula) through steps of an argument provide what is called analytic truth - the truth of which is entirely dependent upon the definitions of the terms and inferences involved, with no reference to reality whatsoever. When we reference reality, then we can move from analytic truth to what is called synthetic truth. Unfortunately these are easily confused, and have been confused quite a lot in the history of logic (the language of logic actually makes such confusion more likely). Perhaps it all started with the idiot who first called truth values "truth values." One for equivocation...

That reflects the hopes and dreams we had for logic throughout history. We dreamed of a systematic way of deriving "novel" (previously unknown) knowledge from already "known" premises. Sadly, the results are not living up to the hopes. We get knowledge, sure, but it is what was already contained in the premises - it is "trivial."

In computer programming a Boolean relationship can be expressed in two ways really. It can use the terms true and false or it could use something else, some other terms, say 0 and 1. To the computer, whether we use "true" and "false" or "0"and "1" is irrelevant. However, to we poor humans, the use of "true" and "false" has content beyond the calculation itself and this leads to error.

### The Sordid Habits of Snufflegrorfts

So, let's look at another example of a categorical syllogism that is valid.

All snufflgrorfts flooft.
Rufus is a snufflegrorft.
Therefore Rufus floofts.

Now what do we know from this argument itself? Pretty much nothing, actually. We do not know if there is such a thing as a snufflegrorft much less that there is one named Rufus who/that floofts. Generally we do not speak of floofting in polite company, but we are dealing with an important point so we'll check our petty sensitivities at the door just this once. However, because the argument is valid (a wff), we do have an analytic truth: if there are snufflegrorfts, and if all snufflegrorfts flooft and if Rufus is a snufflegrorft, then Rufus floofts. See all those "ifs?" The wise person doesn't confuse the validity of the argument with it's claims about snufflegrorts, Rufus, or floofting - it's claims about reality.

### The Ontological Argument

So why is this talk about analytic truth, synthetic truths, and snufflgrorfts personal habits interesting? Why have I gone to such pains to write all this? It has impact on some arguments involving the existence of God, especially the ontological argument. The ontological argument suggests that God is a perfect being, and that perfection entails existence as a matter of definition.

Perfection entails existence.
God is perfect.
Therefore, God exists.

The astute will have noticed that these are all matters of definition, and deductive rules of inference, with no reference whatsoever to reality except for the claim at the end. At no time do we reference reality with the claim that perfection entails existence. At no time do we have a synthetic reference for God being perfect, even if it is worded like there is one. Incidentally, we have no synthetic verification of God existing. The argument represents an attempt to shift from analytic truth to synthetic truth, but at no time is reality actually consulted. Garbage in = garbage out.

Unless it touches reality, the argument is merely an elaborate web of fabrications. The definitions involved may seem convincing, but they are still definitions only.

The Cosmological arguments suffer from the same fatal flaw. They seem plausible only because it seems reasonable to assume every effect has a cause, and the argument uses that "seeming" to (entertainingly enough) claim an uncaused cause (a cause that is not an effect), in order to avoid a infinite regress. Then assumptions are made about the nature of that uncaused cause - namely that it is God. No matter how you look at it, however, these arguments rely on unsubstantiated (read: analytical) claims to attempt to prove a synthetic claim. Again, without reference to reality, the move from analytic to synthetic is unwarranted and indefensible.

Garbage in = garbage out.