Of Science and Faith: Deductive Logic

By Jim Pemberton

In discussing science and faith, the word “reason” is often bandied about without much to say as to what it actually is. Those with any education in philosophy know of the three laws of logic and how to construct syllogisms. That excludes most of the world. So in one short blog article, I intend to lay it out as simply as possible. The reason is that you can’t understand the scientific method without understanding how syllogisms work. Unfortunately, many scientists, while they know how to use the scientific method, don’t understand how it works logically. That’s the reason for discussing it. Hopefully by the end of the next article you will have at least an inkling on how it works and have a leg up on most scientists.

Of Science and Faith

The Three Laws of Logic

There are three important laws of logic that philosophers have discovered over the millennia. They will seem simple at first, and they are. But many simple things get overlooked if they aren’t discussed, and it’s a big deal if these simple things are important.

The first law is called the Law of Identity. The law stated is basically that something is itself. You’re asking yourself why it took a philosopher to figure this out. The fact is that it’s so simple that no one thought to write it down. But it’s important because more difficult things are based on this simple fact and we will forget this simple fact in understanding the more complex thing if we don’t make a note of it. The Law of Identity can be written as a simple formula:

A=A

The second law is built on this fact and is almost as simple. It’s called the Law of Non-Contradiction. This basically means that something is not everything that the something is not. It can be written as this formula:

A≠~A

The tilde (~) means “not”. So you might say this: “A is not not A” Simple?

The third law is only slightly more difficult and is based on the first two. It is called the Law of Excluded Middle. This means that everything must either be or not be. There is no third way where it kind of is, but not really. It can be written as a formula also:

For all A: A or ~A

The three laws are the basis for mathematics. They are also the basis for logic.

Deductive Logic

Deductive logic is the area of logic where the three laws are employed to analyze the logical relationships between things. It is epistemological1 in nature in that it involves the discovery of reality by observing consistent patterns in reality. It isn’t ontological in nature in that it is limited to discovery by relationship rather than by directly apprehending static properties of things.

Since it involves relationships between things, it stands to reason that we need two things rather than one. Thw laws observe two things: A and ~A. For developing syllogisms our two things will be ‘A’ and ‘B’. ‘B’ could be ‘A’ or it could be in the set of ‘~A’. That is that ‘B’ is generally not everything that is ‘~A’, but it is at least one thing that is ‘~A’. Since ‘B’ could be ‘A’, we have at least one kind of possible logical relationship before us:

A=B

This is analogous to the law of identity, but is not exactly the same. This logical relationship doesn’t mean to say that A and B are identical, but that their occurrences are always logically simultaneous. That is, if you experience one of these things, you can be sure that the other one is occurring even if you don’t experience it. For example, we can say that the clock always strikes twelve when both hands are pointing directly at twelve. If the clock is striking twelve, then the hands are both pointing at twelve. If the hands are both pointing at twelve then the clock is striking twelve. By the time the clock is finished striking twelve, we can expect the hands to no longer both exactly point to twelve. Right before the clock strikes twelve, we can expect the hands to not exactly be pointing at the twelve. So we have two things: A=”the clock striking twelve” and B=”the hands pointing to twelve”. (This assumes, of course, that you have a clock that has hands and that the clock also strikes twelve at the right time.)

But there is a far more common logical relationship called a conditional statement. This is most easily explained in terms of causation. That is to say that we notice when something causes something else. Now a conditional relationship may involve two things where one is not a cause of the other. For example, both things may be caused by a third thing. Or there may be some other mysterious mechanism at work. But occurrences of the two can be tested for this relationship. Remember that we are talking about a relationship that is epistemological in nature. So we talk about the relationship necessarily in terms of one thing causing the other, but we use words that remind us how we know something. The two important words that refer to each side of the conditional statement are “sufficient” and “necessary”. That is to say that if we know one thing, it is sufficient, but not necessary to know the other thing. On the other hand, if we know the other thing, it is necessary, but not sufficient to know the first thing. That which is sufficient is called the “antecedent”. That which is necessary is called the “consequent”. The formula looks like this:

A∩B

You would say “If A, then B”
Note that A is the antecedent and is sufficient but not necessary to know B.
Note that B is the consequent and is necessary but not sufficient to know A.

Let me explain the words better. The words “antecedent” and “consequent” can be misleading. Normally when the syllogism represents a causal relationship, there is one thing that is caused and many things that work together to cause it. The thing that is caused is the “antecedent” and one of the causes is the “consequent”. So you can say: “If I know the thing that is caused, then I know one of the things that caused it.” So knowing that thing that was caused is sufficient to conclude all of the things necessary to cause it. But the thing that was caused is not necessary to know the thing that caused it. There would be other ways to determine that.

But you would have problems if you said “I know one of the things that cause something, therefore I know that what it causes happened.” The problem is that one of the other things necessary to cause it might not have happened. I’ll give you an example:

Suppose you have a light in the ceiling and a common light switch wired up in the normal way into the wall such that the idea is to flip the light switch up in order to turn the light on. The light switch being up causes the light to come on. So you would explain the logical relationship this way: “If the light is on, then the light switch is up.” Observing that the light is on, you can conclude that the light switch is up without looking at it. But a blind person could not flip the light switch up and conclude that the light went on. The electricity might be out or the light bulb might be burned out. A rat may have chewed through one of the wires. But for the same reasons you cannot conclude that the light switch is down by observing that the light is off. However, a blind man can conclude that the light is off if he finds the light switch and ensures that the switch is down.

So in relatively short order we have established four logical patterns. Two are valid and two are invalid. The two valid ones are thus:

————————–

If A, then B      (relationship stated)
A                         (observation)
Therefore B2    (conclusion)

————————–

If A, then B
~B
Therefore ~A3

 ————————–

The two invalid patterns are thus:

————————–

If A, then B
~A
Therefore ~B4

————————–

If A, then B
B
Therefore A5

————————–

These patterns are called “syllogisms”. Knowing how to build a syllogism without falling into the invalid patterns is necessary to doing science correctly. You can see that if you have two things and you want to test their relationship, you have to know which one to make the antecedent and which one to make the consequent. If you get them backwards, then your syllogism will be invalid.

In the next article, I will talk about how the scientific method is used to test syllogisms.

 


1Remember that epistemology answers the question, “How do we know?”
2This pattern is called “modus ponens” Latin for “the way of affirming”.
3This pattern is called “modus tollens” Latin for “the way of denying”.
4This pattern is a formal logical fallacy called “denying the antecedent”.
5This pattern is a formal logical fallacy called “affirming the consequent”.

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