Logical Proofs

Unfortunately, determining logical entailment by checking all possible worlds is impractical in general. There are usually many, many possible worlds; and in some cases there can be infinitely many.

The alternative is logical reasoning, viz. the application of reasoning rules to derive logical conclusions and produce logical proofs, i.e. sequences of reasoning steps that leads from premises to conclusions.

The concept of proof, in order to be meaningful, requires that we be able to recognize certain reasoning steps as immediately obvious. In other words, we need to be familiar with the reasoning "atoms" out of which complex proof "molecules" are built.

One of Aristotle's great contributions to philosophy was his recognition that what makes a step of a proof immediately obvious is its form rather than its content. It does not matter whether you are talking about blocks or stocks or sorority girls. What matters is the structure of the facts with which you are working. Such patterns are called rules of inference.

As an example, consider the rule of inference shown here.

Here is a similar example. We know that all Accords are Hondas, and we know that all Hondas are Japanese cars. Consequently, we can conclude that all Accords are Japanese cars.

Now consider another example. We know that all borogoves are slithy toves, and we know that all slithy toves are mimsy. Consequently, we can conclude that all borogoves are mimsy. What's more, in order to reach this conclusion, we do not need to know anything about borogoves or slithy toves or what it means to be mimsy.

What is interesting about these examples is that they share the same reasoning structure, viz. the pattern shown below.

The existence of such reasoning patterns is fundamental in Logic but raises important questions. Which patterns are correct? Are there many such patterns or just a few?

Let us consider the first of these questions. Obviously, there are patterns that are just plain wrong in the sense that they can lead to incorrect conclusions. Consider, as an example, the (faulty) reasoning pattern shown below.

Now let us take a look at an instance of this pattern. If we replace x by Toyotas and y by cars and z by made in America, we get the following line of argument, leading to a conclusion that happens to be correct.

On the other hand, if we replace x by Toyotas and y by cars and z by Porsches, we get a line of argument leading to a conclusion that is questionable.

Here is an example of bad reasoning in Sorority World.

What distinguishes a correct pattern from one that is incorrect is that it must *always* lead to correct conclusions, i.e. they must be correct so long as the premises on which they are based are correct. As we will see, this is the defining criterion for what we call deduction.

Now, it is noteworthy that there are patterns of reasoning that are sometimes useful but do not satisfy this strict criterion. There is inductive reasoning, abductive reasoning, reasoning by analogy, and so forth.

Induction is reasoning from the particular to the general. The example shown below illustrates this. If we see enough cases in which something is true and we never see a case in which it is false, we tend to conclude that it is always true.

Abduction is reasoning from effects to possible causes. Many things can cause an observed result. We often tend to infer a cause even when our enumeration of possible causes is incomplete.

Reasoning by analogy is reasoning in which we infer a conclusion based on similarity of two situations, as in the following example.

Of all types of reasoning, deduction is the only one that guarantees its conclusions in all cases, it produces only those conclusions that are logically entailed by one's premises. For this reason, in what follows, we concentrate entirely on deduction and leave these other forms of reasoning to others.