------------------------------------------------------------------------------

-----------------------------------------------

A syllogism (short for categorical syllogism) is a specific argument form. It consists of three categorical sentences that contain precisely three different terms, each one of which appears in two different sentences in the group.

The definition leaves room for a number of different combinations of categorical sentences, but let's take one combination as an example. We'll use the categorical forms: All P are Q and No P are Q. We'll take the latter to be 'No M are Q,' in which case the syllogism is as follows:

There are three categorical sentences and three terms, M, P, and Q, and each term appears in two of the sentences. So we have a legitimate syllogism. The first two sentences are premises, and the third one is the conclusion. Read the dashed line as 'therefore.'

The basis for the validity of this argument is that all P's lie inside Q, but no M's are inside Q. So any M that exists has to be outside Q and therefore outside P. The conclusion, therefore, is that no M's are P's. Get the picture?

Back to Index

--------------------------------------------

Another approach to the validity question is to use truth tables to tie down the argument forms. The approach draws on logical functions to define argument conditions and isolate valid from invalid conclusions. (Logical functions are like arithmetic functions except they use truth-values instead of numbers.)

Different from the syllogism, this group of arguments uses simple and compound sentences as the logical parts. (When viewed systematically -- i.e., as a deductive system -- the approach is known as the sentential (or propositional) calculus, and is one of a variety of logical systems.)

To get a better perspective of the situation, consider all statements that could possibly be made and split them into separate piles according to their construction. Simple sentences therefore go in one pile. Conjunctions made up of simple sentences -- joining them by AND -- go in another pile. A third pile might have the simple disjunctions -- joining them by OR. A fourth might have the basic negations.  A fifth might have the simple conditionals. And another might have all the rest, including the more complex statements made up of combinations of conjunctions, disjunctions, negations, and conditionals.

Now we assign truth-value possibilities to the sentences in each group, by using truth-value functions (or simply truth functions) -- also presented as truth tables.

For each simple sentence, it's enough to say directly that it's either true or false.

A simple conjunction is true if both of its simple sentences are true, but otherwise it's false.. In table form, using symbols for simple sentences, the function would be as follows:

On line 2 (colored yellow): P is true and Q is false, so the compound sentence, P and Q, is false.

For disjunctions we have the truth-value function:

In this case, a disjunction is false (blue) if both simple sentences are false, otherwise it's true.

For negations the function table would say the negation is false if the simple sentence is true, otherwise it's true, as follows.

Finally, the conditional could be regarded as false if the consequent (the "then" part) is false and the antecedent (the "if" part) is true, otherwise it's true, as in the following table:

This interpretation (highlighted in green) might raise some eyebrows, as indeed it has in the past. But it's generally accepted, these days. For instance, whenever the antecedent (P) is false, the sentence is true: If the moon is made of green cheese, then cats are reptiles.

Finally, these functions would be used to form the more complex truth-value functions for the sentences in the pile of complex, possibly mixed sentences.

Back to Index

---------------------------------------------- 

Let's consider an example of an argument to see how truth tables can be used to mechanize your logical intuition.

By replacing the content of this argument with symbols, we can eliminate the meaning and get a strictly formal or structural argument.  So we can

And we have the symbolic form:

Using T for true and F for false, our truth table then becomes:

Our conclusion fails if both P and R are false, as in the purple row. That is, it fails when the street is not dry and it is raining, so you might expect the argument to be perfectly all right.

However, in that case, our two premises are true, as you can see in the 4th and 5th columns of the table. And that's bad news, because we're especially interested in the cases where the premises are true. We want to know if the conclusion follows when that's the case. But the argument fails at that point. This shows the argument is not truth-functionally valid.  The point is, you might go shopping even if it's pouring.

Back to Index

--------------------------------------

Top of Page