Propositional Logic is one attempt to symbolize without remainder our ordinary, propositional language. As noted in the last lesson, it does not try to capture the internal logic of propositions. Nor does it try to handle all utterances. It limits itself to statements that are either true or false. But for these sorts of statements it provides a way to symbolize them completely by using variables (capital letters) and constants (the symbols for the logical operators). In the last lesson we defined these constants in a rough sort of way. We will now do so more precisely by making use of truth tables.

A truth table is a device to specify all of the possible truth-values of a given atomic or compound proposition. The truth table for an atomic proposition is a simple affair, since there are only two possibilities--true or false. Thus a given proposition, "The sun is shining", can be symbolized by "S" and its truth-value possibilities can be displayed in a two-line, one column table:

S
______

True

False

A truth table for a compound proposition, as we shall see, can be a complex affair involving many columns and lines. There is no hard-and-fast rule for the number of columns, but there will be at least one for each simple proposition and one for each compound proposition. Sometimes, for clarity's sake, there will be additional columns. (This will be made clearer below.) The formula for determining the number of lines (or rows) in a truth table is: The number of lines is 2 lines to the nth power (2ⁿ), where n is the number of variables (distinct letters). Thus, if there are two distinct letters, say, p and q, then there will be four lines. The formula for determining how many "Trues" and how many "Falses" to enter is: Column 1: half and half; column 2: quarters; column 3: eighths, etc. Using "T" to stand for "True" and "F" to stand for "False", we can display a four-line truth table as follows:

p	q
T	T
T	F
F	T
F	F

But note: all we have done thus far is indicate the possible truth-values of p and q separately. We have treated them as free-standing atomic propositions. If we want to show the truth-value possibilities of the compound, p & q, then we need to add a column:

p	q	p & q
T	T	T
T	F	F
F	T	F
F	F	F

Now we have a complete truth table for p & q. We have shown all the truth-value possibilities for each atomic proposition and the truth value of the compound. (Note that the two columns on the left, the guide columns, are in bold.) We are showing that in the four possible situations involving p and q, p & q will be true in the first situation only and false in the other three. This is so because we are defining conjunction as the simultaneous assertion of two propositions. The compound is true, if, and only if, both atomic propositions are true. In other words, if either or both atomic propositions are false, then the conjunction is false. Moreover, we are saying there are no other truth-value situations than these four. No more, no less.

You will be expected to use the formulas in this lesson to construct tables; you will also be expected to memorize the truth tables of each of the logical operators.

Exercise 5 provides you with an opportunity to answer some questions about truth tables.

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Copyright © 1999, Michael Eldridge