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Statements, truth values and truth tables

A statement is an assertion that can be determined to be true or false. The truth value of a statement is T if it is true and F if it is false. For example, the statement ``2 + 3 = 5'' has truth value T. Statements that involve one or more of the connectives ``and'', ``or'', ``not'', ``if tex2html_wrap_inline76 then'' and `` tex2html_wrap_inline76 if and only if tex2html_wrap_inline76 '' are compound statements (otherwise they are simple statements). For example, ``It is not the case that 2 + 3 = 5'' is the negation of the statement above. Of course, it is stated more simply as ``2 + 3 tex2html_wrap_inline82 5''. Other examples of compound statements are:

If you finish your homework then you can watch T.V.
This is a question if and only if this is an answer.
I have read this and I understand the concept.

In symbolic logic, we often use letters, such as p, q and r to represent statements and the following symbols to represent the connectives.

tabular20

Note that the connective ``or'' in logic is used in the inclusive sense (not the exclusive sense as in English). Thus, the logical statement ``It is raining or the sun is shining '' means it is raining, or the sun is shining or it is raining and the sun is shining.

If p is the statement ``The wall is red'' and q is the statement ``The lamp is on'', then tex2html_wrap_inline110 is the statement ``The wall is red or the lamp is on (or both)'' whereas tex2html_wrap_inline112 is the statement ``If the lamp is on then the wall is red''. The statement tex2html_wrap_inline114 translates to ``The wall isn't red and the lamp is on''.

Statements given symbolically have easy translations into English but it should be noted that there are several ways to write a statement in English. For example, with the examples above, the statement tex2html_wrap_inline116 directly translates as ``If the wall is red then the lamp is on''. It can also be stated as ``The wall is red only if the lamp is on'' or ``The lamp is on if the wall is red''. Similarly, tex2html_wrap_inline118 directly translates as ``The wall is red and the lamp is not on'' but it would be preferable to say ``The wall is red but the the lamp is off''.   Click on the microscope for a more extensive list of English equivalents.

The truth value of a compound statement is determined from the truth values of its simple components under certain rules. For example, if p is a true statement then the truth value of tex2html_wrap_inline122 is F. Similarly, if p has truth value F, then the statement tex2html_wrap_inline122 has truth value T. These rules are summarized in the following truth table.

tabular29

If p and q are statements, then the truth value of the statement tex2html_wrap_inline110 is T except when both p and q have truth value F. The truth value of tex2html_wrap_inline142 is F except if both p and q are true. These and the truth values for the other connectives appear in the truth tables below.

tabular34

From these elementary truth tables, we can determine the truth value of more complicated statments. For example, what is the truth value of tex2html_wrap_inline118 given that p and q are true? In this case, tex2html_wrap_inline166 has truth value F and from the second line of the tables above, we see the truth value of the compound statement is F. Had it been the case that p was false and q true, then again tex2html_wrap_inline166 would be false and from the fourth row of the above table we see that tex2html_wrap_inline118 is a false statement. tvalq1"   To consider all the possible truth values, we construct a truth table.

tabular39

The lower case t and f were used to record truth values in intermediate steps.   Note that while a truth table involving statements p and q has 4 rows to cover the possibility of each statement being true or false, if we have additional information about either statement this will reduce the number of rows in the truth table. If, for example, the statement p is known to be true, then in constructing the truth table of tex2html_wrap_inline188 we will only have 2 rows. Truth tables involving n statements will have tex2html_wrap_inline192 rows unless additional information about the truth values of some of these statements is known.  

A statement that is always true is called logically true or a tautology. A statement that is always false is called logically false or a contradiction. Symbolically, we denote a tautology by 1 and a contradiction by 0.


next up previous
Next: Logical equivalence and implication Up: NOTES ON SYMBOLIC LOGIC Previous: NOTES ON SYMBOLIC LOGIC

Peter Williams
Mon Sep 2 15:51:33 PDT 1996