In this lesson we define by means of truth tables three of the other four logical operators: disjunction, conditional and biconditional.
The truth table for disjunction is:
p q p Ú q T T T T F T F T T F F F
Once again, because disjunction is a relationship involving a compound statement, it will have two variables. So, we used four lines. Note that the only situation in which p Ú q is false is the fourth one. This is because we are defining disjunction as "either . . . or . . . , or possibly both". In line four neither p nor q is true, so the compound is false. This is the inclusive use of disjunction.
The inclusive use does not exactly coincide with ordinary language use. There is an exclusive use of disjunction: Either this or that but not both. But we will not be symbolizing or using this. But the inclusive use does match up in most cases with the way we ordinarily use the "either . . . or" expression. For instance, if your boss were to say that you would be paid for overtime if you were to work either extra hours or on weekends, then you would be upset if you did both and got paid for the extra hours through the week but not the weekend work.
The truth table for the conditional compound is not easy to grasp, because the conditional in propositional logic is counter-intuitive. It does not exactly match up with the use of "if . . . then" in ordinary language. For a discussion of the problem involved in this mismatch, see the reading where this problem is discussed.
The conditional, like the other compounds, is defined by its truth table (its truth-value is a function of the atomic statements and the defined relationship) and not ordinary language. But the preceding ones are more like our ordinary usage than is the conditional:
p q p É q T T T T F F F T T F F T
Note that the only situation in which a conditional statement is false is when the antecedent is true and the consequent is false. Thus, the following statement would be true, even if Mary were not at the beach:
If Mary is at the beach, then she is swimming.
Moreover, if she were not at the beach and she were not swimming, the statement would still be true. Even if the antecedent and the consequent, separately, were false, the conditional statement as a whole would be true.
The biconditional is defined as two conditional statements, which have been reversed but involving the same propositions, conjoined together, such as (p É q) & (q É p) and is represented as "p º q". Its truth table is:
p q p º q T T T T F F F T F F F T
An example of a biconditional statement is this: "The lights are on if and only if John is at home." That this is a true statement depends on the necessity of John being home for the lights to be on. We can well imagine that his being home would be sufficient for the lights to be home. This idea could be expressed in the proposition: "If John is home, then the lights are on." But we can also easily imagine some other cause that would be sufficient for the lights to be on--someone else turned them on. For the other part of the biconditional to be true--if the lights are on, then John is home--we need to imagine a situation in which John is the only one that could turn them on. This is not a very likely situation. Nor is the next one, but it provides an additional illustration of the biconditional.
Let's say we had a circumstance in which either John or Mary were the only ones who could turn on the lights. Then we could have the following: "The lights are on if and only if either John or Mary are at home." This proposition would be symbolized as:
L º (J Ú M)
We will encounter less contrived examples later.
Exercise 6 provides practice opportunities.
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Copyright © 1999, Michael Eldridge