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Symbolic Logic - Disjunctions

statement true account doe

Another word used in both ordinary English and in logic is "or." Someone who says, "Either he did not hear me, or he is being rude," is saying that at least one of those two possibilities is true. By connecting the two possibilities about which he or she is unsure, the speaker can make a statement of which he or she is sure.

In logic, "or" means "and/or." If p and q are statements, p V q is the statement, called a "disjunction," formed by connecting p and q with "or," symbolized by "V."

For example if p is the statement, "Mary Doe may draw money from this account," and q is the statement, "John Doe may draw money from this account," then p V q is the statement, "Mary Doe may draw money from this account, or John Doe may draw money from this account."

The disjunction p V q is true when p, q, or both are true. In the example above, for instance, an account set up in the name of Mary or John Doe may be drawn on by both while they are alive and by the survivor if one of them should die. Had their account been set up in the name Mary and John Doe, both of them would have to sign the withdrawal slip, and the death of either one would freeze the account. Bankers, who tend to be careful about money, use "and" and "or" as one does in logic.

One use of truth tables is to test the equivalence of two symbolic expressions. Two expressions such as p and q are equivalent if whenever one is true the other is true, and whenevet one is false the other is false. One can test the equivalence of ~(p V q) and ~pΛ ~q (as with the minus sign in algebra, "~" applies only to the statement which immediately follows it. If it is to apply to more than a single statement, parentheses must be used to indicate it):

p q ~p ~q pVq ~(pVq) ~pΛ~q
T T F F T F F
T F F T T F F
F T T F T F F
F F T T F T T

The expressions have the same truth values for all the possible values of p and q, and are therefore equivalent.

For instance, if p is the statement "x > 2" and q the statement "x < 2," p V q is true when x is any number except 2. Then (p V q) is true only when x = 2. The negations p and q are "x 2" and "x 2" respectively. The only number for which ~ p Λ ~ q is true is also 2.


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