BasicsWhen making precise arguments, we often need to make conditional statements, like
Important note: The statement A ⇒ B does not make any claim about whether B is true if A is NOT true! It says only that if A is true, then B is true. While this point may seem obvious, it is sometimes a source of error, partly because in everyday communication we do not always adhere to the rules of logic. For example, when we say “if it's fine tomorrow then let's play tennis” we probably mean both “if it's fine tomorrow then let's play tennis” and “if it's not fine tomorrow then let's not play tennis” (and maybe also “if it's not clear whether the weather is good enough to play tennis tomorrow then I'll call you”). When we say “if you listen to the radio at 8 o'clock then you'll know the weather forecast”, on the other hand, we do not mean also “if you don't listen to the radio at 8 o'clock then you won't know the weather forecast”, because you might listen to the radio at 9 o'clock or check on the web, for example. The point is that the rules we use to attach meaning to statements in everyday language are subtle, while the rules we use in logical arguments are absolutely clear: when we make the logical statement “if A then B”, that's exactly what we mean—no more, no less.
We may also use the symbol “⇐” to mean “is implied by”. Thus
Finally, the symbol “⇔” means “implies and is implied by”, or “if and only if”. Thus
If A is a statement, we write the claim that A is not true as
If A and B are statements, and both are true, we write
- Rule 1
- If the statement
A ⇒ Bis true, then so too is the statement(not B) ⇒ (not A).The first statement says that whenever A is true, B is true. Thus if B is false, then A is false—hence the second statement.
- Rule 2
- The statement
not(A and B)is equivalent to the statement(not A) or (not B).Note the “or” in the second statement. If it is not the case that both A is true and B is true (the first statement), then either A is not true or B is not true.
QuantifiersWe sometimes wish to make a statement that is true for all values of a variable. For example, denoting the total demand for tomatoes at the price p by D(p), it might be true that
Important note: We may use any symbol for the price in this statement: “p” is a dummy variable. After having defined D(p) to be the total demand for tomatoes at the price p, for example, we could write
Another type of statement we sometimes need to make is