A Gentle Introduction To Mathematics - Implications

sinbad989 (52)in #mathematics • 7 years ago (edited)

Previously, we’ve covered the idea of predicate and conjunctions. Today, we’re going to cover implications.

Suppose a mother makes the following statement to her child:

“If you finish your peas, you’ll get dessert.”

This is a compound sentence made up of two simple sentences, P = “You finish your peas” and D=”You’ll get dessert.” This is an example of a conditional. Conditionals are if-then type statements. If we rephrase mother’s statement it would sound like this: “If P then D.”

The relationship of P and D is an example of a implication, in other words, P implies D. If you have a mathematically inclined mother she would sound something like this, “Finishing your peas implies that you will receive dessert.”

Fortunately, we don’t have that kind of mother.

A conditional statement’s components are called the antecedent (e.g. “finish your peas”) and the consequent (e.g. “you’ll get dessert”). The conditional involving an antecedent A and a consequent B is expressed symbolically (mathematicians love symbols) using an arrow:

The truth table for this connective is given as,

Note that this truth table compared with disjunction from our previous post, has only a single row with phi. For implication, it’s on the second row, while on disjunction it’s on the fourth row.

Conditional as Threats

Conditionals are commonly used to express threats. Another way to express a threat is to use disjunction – “Finish your peas, or you won’t get dessert.” This expression has the same logical content as “If you get dessert then you finished your peas.” (Notice that the roles of antecedent and consequent have been switched.)

The operator that results if we do make this modification is called the biconditional and is expressed in English using the phrase, “if and only if” (abbreviated by mathematicians as “iff” much to the consternation of spell-checking programs everywhere). The biconditional is symbolized by using an arrow that points both ways. Its truth table follows.

While we strive for precision, we do not necessarily recommend the use of phrases such as “You will receive dessert if, and only if, you finish your peas” with children.

To avoid confusion of the role of antecedent and consequent in biconditional, the switched-around sentence has been given a name: it is the converse of the original statement. The converse of an implication has the pieces switched about. Another conditional that is distinct from a given conditional is its inverse. Neither of these is the same as the original implication.

If you start with an implication, form its converse, then take the inverse of that, you get a statement having exactly the same logical meaning as the original. This new statement is called the contrapositive. Oddly, this is one of those times when two wrongs do make a right.

One final advice on conditionals: don’t confuse logical if-then relationships with causality. Most of the if-then sentences we encounter in ordinary life are described as cause and effect.

“If you cut the green wire the bomb will explore” (Okay, that one is an example from the ordinary life of a bomb squad technician)

It is best to think of if-then relationships we find in logic as divorced from the flow of time.