Chapter 6, A (technical) Language Change

Implication rules can be stated in several ways. We need to recognize them.

One-Way Implication Rules

In the chapter Implication Rules , we met the rule

When Aunt Jane visits her nephew Tom's home, Tom goes out to play

Rules like this can be said in different ways. This gives variety and choice in the way in which we write rules. The form of a rule does not matter, if we understand exactly what it says. The above one-way rule can also be rewritten (or restated, again without changing its meaning) using the words IF and THEN as follows.

IF Aunt Jane visits her nephew Tom's home THEN Tom goes out to play.

The word IF introduces a condition, namely Aunt Jane's visit to her nephew Tom's home. The word THEN introduces the consequence, what should occur, when the condition is satisfied. Here the consequence is Tom goes out to play. Since the original rule can be rewritten in the IF condition THEN consequence form, we say the original rule and the if-then form are conditional statements.

Note that a statement If A then B is only false when the situation or condition A occurs, but the anticipated consequence B does not.

Another way of writing the above one-way Aunt Jane and nephew Tom rule (with no change in meaning) is given by:

Aunt Jane's visit to her nephew Tom's home IMPLIES Tom goes out to play.

The words forces or makes may be used instead of the word implies. We could also use the word suggests, but in everyday use, a suggestion is optionally obeyed or followed while a rule (when it is correct) should or must be obeyed or followed. In talking about rules, we use the words implies, forces or makes for those rules we expect will be obeyed, or more precisely will never be disobeyed in the circumstances at hand. The explicit identification of such circumstances is exhaustive unless the circumstances in question are understood from a context, an obvious one, we hope.

Postscript : Instead of writing If A then B we may write B if A. The latter states that the situation B will happen if the situation A happens. That being said we cannot say that

B if and only if A

holds when there is a third situation C different from A, a situation which may occur when A does not, such that B if C also holds.

In the case

B if A
and also
B if C

the situation B may occur because of situation A or situation C, that is, due to A OR C. So when situation B occurs, the occurrence may be implied by A, C or another situation.

However, we can assert or state B if and only if A holds when B follows from the occurrence of A and whenever B occurs, so must A.

Two-Way Implication Rules

In the previous chapter Implication Rules, we met the rule

Tom goes out to play
when and only when
Aunt Jane visits his home.

This is an example of a two-way rule. Two-way rules can also be said or presented in different ways. Again the form of a rule does not matter, provided we recognize exactly what is meant. The above rule also can be rewritten (or restated, again without changing its meaning) in the if-and-only-if form:

Aunt Jane visits her nephew Tom's home
if and only if
Tom goes out to play.

This form suggests we call such rules biconditional statements. The prefix bi- here signals two ways. Whenever the condition (or situation) Aunt Jane visits her nephew Tom's home occurs, the other condition (or situation) Tom goes out to play must also occur, and vice-versa, if this rule is to be never-disobeyed.

You may prefer to say if and only if instead of when and only when. For instance, I might say or suggest to you: I will do that for you if and only if you do this for me. Alternatively, I might say or suggest to you: I will do that for you when and only when you do this for me. Tone provides the only difference between the two suggestions. Both of these suggestions represent a two-way obligation to which we might agree. Confusion or disappointment or false expectations may happen when suggestions such as these are not explicitly accepted or rejected.

Two-Way or Two One-Way Rules

The two-way Aunt Jane and nephew Tom rule above is rewritten (with no change in meaning) as

Aunt Jane visits her nephew Tom's home implies Tom goes out to play,

and also that

Tom goes out to play implies Aunt Jane visits her nephew Tom's home.

In this form, the two-way rule is seen to be the same as two one-way implication rules, each going in the opposite direction.

Equivalent Conditions (or Situations)

Two situations or conditions A and B, each of which must happen whenever the other does, are said to be equivalent to each other. So when a first situation is equivalent to a second, each situation implies and is implied by the other.

Conditional versus Biconditional

One-way and two-way implications are called conditional and biconditional statements (or rules), respectively.

The Abbreviation Iff

The terms and phrases

• if and only if
• when and only when
• iff (shorthand for if and only if)

can all be used instead of each other. They are interchangeable. No matter what term or phrase is used to indicate a two-way implication, the difference between one-way and two-way needs to be remembered. Otherwise, statements, definitions and assertions will be read incorrectly.

Selby A, Volume 1A, Pattern Based Reason, 1996.

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