Chapter 8, A Language Change (if and only if, if then), Pattern Based Reason

Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com) \|\|Définition d'une variable \|\| Algèbre \|\| Arithmetique \|\| Logique \|\|La raison basée sur les règles et modelés\|\|
Online Volumes 1, Elements of Reason. 1A. Pattern Based Reason 1B. Math Curriculum Notes 2. Three Skills for Algebra 3. Why Slopes & More Math (Optional Book Orders)	More Site Areas 1. Help Your Child or Teen Learn 2. Solving Linear Equations 3. Fractions Ratios Rates Proportions & Units 4. Euclidean Geometry 5. Analytic Geometry/Functions 6. Number Theory. 7. More Calculus	More Site Areas 8. Complex Numbers 9. Qc Maths Education 10. Secondary IV(?) maths 11. Real Analysis 12. LaTeX2HotEqn: 13. Electric Circuits Etc 14. Français 15. Algebra, Odds & Ends, Etc	More Site Areas 16. Math Education Essays 17. Telling & Working with Time 18. Maps, Plans & Drawings 19. Quantitative Skills for home, shopping and work 20. Statistics Useful, or Not. Try the Twiddla Whiteboard to work online with others.
[Site Entrance & Hub][ Back ] [ Book Entrance ] [ Next ][Site Exit]

A Language Change
Chapter 8, Part I

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

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 (Not in Printed Version)

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.

A Language Change Chapter 8, Part I

One-Way Implication Rules

Postscript (Not in Printed Version)

A Language Change
Chapter 8, Part I