Doctors often must schedule and carry out a series of DIAGNOSTIC TESTS on a patient. A mechanic must often schedule and carry out a series of DIAGNOSTIC TESTS on an automobile motor. Why? And how does this relate to SIEVING?The answers invoke simple rules of LOGIC which every citizen should know! Hence, should be taught in school. (Elsewhere, I display 100 STANDARD TASKS, of the form I required of my college students to SURVIVE in SOCIETY. These rules appear in some of those TASKS.)
Consider a statement, such as, "If you have a veneral disease, you will fail a Wassermann test." (This is of the abstract form, "If A, then B", and is called an "implication", although the correct term is "a logical conditional", a form used to deal with implications.)
The CONVERSE of the above CONDIITONAL is, "If you fail a Wassermann Test, you have a venereal disease." Is that true? NOT NECESSARILY. And THE CONVERSE OF A CONDITIONAL DOES NOT NECESSARILY HAVE THE TRUTH-VALUE OF THE CONDITIONAL FROM WHICH IT IS DERIVED. (We'll see why -- anon!)
The CONTRAPOSITIVE of that CONDITIONAL is, "If you test negatively on a Wassermann test, you do not have a venereal disease."
Before going on, let's learn about these forms by using Venn diagrams. (Elsewhere, I use VENN diagrams to explain the combinatorial algorithm, "Inclusion-Exclusion".)
A Venn diagram is usually drawn as circles within a rectangle. It's convenient here to draw rectangles within rectangles.
The forms, "If A, then B" and "If you have a venereal disease, your Wassermann will be positive", are diagrammed as follows:
__________________________________ | ____________________________ | | | B | | | | _______________________ | | | | | | | | | | | A | | | | | |______________________| | | | |___________________________| | |__________________________________|__________________________________ | ____________________________ | | | POSITIVE WASSERMANN | | | | _______________________ | | | | | | | | | | | VENEREAL DISEASE | | | | | |______________________| | | | |___________________________| | |__________________________________|Charlotte? Saying "If A, then B" is, diagramatically saying "If it's in A, then it's in B". And you can see that in the diagram. But the CONVERSE, "If B, then A", says "If it's in A, then it's in B". But, clearly, a point can be INSIDE B AND OUTSIDE A, ALTHOUGH A POINT CAN BE IN BOTH. So, the CONVERSE OF A CONDITIONAL DOES NOT SAY THE SAME AS THE CONDITIONAL!But the CONTRAPOSITIVE says "If it's not in B, then it's not in A", which is correct about our diagram. Hence, THE CONTRAPOSITIVE OF A CONDITIONAL SAYS THE SAME AS THE CONDITIONAL! (This is part of the sense of "Whitehead's NO-NO PRINCIPL": ITS DIFFICULT TO KNOW "WHAT IS SO", BUT OFTEN IT'S EASY TO KNOW "WHAT IS NOT SO".)
So the TRUTH of "If you have a venereal disease, you'll have a positive Wassermann" DOES NOT GUARANTEE THE TRUTH of "If you have a positive Wassermann, you have a venereal disease". In fact, "If you've had rheumatic fever, you'll have a positive Wassermann". (Elsewhere, we learn about two TYPES OF FUNCTION: MANY-ONE FUNCTION and ONE-ONE FUNCTION. This present is the TRUTH-FUNCTIONAL case of a MANY-ONE FUNCTION: MANY diseases -- venereal, rheumatic fever, and others -- lead to THE SAME RESULT: POSITIVE WASSERMANN.)
There's an old saying, "All roads lead to Rome.". As Romans extended their domain, they arranged, whenever possible that any road they built would connect to a road to Rome. SO, GET ON A ROAD AND KEEP GOING AND YOU'LL ARRIVE IN ROME. BUT BE CAREFUL GOING HOME!!!
This is how I taught students to understand MANY-ONE FUNCTIONS and CONDITIONAL STATEMENTS. To achieve an INVERSE FUNCTION from a MANY-ONE FUNCTION, you must RESTRICT THE DOMAIN. To turn a CONDITIONAL AROUND, YOU MUST ALSO NEGATE BOTH PARTS, that is, FORM A CONTRAPOSITIVE, NOT A CONVERSE!
To look at this another way, writing "If A, then B" as "A Þ B", this has the form "SUFFICIENT Þ NECESSARY". "A is SUFFICIENT for B"; "B is NECESSARY for A". Look at the diagram above. It is SUFFICIENT TO BE IN RECTANGLE A to also be in B. It is NECESSARY to be in B to be in A, that is, OUTSIDE B IS OUTSIDE A.
__________________________________ | UNIVERSE OF DISCUSSION | | ____________________________ | | | NECESSARY | | | | _______________________ | | | | | | | | | | | SUFFICIENT | | | | | |______________________| | | | |___________________________| | |__________________________________|BANG!!!! That's "why the diagnostic test series was born -- along with the Blues". THE WORLD PRESENTS US WITH LOTS OF MANY-ONE FUNCTIONS and LOTS OF SUFFICIENT Þ NECESSARY FORMS.Any student who has worked in QUALITATIVE ANALYSIS IN CHEMISTRY learns that SEVERAL DIFFERENT CHEMICALS WILL GIVE THE SAME REACTION TO A GIVEN REAGENT. The only POSITIVE RESULT is that LOTS OF DIFFERENT CHEMICALS WON'T GIVE THIS REACTION WHEN EXPOSED TO THIS GIVEN REAGENT -- AND ARE ELIMINATED. So, the student schedules a SERIES OF QUALITATIVE TESTS TO ELIMINATE AS MANY CHEMICALS AS POSSIBLE -- MOST DESIREABLY, ALL BUT ONE CHEMICAL.
Similarly, the physician schedules A SERIES OF DIAGNOSTIC TESTS TO ELIMINATE MANY DIFFERENT MALEVOLENT CONDITIONS.
Children can learn about this, starting with the First Grade, by my method of "flip card
Teaching the child about SIEVING as soon as possible prepares the child for DIAGNOSTIC TESTING IN MEDICINE or CHEMISTRY or in AUTO REPAIR. The same education for children who may later be placed on the "vocational track" or on the "college track". Hence, I say this kind of education eliminates one form of present SEGREGATION (which I suffered as a child).
Besides the CONDITIONAL STATEMENT, there is a very desirable form, THE BICONDITIONAL, "A if, and only if, B", for A « B. In the Venn diagram, A and B are represented by the same circle or rectangle. The CONVERSE IS THE SAME AS THE BICONDITIONAL. The "best" theorems in logomath are BICONDITIONAL. But this is rare outside of logomath.
SUMMARY SONG (Tune: "London Bridge") If SUFFICIENT, then NECESSARY. If SUFFICIENT, then NECESSARY. If SUFFICIENT, then NECESSARY -- The form of THE CONDITIONAL. CONVERSE CONDITIONAL doesn't equal CONDITIONAL! Doesn't equal CONDITIONAL! Doesn't equal CONDITIONAL! CONVERSE CONDITIONAL doesn't equal CONDITIONAL! That's the Bad News! CONTRAPOSITIVE equals its CONDITIONAL! Equals its CONDITIONAL! Equals its CONDITIONAL! CONTRAPOSITIVE equals its CONDITIONAL! That's the Good News!Connection to Other STRATEGIES.