All of the FOUR INVALUABLE PURPOSES (of "The Deal" and "MP" files) apply below:

1. Determines when two ASSERTIONS can be EQUIVALENT in TRUTH-VALUE.
2. Provides a "nonleaky vessel": If you put TRUTH into it, it won't "leak out".
3. Provides a form -- THE CONDITIONAL ASSERTION -- that extends and "dynamizes" the formalism. This is the form "if A, then B" (symbolized A → B) where A, B are different ASSERTIONS. (IF YOU KNOW "ONE THING", THEN YOU KNOW "ANOTHER".)
4. Provides a TAUTOLOGY, "modus ponens" (MP) -- using a CONDITIONAL ASSERTION that PROVES TRUTH WHEN PRESENT.

MP is of the form: "If A, then B; & A is TRUE. Then B is TRUE." Using → for THE CONDITIONAL OPERATOR, & for CONJUNCTION OPERATOR, then MP can be symbolized thus:

		((A → B) & A) → B.

In fact, MP PROVIDES THE FORMAT FOR FAC (FALLACY OF ASSERTING THE CONSEQUENT), although those picky-pickies who call it a "fallacy" don't seem to realize this.

In perspective, MP should be called "Tautology of Asserting The Precedent". For, given the CONDITIONAL STATEMENT, (A → B), MP then CONJUNCTS ("ANDS") this CONDITIONAL with ASSERTION A, DECLARING THAT ASSERTION A IS TRUE. But ASSSERTION A is the PRECEDENT ("party of the first part") of CONDITIONAL, (A → B), so this is ASSERTION OF THE PRECEDENT, and leads to a TAUTOLOGY. -- with A TRUTH-TABLE CONSISTING OF ALL 1'S (four 1's), meaning all TRUES, as we see below:

A	B	A → B	(A → B) & B	((A → B) & B) → A
0	0	1	0	1
0	1	1	1	1
0	0	0	0	1
1	1	1	1	1

A TAUTOLOGY. And, remember, TAUTOLOGIES "are not about REALITY but about LANGUAGE USAGE".

But suppose WE INTERCHANGE A (CONDITIONAL PRECEDENT) with B (CONDITIONAL CONSEQUENT, "party of the second part") in the rest of the form. Then, we have:

  
      ((A → B) & B) → A.

We'll see below that FAC has a TRUTH TABLE (a.k.a. INDICATOR TABLE) NOT OF FOUR 1's (as with MP), but THREE 1'S and a SINGLE ZERO (for "FALSE").

Three "things happen" here:

1. FAC fails (by the SINGLE ZERO) to be a TAUTOLOGY.

2. This FAILURE plucks FAC out LANGUAGE and gives it REALITY potential. IT CAN BE FALSIFIED: EXPECTED EVENTS CAN FAIL TO OCCUR.

3. But the SINGLE FAILURE MINIMIZES THE DEVIATION FROM TAUTOLOGY -- which gives FAC its USEFULNESS (and, as we'll see, its POWER TO GENERATE BONUSES).

Here is the TABLE for FAC:

A	B	A → B	(A → B) & B	((A → B) & B) → A
0	0	1	0	1
0	1	1	1	0
1	0	0	0	1
1	1	1	1	1

Please notice the "0" in the 2nd Row of the LAST COLUMN OF THE TABLE. This is the "deviant" -- which gives FAC its REALITY potential; but its SINGULARITY MINIMIZES DEVIATION FROM TAUTOLOGY.

Please notice something else. The 3rd COLUMN (TABLE) above is THE SAME AS the 2ND COLUMN (TABLE) above. What does that mean?

It means that (A → B) & B) is EQUIVALENT TO B. Hence, ((A → B) & B) → A is EQUIVALENT TO THE CONDITIONAL statement, B → A! That is, WE ARE GIVEN THE INFORMATION OF THE CONDITIONAL STATEMENT, A → B, and we claim to derive from it the CONDITIONAL STATEMENT, B → A.

We can learn two "things" from this:

1. The form B → A is labeled "THE CONTRARY" of A → B. So FAC can also mean "Fallacy of Asserting the CONTRARY".
2. To make this formalism useful, we need to REPLACE "A" (IN "A → B") by A CONJUNCTION OF MANY ASSERTIONS or HYPOTHESES, taking the form &_iH_i, where i = 1,2,...; and we need to REPLACE B (in A → B) by A CONJUNCTION OF MANY OTHER ASSERTIONS or PREDICTIONS, taking the form, &_jP_j, where j = 1,2,....

S00000! Our finding about THE CONTRARY makes it EASIER to WRITE and PROVE A FORMULA for this GENERALIZATION of FAC. We simply work with (&_jP_j) → (&_iH_i).

The DEAL was that you will be given a FORMULA involving only FRACTIONS to be added and subtracted. If you've done any PROBABILITY problem, you've done the same thing. So the TACTICS haven't changed -- only the STRATEGY, since the PROBABILITY MEASURES refer to EVENTS, while the ASSERBILITY MEASURES refer to ASSERTIONS. (Homologously, ASSERBILITY relates to ASSERTIONS in the way that PROBABILITY relates to EVENTS. Dig?)