Here are the other two INVOLUABLE PURPOSES of Standard Logic, mentioned in "DEAL".

• Logic 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. That is, "if A, then B" says "IF YOU KNOW ONE THING, THEN YOU KNOW ANOTHER".
• Logic provides a TAUTOLOGY, "modus ponens" (MP) -- using a CONDITIONAL ASSERTION -- that PROVES TRUTH WHEN PRESENT. It is of the form: "If A, then B; & A is TRUE. Then B is TRUE." Using → for the CONDITIONAL OPERATOR and & for CONJUNCTION, MP can be symbolized thus:

  
  		((A → B) & A) → B.

  The tautological nature of MP can be easily DEMONSTRATED in two different ways.
  1. By INDICATOR or TRUTH TABLES:
     1. Write COLUMNS for all POSSIBLE TRUTH SUBTABLES of ASSERTIONS A, B, ENCODING TRUE AS "1", FALSE AS "0":

        A	B
        0	0
        0	1
        1	0
        1	1

     2. A CONDITIONAL ASSERTION (such as A → B is CONSIDERED FALSE ONLY WHEN PRECEDENT (here, A) IS TRUE and CONSEQUENT (here, B) IS FALSE -- the 3rd ROW in above Table. A CONJUNCTION -- such as (A → B) & A -- IS TRUE ONLY WHEN BOTH CONJUNCTS ARE TRUE -- 4th ROW in Table. So we can complete the Subtables for MP, repeating those columns above:

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

        The totality of "1's" in the last SUBTABLE reveals the TAUTOLOGICAL nature of MP: IT CANNOT BE FALSIFIED.

     3. The other way to prove MP uses these CLUES: A → B is EQUIVALENT TO "B INCLUDES A"; and saying "A is TRUE" is EQUIVALENT TO "A IS NOT EMPTY, as FALSITY WOULD BE". If you draw a CIRCLE (or rectangle) for "A", putting it inside the CIRCLE (rectangle) for "B", then a point (or asterisk) in CIRCLE (rectangle) A is NECESSARILY in CIRCLE (rectangle) B -- as simple and obvious as that.

        		_________________________________________________B
                        |                                               |
                        |                                               |
                        |      __________________________________A      |
                        |      |                                |       |
                        |      |                                |       |
                        |      |              *                 |       |
                        |      |________________________________|       |
                        |                                               |
                        |_______________________________________________|

        WARNING! The form ((A → B) & A) → B uses the CONDITIONAL OPERATOR ( → )
        twice, but in TWO DIFFFERENT WAYS, which is IMPLIED BY WHAT IS PLACED
        PARENTHETICALLY and WHAT IS NOT. The part ((A → B) & A) says (A → B)
        IS TRUE, and also ASSERTION A IS TRUE. SO, THESE TWO TRUTHS IMPLY TRUTH-HOOD
        OF ASSERTION B.

        Anyone who has studied the form of a SYLLOGISM can understand the above, by
        homology. Take the most famous of Syllogisms: "All men are mortal. Socrates
        is a man. Therefore, Socrates is mortal." The first two ASSERTIONS are
        PREMISES of the SYLLOGISM. The third ASSERTION is the CONCLUSION of the
        SYLLOGISM. This is sometimes shown by the following form:

        
        		All men are mortal.
        		Socrates is a man.
        		__________________
        		Socrates is mortal.

        Note how this resembles, say, a SUM:

        			3
        			2
        			_
        			5

        Then you can put MP in a similar form:

        
        			A → B
        			A
        			______
        			B

        That is, the PREMISES are A → B and A; the CONCLUSION is B.
        (This will be critical in what follows.)