Let's Say: A Child's First Calculus



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Topic 4b.       Propositional Logic in Pancake, continued

 
4.30       Modus ponens
4.31       Extended modus ponens
4.40       So-Whattees(sorites)






4.30       modus ponens

There is a proof technique --a logical grand spell, if you like-- that has been known since Aristotle and company worked out logic in syllogisms. The Latins later named it modus ponens, the "way of putting." It has also been called "the law of the excluded middle." Modus ponens says

if A implies B and B implies C, then A implies C.

From this short chain of universal implications the middle term, B, can be eliminated, claims this "way." Maybe you think this is a valid technique in argumentation, or maybe you think it's "plausible but not certain," or perhaps "interesting if true."

Let us review what tools we have for expressing and testing this claim in pancake.

There are several logical connectives in modus ponens: if-then, implies, and and. We have discussed implication already, and realize it's merely an abbreviation for if-then. It remains, then, to discuss the logical connective and.

As usual, we have a modeling choice to make.

Thinking a bit about what we mean by "and," we see that we want a form of arrangement equivalent to TRUE if-and-only-if all the parts, pN below, are true, and FALSE otherwise:

p1 p2 p3 p4 ... pN

If we let that arrangement stand as-is in an unrepresented showdown plaza, it should be clear by now that if any part, say, p3, happens to evaluate to ( ) vizzo, the whole arrangement must evaluate to ( ) vizzo, by the tromp identity.

This should in turn suggest that a ( )=FALSE model would be easy to use in this problem.

So, for example, R AND S AND U would be rendered R S U in ( )=FALSE modeling. We call this the "understood AND" of parts, and it presupposes ( )=FALSE modeling.

The model, then, will be to form the AND of two universal implications, and make the result the implier in an outer, larger, "grand" implication --a meta-implication.

In vizzo=FALSE modeling, we know, general implication is rendered by the form (p(q)). We now let p be made up of two parts, p1, p2:

(p1 p2(q))

giving us a pancake model for modus ponens:

((A(B)) (B(C)) ((A(C)) ))


with the paren-pairs rendering the grand implication in bold. Applying some pancake identities --doing algebra-- we proceed to resolve:
=   (A(B)) (B(C)) A(C)),     doubleflip
=   ((B)) (B) A(C)),     outfect a,(c)
=   ( ) (B) A(C)),     outfect (b)
=         ,     exile
Simple invizzo, interpreted TRUE by our vizzo=FALSE transcription choice.

The value TRUE that we transcribed from pancake resolution, we interpret yet again: "the argumentation technique modus ponens is valid."


4.31       Extended modus ponens

Do you think an extended modus ponens would be valid? That is, can we extend the chain of linques and still have a valid conclusion?

This is how it might read:

If
A implies B
and B implies C
and C implies D
and D implies ...
...
and ... implies pN
and pN implies q,
then
A implies q.


4.31.1       hooque, linques, conclusion of modus ponens

In calling modus ponens "excluded middle," evidently the middle might aptly be called a link. We are about to discuss a chain of "middles," A m1 m2 m3 ... mN C and so we want names for each major part of an extended modus ponens.

Let's call the propositions A, m, C the hooque, the linques, and the conclusion, respectively, of modus ponens. Among the linques m, mN itself is special in that, like the hooque, it involves a term of the conclusion. Call it the last linque.

In our extended nomenclature, then, the original three-part modus ponens is made of hooque, last linque, conclusion.

Again, our extended modus ponens:

If
A implies B           hooque
and B implies C       linque
and C implies D       linque
and D implies ...       linque
...                     more linques
and ... implies pN         linque
and pN implies q,       last linque
then
A implies q.       conclusion


With hindsight gained from experimentation, I now choose a ( )=TRUE model. The purpose is to reveal a shorthand technique for ferreting-out a conclusion from an unordered collection of linques.

In a ( )=TRUE model, any part directly on the unrepresented plaza can cause the evaluation TRUE by being TRUE, from that same tromp consideration we now are so familiar with. We call this the "understood OR" of parts, and it presupposes ( )=TRUE modeling.

But we need a "formal AND" of parts to model our grand implication, so we introduce "synthetic AND."

DeMorgan's Law gives us a way of synthesizing whichever logical connective (AND, OR) is needed when the understood connective in the model we have chosen is not immediately what we want:

understood ____ <--deMorgan-->> ____ synthetic

a AND b AND c <--deMorgan-->> ((a) OR (b) OR (c))

a OR b OR c <--deMorgan-->> ((a) AND (b) AND (c))



We are equipped now to express the extended grand implication in a ( )=TRUE model:

( )=TRUE; "p implies q:" (p)q; synthetic AND:

(( ((A)B) ((B)C) ((C)D) ((D)pN1) ...((pN)q) )) (A)q
    =
((A)B) ((B)C) ((C)D) ((D)pN1) ...((pN)q) (A)q ,     doubleflip
    =
(B) ((B)C) ((C)D) ((D)pN1) ...((pN)q) (A)q ,     outfect (A)
    =
(B) (C) ((C)D) ((D)pN1) ...((pN)q) (A)q ,     outfect (B)
    =
(B) (C) (D) ((D)pN1) ...((pN)q) (A)q ,     outfect (C)
    =
(B) (C) (D) (pN1) ...((pN)q) (A)q ,     outfect (D)
    =
(B) (C) (D) (pN1) ...(q) (A)q ,     outfect (pN)
    =
( ) ,         self-viz : (q)q = ( )
simple vizzo, which we score TRUE in ( )=TRUE modeling.


Look again at just the linques without the deMorgan shell (( )( )...) and without the grand implication pancake, and without the conclusion:

(A)B (B)C (C)D (D)pN1 ...(pN)q

and note all the chained tautological forms, (t)t, that now appear:

(A) B(B) C(C) D(D) pN1...(pN) q

See this chain, not as a proper pancake expression on an unrepresented plaza, but merely as a list, outside of showdown space -- in scratch space, we might call it.

In scratch space, then, note that we might do a formal cancellation procedure, "canceling out" all the revealed tautological forms:

(A)q

and note that what remains is just the conclusion, isolated.

We will use this procedure to "scratch and sniff out" a conclusion -- a grand conclusion -- from a jumbled chain (sorites) of propositions, avoiding the necessity of ordering them (putting them in serial order).

What we canceled out in the procedure were not tidy tautological forms -- (t)t -- but the separate impliers and implicands of the propositions in the chain: B canceling-out (B) somewhere, D canceling-out (D) somewhere, and so on.

This procedure, then, does not require tautologies -- that would indeed make it useless. Rather, it handles any jumbled chain of general implications in a grand extended modus ponens implication, such as

b(g) f(m) d(r) r(b) m(x) g(f)

to reveal --in this example-- d(x).

















4.40       So-Whattees (sorites)

A set of propositions about a set of classes may possibly support a valid conclusion, if they are all accepted as true. Modeling propositions formally in pancake makes it easier to say whether a conclusion is supported under the conditions -- valid.

A sorites (pronounce the vowels as in "(dino)saur nighties") is a chain of propositions presented jumbled as a puzzle.

Except for one small thing, this example would be merely an extended modus ponens chain. Do you see the difference?

      No gamblers are lucky.
      All crabby people are lucky.
      All sailors wear blue for luck.
      All ninnies are crabby.
      Those who wear blue for luck are all ninnies.

"So what?"  Assuming all these propositions to be true, can you draw any conclusion from this jumble?

The procedure illustrated in our discussion of extended modus ponens is worth trying, even despite the wrinkle you have noticed:

First, we identify the classes with one-letter symbols and represent the propositions using the ( )=TRUE model. . Write the propositions in scratch space together:

    (g)(l)   (c) l   (s) b     (n) c     (b) n

Then rearrange to gather self-viz forms:

   (l) l     (c) c    (n) n     (b) b    (g) (s)

Cancel self-viz forms:

          (g) (s)

What remains is the conclusion,

(g) (s),     No gambler is a sailor,

which is isotropic with

(s) (g),     No sailor is a gambler.

This is a fair illustration of how an abstraction model may fail to reflect reality.

4.40.9       Roll-your-own sorites

You now have the ingredients for constructing a sorites.

More important, you have the tools to prove (test) whether a chain of general implications (possibly including general exclusions ) supports a valid conclusion.

Then, having done such a proof, throw in a particular implication and see if a general conclusion is supported.

Lewis Carroll constructed a sizable collection of sorites.

   


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