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Topic 4. Implication, Root of Logic
4.0 Implication
4.1 Propositions
4.2 General/universal implication
4.3 General/universal exclusion
4.4 Just-sufficient Contradiction
4.5 Quantifier
4.6 Some
4.7 Summary of implication models
4.8 DeMorgan transformation and the Modeling Choice
4.9 Tautology
4b modus ponens and So-Whattees (sorites)
All these parts are junk. (If a part is among these, then it is junk.)
Cool people drink Spritzenblitzenwitz.
(If a person is Cool, then that person drinks Spritzenblitzenwitz.)
None of the people on Team Charlie have janissary training.
(If a person is on Team Charlie then that person has no janissary training.)
Propositional logic is about statements of class or set membership. Note that the "if-then" --implication-- examples already given may be seen as examples of such set-membership statements:
(If a part is found among the set, {these parts}, then that part is in the set, {junk}.
(If a person is found among the set, {Cool people}, then that person is a member of the set, {those who drink Spritzenblitzenwitz}.
(If a person is a member of the set, {those on Team Charlie}, then that person is outside the set, {those who have janissary training}.
All the parts in Bin 12 are defective.
All men are created equal.
All men are pigs.
All pixies are quick.
Some boarding students are slobs.
Some of Nerkely's students did well enough to go to the regional contest.
4.2
General/universal Implication: all p is q
In propositional calculus, we do a lot of naming of classes, or sets. The name k4 might in one investigation represent the set of all parasites; in another, the set of all genetic codons; in another, the set of all propositions.
To model propositional logic in pancake, we need to find some formal way of saying "if ... then ..." When we say "if p then q" we are making a universal, or general, implication. The same formalism will express set membership as well: "all p is q" -- "all the members of set p are members of set q."
From a truth table involving
pancake arrangements of p, q and what
we understand implication to mean,
we will figure out how to express universal
implication in the pancake calculus.
4.2.2 Choosing a Modeling Sense
Pancake has given us two values, and an operator
capable of flipping one into the other.
To model a two-valued logic such as propositional
calculus, then, we shall obviously be mapping
the logic values TRUE, FALSE to the pancake values
( )vizzo, invizzo.
But which of the two possibilities are we
to choose --
( )=TRUE modeling?
or
( )=FALSE modeling?
You probably suspect it makes no difference, and
that is indeed what we shall discover for
modeling propositional calculus.
4.2.3
Truth table
A truth table systematically lists input combinations --valuations, that is-- to the left, and (evaluation) outputs to the right of each valuation.
In set membership propositions we distinguish a subject set and a predicate set. Here, the respective set roles are symbolized by p, q.
For the truth table of implication we shall use vizzo, ( ), for the value "all members of set p are members of set q." Guess what value invizzo is to represent?
Don't look for the answer on my face. It's not there. I'm not going to tell you right away. You have to think about it.
Don't be hasty! Invizzo will not represent the value, "no members of set p are members of set q."
p
q
all p is q
( ) ( )
( )
( )
( )
( )
( )
truth table for implication, vizzo=TRUE model
The only valuation which is inconsistent with "all p is q" being true is the valuation {p,q} = {TRUE, FALSE}: some item is found which is a member of the set p but not of the set q.
Can you construct a pancake expression which has the same behavior (input-output characteristic) as the implication truth table? There are not so many that an exhaustive search is out of the question ...
(rummage-rummage, try-this, try-that,
rummage, experiment, try-this, try-that, ...)
Aha!
p
q
( p ) q
( ) ( )
( )
( )
( )
( )
( )
truth table for implication, ( )=TRUE model
The expression (p)q evidently has exactly the behavior we require for general/universal implication in a vizzo=TRUE model.
4.3
General exclusion
An alternative way of saying "all members of set p are members of set q" is, "p is a subset of q."
What if q contains all of p's members and more besides? Then, p is a proper subset of q.
What if p is a subset of q and q is a subset of p? This can only be true if sets p, q have exactly the same members; in this case we call the sets equal: p=q.
4.3.2 Complement the predicate set
We discover another variant of implication, general exclusion, "no p is q", by looking at the predicate set, q, and its complement, ~q -- that is, all that is excluded from q.
Every set, that is, has its complement, which set theory notates as ~q (read "not-q") and pancake calculus will model, of course, as (q).
Thus, another way to say "no member of p is a member of q," is "p is a subset of ~q."
Recalling our discovery of (p)q as a model for "p is a subset of q," --(is that what "all p is q" means?)-- we now have a ( )=TRUE model for "no p is q:"
(p)(q)
The two general implications we have in hand as ( )=TRUE models are the only general/universal implications:
general inclusion: (p)q all p is q
general exclusion: (p)(q) no p is q
4.4 Just-sufficient Contradiction
There is an old aphorism in English, "All that glitters is not gold." On the face of it, this seems to go against the modeling we have developed. "...not gold" might seem to be complementing the predicate set {all that is raw gold or made of gold}, so that "nothing that glitters is gold" would be accurate periphrasis.
I presume, though, that no-one reading this has any doubt what is being said. The common understanding is that what is really being negated by "not" is the "all" of the proposition, "all that glitters is gold," giving "not all that glitters is gold."
All is/are can be just sufficiently contradicted by some is/are not -- "some things that glitter are not gold."
"None/nothing" would overshoot by being too strong: "nothing that glitters is gold" is not an exact, or just-sufficient, contradiction of "all that glitters is gold."
In what is to come, we shall pay close attention
to the idea, just-sufficient contradiction.
Now that we have vizzo=true models for general or universal implication, we look at variants of implication generated by complementing the whole expression. This will affect what propositional calculus calls the quantifier, which we have modeled so far in just the flavors "all" and "no/none," the universal quantifiers.
There will be one more quantifier in propositional calculus, the particular quantifier "some."
Predicate calculus will be distinguished by having a way of expressing "there is a ...", "there exists some ..." and calling it an existential quantifier.
For modeling predicate calculus, we shall find it more convenient to settle on the vizzo=TRUE modeling sense, expressing "there exists some x" by "x" rather than "(x)".
As you see, any "models of quantifiers" we develop in pancake calculus are not built by inventing new symbols. Rather, they are a matter of seeing our one operator, ( ), in new lights and using it in new formal patterns.
Begin the discovery of a formal model of the particular quantifier, "some," by negating general/universal implication as a whole, in, say, the general inclusion variant, "all p is q"
(p)q.
What, then, does the whole negation ((p)q) say? To contradict "all p is q" we might say directly "no. Not all p is q." If our opponent presses us more exactly and says, "How do you know?" we might answer, in just-sufficient contradiction, "Some p is not q."
This gives us immediately an interpretation of the ( )=TRUE model ((p)q).
Because of the "some" in this interpretation we call this kind of implication particular rather than general/universal.
We have "some is not;" we now would like to see "some is." Where would you start looking?
"Some is not" came from negating "all is." Maybe "some is" would come from negating some similar form, perhaps the other universal implication?
So do it.
4.7 Summary of implication models
Good! you did it.
This observation completes our required vizzo=TRUE set of implication models for propositional logic:
general implication particular implication
all p is q: (p)q
some p is not q;
not all p is q: ((p)q)
no p is q:
(p)(q)
some p is q:
((p)(q))
Note that the forms for particular implication are whole negations of forms for general implication.
Note also that two of the implications are isotropic (they mean the same if we exchange the terms). For example, "some p is q" means quite the same as "some q is p."
Do you think that general exclusion, (p)(q), is likewise isotropic?
Can you give a simple inspection test of a formal model to say whether it expresses an isotropic implication?
4.8
DeMorgan transformation and the Modeling Choice
We shall soon be requiring pancake forms for modeling in ( )=FALSE sense. Instead of re-developing these from truth-table considerations as we just did for ( )=TRUE models, we note that DeMorgan's Law has a reflection in pancake forms.
DeMorgan's Law was first stated for boolean B2 calculus, which we have not discussed yet. Nevertheless, its reflection in pancake calculus is easily illustrated. It amounts to noticing that a ( )=TRUE model can be transformed into a ( )=FALSE model by complementing both the expression as a whole, and its terms severally. Such transformations may be done in either direction.
( )=TRUE model <-- --> ( )=FALSE model
a b <--deMorgan--> ((a)(b))(a b) <--deMorgan--> (a)(b)
(a) b <--deMorgan--> (a(b))
((a) b) <--deMorgan--> a(b)
The general implication where both terms are the same is call a tautology: "all m is m" -- "all mortals are subject to death," "all winged animals have wings." A tautology is necessarily true, so we would be dismayed if our pancake models of tautology failed to have the same property.
Let's investigate this first in ( )=TRUE modeling and show tautology ALWAYS TRUE:
(f) f , model: ( )=TRUEWe have said above that a ( )=FALSE model is immediately available by a deMorgan transformation. Let's form tautology in vizzo=FALSE and show it ALWAYS TRUE:
=
( ) , self-viz identity; TRUE; model
(f(f)) , model: ( )=FALSE
=
, self-inviz identity; TRUE; model
"It is what it is."