out to Contents
next Topic
Topic 5. Propositional Logic in Pancake: Antilogisms
5.0 Antilogism, a natural form of argument
5.1 Emily's Fork
5.2 The logical connective "but"
5.3 Four antilogism types
"Nobody eats soup with a fork, Emily!"
"But I do, and I'm somebody!"
Constance Ladd-Franklin, in her paper in the journal Mind, claims that she hardly ever hears people arguing in syllogisms, but often hears arguments shaped in what she calls antilogisms.
The antilogism is essentially refutation -- proof of the falsity of a proposition. A proposition is countered with two other propositions which are intended to make it clear that the countered proposition cannot be true.
Ladd-Franklin's first example posits a
general implication, namely the general
exclusion, "Nobody eats soup with a fork, Emily!"
Emily refutes this with
two particular propositions which she regards
as obviously true to an impartial observer:
"But I do, and I'm somebody!"
The unspoken overall structure of Emily's
argument is:
Here are three propositions, one yours and two mine.
They cannot all be true.
Since you cannot but admit that mine are obviously true, the false proposition must be yours.
The same structure characterizes all four types of antilogism and will guide our investigation of them in the pancake calculus.
Identify the sets:
Let p= {those who are somebody} = {everybody}
Let f= {those who eat soup with a fork}
Let i= {Emily}
The antilogism argument structure calls for arranging the three implications in an expression which can only be true if all the parts, or terms, are true. We need only identify such a form to have a clear formal test for the validity of Emily's refutation: it succeeds if the model evaluates FALSE.
Such an expression, in ( )=FALSE modeling, is A B C.
In ( )=TRUE modeling we would need to use the synthetic "deMorgan shell:" ((A)(B)(C)).
Evidently we can save some work, and avoid occasion for errors in calculation, by choosing ( )=FALSE modeling.
Accordingly, we write, substituting in A B C the three implication models:
(p f) i f i p, model; subst in A B C
( ) i f p, outfect
( ), tromp
FALSE by ( )=FALSE model
The model reduces all the way to ( ), which we score as FALSE in ( )=FALSE modeling.
Just as Emily argued, the model shows that the three propositions cannot all be TRUE, and the fly in the ointment has to be her challenger's proposition.
5.2 The logical connective "but"
Emily uses the word "but" in her antilogism. What is the difference between "but" and "and"?
What -- is your test?Replace "but" with "and" in the argument? Excellent, Hermione!
Aside from the rhetorical emphasis, and the grouping it seems to do in this argument -- grouping her two propositions in opposition to the proposition she is refuting -- there is no logical difference.
Since there are four implication types which we might refute by antilogistic argument, we seek a pair of implication types to refute each.
Emily's challenge came from a general exclusion, the "no p is q" implication.
How would one refute a general inclusion implication? Or one of the particular implications?
Since vizzo=FALSE modeling is easily the more convenient for this problem, we carry it through.
We have characterized Emily's challenge; call it A2. If we are to refute an implication, A, with a B and a C implication, the challenges come from
A1= general implication, "All p is q:" (p(q))
One might refute this with
Let B= "all m is p," and C= "some m is ~q."(p(q)) (m(p)) m(q), model; subst in A B C
(p) ((p)) m(q), outfect
(p) p m(q), doubleflip
( ) m(q), self-viz
( ), tromp
FALSE, by model choice.
A2= general exclusion, "No p is q:" (p q)
Emily refutes this with
Let B= "some m is q," and C= "some m is p."
(Argument detailed above.)
A3= particular implication, "Some p is q:" p q
One might refute this with
Let B= "all p is m," and C= "no m is q."p q (p(m)) (m q), model; subst in A B C
p q ((m)) (m), outfect
p q ( ) (m), outfect
( ), tromp
FALSE, by model choice.
A4= particular exclusion, "Some p is not q:" p(q)
One might refute this with
Let B= "all p is m," and C= "all m is q"p(q) (p(m)) (m(q)), model; subst in A B C
p(q) ((m)) (m), outfect
p(q) ( ) (m), outfect
( ), tromp
FALSE, by model choice.