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Refuting Dilemmas

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A Study of Dilemmas             Dr. Peter Krey for Critical Thinking

1. If God is perfectly loving, God must wish to abolish evil
and if God is all-powerful, God must be able to.
But evil exists.
God cannot be both omnipotent and perfectly loving.

(If p, then not q) and (if r, then not q)
q, therefore not p v not r.

Here it is         (P → not Q)   &   (R →  not Q)
In notation:               Q
—————-   ∴ not P     v     not R

Now unless we are Christian Scientists, we will not hold
that evil does not exist. Thus we have to grasp the dilemma by
the horns. This expression means that we have to refute only one
of the conjuncts and the conditional premise falls and then we
can argue that the dilemma may be valid, but because the
conditional premise is false, the conclusion need not be true.

Thus for example we could argue that although God is all-
powerful, God restricts his/her power voluntarily to allow for
human freedom.
∴ we are not forced to accept this very negative

2. If a student is fond of learning, s/he needs no stimulus and
if s/he dislikes learning, no stimulus will be of any avail.
But any student is either fond of learning or dislikes it.
a stimulus is either needless or of no avail.

(If p, then q) and (if not p, then r)
p v not p, therefore q v r.

Here it is         (P →  Q)   &   ( not P →   R)
In notation:            P    v   not P
——————   ∴ Q     v      R

Now by challenging the disjunctive premise we can claim it
to be false and thus go between the horns of the dilemma.

We argue that students have all kinds of attitudes to
learning: fondness, dislike, and indifference. Thus the
conclusion is not false, but the argument does not constitute
adequate grounds for accepting the conclusion.

3. The third way to give a rebuttal to a dilemma, which is a very
devastating kind of argument, is to oppose the dilemma with a

Protagoras tutored Euathlus in the study of law, and not being
able to pay tuition, Euathlus promised to pay from the earnings
of his first case. But then he never practiced law. Protagoras
had to take him to court for his money and his charge took the
form of a dilemma.

If Euathlus loses his case, then he must pay me (by the
judgment of the court); if he wins this case he must pay me (by
the terms of the contract). He must either lose or win the case.
Euathlus must pay me.

Euathlus countered Protagoras as follows:

If I win this case, I shall not have to pay (by the judgment
of the court); if I lose this case, I shall not have to pay
Protagoras (by the terms of the contract). I must either lose or
win the case.    I do not have to pay Protagoras.

Protagoras                          Euathlus

If L, then P (by judgment)       If L, then not P (by contract)
If W, then P (by contract)    If W, then  not P (by judgment)
———–L   v  W                         L   v  W

————∴ P                            not  P

Had you been the judge, how would you have decided?

A Constructive Dilemma:

(If p, then q) and (if r, then s)
p v r, therefore  q v s.

——- Here it is         (P → Q)  &     (R → S)
In notation:           P v   R
——————       Q v  S

A Destructive Dilemma:

(If p, then q) and (if r, then s)
not q v not s , therefore  not p v not r  .

Here it is ————-        (P → Q)  &     (R → S)
In notation:                   not Q   v   not  S
———————–     not P v not  R


Written by peterkrey

January 18, 2009 at 2:34 am

Posted in Logic, Philosophy

2 Responses

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  1. Above the name must be Euathlos (Latin form Euathlus, meaning the good athlete), not Eulathus.


    November 27, 2012 at 12:16 am

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