peter krey’s web site

The DUAL of

(P • ~Q) v (~P • Q), that is,

(P and not Q) or (not P and Q) ,

is

(P v ~Q) • ( ~P v Q), that is,

(P or not Q) and (not P or Q).

De Morgen’s laws are essentially duality principles.

1.      (P • ~Q) v (~P • Q)

2. ~ [(P • ~Q) v (~P • Q)]

3.      (~P v Q) • (P v ~Q)

See my post on the duality of the “XOR” and “IFF”, that is, the exclusive “or” and the “if and only if”, where  I derive the two definitions of the strong “or” from one anther.

Philosophy of Religion, Diablo Valley College, Dr. Peter Krey – July 20, 2004

David Hume lays some heavy skepticism on people who believe in God. He writes in Dialogues Concerning Natural Religion:

His syllogism:

Our ideas reach no farther than our experience:

We have no experience of divine attributes and operations:

I need not conclude my syllogism: you can draw

the inference yourself.

Now after the first shock of reading such an argument, the question arises whether it is valid. First, it came as a relief to notice that there were two negative premises, and Hume may have been counting on the fact that few people know the rules that determine the validity of syllogisms. None are valid with two negative premises.

But that proves too easy a solution, because the first premise really needs to be translated into a positive universal.

No ideas are thoughts that reach farther than experience.

All ideas about divine attributes and operations are thoughts that reach farther than experience.

Therefore no ideas are ideas about divine attributes and operations.

Symbolized it becomes

No I are E.   *****EAE Figure II

All D are E *****Valid Syllogism called Cesare.

No I are D. ******Conclusion

Thus the only way to disagree with Humes’ skepticism is to challenge his premises. The fact that there are a priori ideas show that they can come before experience and be independent of experience. Thus his first premise is untrue, and therefore the conclusion does not follow, nor does it need to be accepted.

Another translation of his syllogism:

All ideas are representations of experience.

No divine attributes and operations are rep. of experience.

Therefore no ideas are about divine attributes and operations.

Deriving the Two Expressions of the Exclusive “Or” from each other and showing that “XOR” is the dual of “iff,” that is, if and only if.

I have done the derivation of the exclusive “or” (XOR) in long hand. Here is the scan of my work:

After taking Aristotelain Logic and studying the syllogisms, we went to Truth Functional Logic. At the end of the class I would just give a preview of the more powerful predicate calculus, as well as other kinds of logic. WordPress des not pick up these symbols, therefore I scan the two pages:

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
conclusion.

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
counter-dilemma.

Protagoras tutored Eulathus in the study of law, and not being
able to pay tuition, Eulathus 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 Eulathus 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.
∴ Eulathus must pay me.

Eulathus 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                          Eulathus

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

Philosophy of Religion, Diablo Valley College , Dr. Peter Krey July 20, 2004

David Hume lays some heavy skepticism on people who believe in God. He writes in Dialogues Concerning Natural Religion:

Our ideas reach no farther than our experience:

We have no experience of divine attributes and operations:

I need not conclude my syllogism: you can draw

the inference yourself.

Now after the first shock of reading such an argument, the question arises whether it is valid. First, it came as a relief to notice that there were two negative premises, and Hume may have been counting on the fact that few people know the rules that determine the validity of syllogisms. None are valid with two negative premises.

But that proves too easy a solution, because the first premise really needs to be translated into a positive universal.

No ideas are thoughts that reach farther than experience.

All ideas about divine attributes and operations are thoughts that reach farther than experience.

Therefore no ideas are ideas about divine attributes and operations.

Symbolized it becomes

No I are E. ———-EAE Figure II

All D are E ———-Valid Syllogism: Cesare.

———-No I are D.

Thus the only way to disagree with Humes‘ skepticism is to challenge his premises. The fact that there are a priori ideas show that they can come before experience and be independent of experience. Thus his first premise is untrue, and therefore the conclusion does not follow.

Another translation:

All ideas are representations of experience.

No divine attributes and operations are representationsof experience.

Therefore no ideas are about divine attributes and operations.

Aristotle’s Practical Syllogism

Dr. Peter Krey, Jan. 29^th 2004

David Hume argues that you cannot get from the “is” to the “ought” via logic. Aristotle, however, works out a practical syllogism that answers to desire and could have a standard of right desire. Is it valid?

Aristotle’s Practical Syllogisms^[1]

The major premise of a practical syllogism is universal, a general rule. The minor is particular, the application of the rule to the case at hand. The application really quires two syllogisms.

Dry food is good for everybody. Syllogism One.

I am somebody.

Therefore, dry food is good for me.

Dry food is good for me. Syllogism Two.

This stale loaf is dry food.

Therefore, this stale loaf is good for me.

Aristotle notes that action has to do with particular things. One universal has to be predicated about the thing itself and the other about the person.

The major premise could be an opinion, while the minor could deal with particular things, which are perceived. ANow when the two premises are combined, just as in theoretical reasoning the mind is compelled to affirm the result of the conclusion, so that in the case of practical premises you are forced at once to do it. For example, given the premises, “All sweet things are to be tasted” and “Yonder thing is sweet” - a particular instance of a general class -, you are bound, if able and not prevented, immediately to taste the thing.” Aristotle continues that you can have a negative universal rule: “Avoid sweet things.” But if a strong desire is present, then you eat the sweet thing. It is the desire not the opinion that is really opposed to the right principle in an unrestrained person.

“All sweet things are to be tasted”

and “Yonder thing is sweet”

Therefore, yonder thing has to be tasted. (As you dish in).^[2]

Does Aristotle’s Syllogism disprove Hume? Observe that two syllogisms are needed to complete the application from principles to actions.