From owner-cse191-sp08-list@LISTSERV.BUFFALO.EDU  Thu Feb 14 10:43:46 2008
Date:         Thu, 14 Feb 2008 10:43:26 -0500
From: "William J. Rapaport" <rapaport@cse.Buffalo.EDU>
Subject: 191: MODUS TOLLENS QUESTION
To: CSE191-SP08-LIST@LISTSERV.BUFFALO.EDU

A student writes:

|    I had a question about Modus Tollens.
|	if, for example, you had the premises:
| 
| 1:   (p^r)-->(s^t)
| 2:    NOT s
| 
| could you use Modus Tollens to arrive at NOT (p^r) since the NOT s would
| make the conclusion false meaning the antecedent would also have to be
| false? If so, is it enough to just state MT as the reason behind this,
| or is there some other steps that need to be done first?

The answer depends on whether you consider yourself a logician or a
mathematician.  (If you consider yourself a computer scientist or a
computer engineer, you get to choose :-)

A mathematician would say that you are correct.

A logician would say that, technically speaking, you could not cite MT
immediately.  Instead, you would have to do derive -(s^t) before you
could apply MT.  To derive -(s^t), you could use addition (our old
friend :-) and DeMorgan:

1. ((p ^ r) -> (s ^ t))
2. -s
3. -s v -t		: from 2, by Addition
4. -(s ^ t)		: from 3, by DeMorgan
5. -(p ^ r)		: from 1,4, by MT

Now, if you were a computer that only knew how to use MT, Add, and DeM,
would you be able to jump immediately from line 2 to line 5, the way a
(human) mathematician could?  Possibly, depending on how you were
programmed.  But most likely, such a computer would have to behave like
a logician, and follow the rules without taking any shortcuts.  

(Possibly, if it was capable of learning, it could learn such a
shortcut, but that would require some very clever AI programming; sounds
like a nice project for anyone looking for one :-)

By the way, for more information on "teaching" logic to a computer by
programming it to do logic, take a look at (a now somewhat outdated, but
still useful text), co-authored by one or two people whom some of you
may have heard of:

Schagrin, Morton L.; Rapaport, William J.; & Dipert, Randall R. (1985),
Logic: A Computer Approach (New York: McGraw-Hill)
    * SCI/ENGR Book Collection BC138 .S32 198