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From: greg@mind.UUCP (Greg Nowak)
Newsgroups: sci.philosophy.tech
Subject: Re: Modus Ponens = NIL ?
Message-ID: <1012@mind.UUCP>
Date: Tue, 14-Jul-87 02:50:17 EDT
Article-I.D.: mind.1012
Posted: Tue Jul 14 02:50:17 1987
Date-Received: Wed, 15-Jul-87 04:04:39 EDT
References: <3587e521.44e6@apollo.uucp> <680@gargoyle.UChicago.EDU>
Reply-To: greg@mind.UUCP (Greg Nowak)
Distribution: world
Organization: Cognitive Science, Princeton University
Lines: 115
Keywords: logic theory belief truth consistency

what follows from these by modus ponens,
>]viz. that if Reagan doesn't win, Anderson will (everyone believed that
>]if Reagan didn't win, Carter would).
>


>  Consider Statment 2:
>	"It's a Republican that will win the election."
>What is one to make of this?  Is it a statement of _FACT_ (i.e. in the same
>category as, say, the Law of Universal Gravitation) or is it merely a 
>_SENTIMENT_ (much like "Dem Bums will never win the Pennant!") ?  It is 

It is not a statement of fact, or a sentiment  ... it is a proposition.
To the extent that we are talking about logic, its content is irrelevant;
we might as well be talking about purple cows. If this curious problem 
demonstrates anything, it demonstrates that modus ponens is valid
only for the truth and falsity of propositions, not the plausibility or
implausibility of beliefs.

>unreasonable.  A much more reasonable statement would be:
>	1A: "If it's a Republican that will win the election, then if Reagan
>	     doesn't win, Carter will win."
>This statement is possible because now the (first) IF clause is no longer a
>statement of fact

Please. We are talking about logic. Remember, Carter is a Democrat; 
so your statement 1A is absurd on the face of it. (Recall Einstein's dictum:
To the extent that mathematics is certain, it does not discuss reality;
to the extent that it discusses reality, it is not certain." Substitute
logic for mathematics, and politics for reality ...)


>Comments?

Please forgive me for taking less-than-kindly advantage of your invitation.

Formal logic:

P: "A Republican will win the election."
R: "Reagan will win the election."
A: "Anderson will win the election."

Formally, proposition 1 is :

P  -->  (~R --> A)

and proposition 2 is simply 

P

There is nothing wrong with deducing (~R --> A) from the above
as a formal system; your unease comes only from the implausibility
of (~R --> A) taken out of context ...

But observe that P is a complex proposition;
we are really saying that we know there are only two Republicans
running, Reagan and Anderson, and either one of them winning will mean a 
Republican will win the election.

Thus :   P  = ( R  \/  A )     [\/ = "or"]

Proposition 1 becomes

(R \/ A) -->  (~R --> A)

And 2:

(R \/ A)

so far this is just formal logic with which I hope no one
will quibble. Now, using the revised versions of the propositions, we
can address the issue of interpretation.


Why do we "Believe"  (R \/ A) ?
Clearly, because we believe R.
Thus

Bel (R) --> Bel (R \/ A).

Bel(R) is a hidden axiom of the system which starts everything off.

thus the derivation goes:

Bel(R)					[axiom]
Bel (R \/ A)				[definition of \/ ]
Bel(R \/ A) --> Bel (~R --> A)		[definition of \/]
Bel(~R --> A)				[modus ponens]

We are left with Bel (R) and Bel (~R --> A).
Bel is a weaselly function; we could have carried out the above
derivation without it by taking R as axiomatic, and ending up with
R  and (~R --> A) . Since the condition of the implication 
is false, we have no problem; (R  /\ (~R --> A)) is true for any
two propositions R, A. It is the "Bel" operator which is the
source of our problem, and it is introduced because the apprehension we
felt about Reagans's impending victory was just shy of certainty.
Here is where the real world enters, and logic says, "I never
promised you a rose garden." ... 

So Modus ponens may not be perfect for "beliefs", but it's still
as solid as a rock for real logic ...


-- 
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