tjpak@athena.mit.edu (Tae J Pak) (07/12/87)
In article <19647@ucbvax.BERKELEY.EDU> kube@cogsci.berkeley.edu.UUCP (Paul Kube) writes:
] It's October 1980. You hold the following plausible beliefs:
] 1. If it's a Republican that will win the election, then if
] Reagan doesn't win, Anderson will.
] 2. It's a Republican that will win the election.
]However, you don't believe 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).
An article on this dilemma (and in fact, citing this very example) appeared
in a recent issue of _Analysis_, a philosophical journal. I would suggest
to those who are interested that they refer to this very fine journal.
In the mean time, I would like to propose a solution of my own.
Now it seems to me that there is a fundamental ambiguity in the two
statements above. 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
crucial that this ambiguity be resolved, else it cannot be certain that
Statement 2 is an affirmation of the IF clause of Statement 1. I find it
highly implausible that Statement 2 is a statement of fact, so for the moment
let us assume it is a sentiment. If that is the case then clearly Statement
2 does not affirm the IF clause of Statement 1, for Statement 1 assumes the
factual interpretation and not the sentimental interpretation. That is to say,
Statement 1 makes the claim that IF a Republican MUST win, then it will either
be Reagan or Anderson. Further more, if in fact the IF clause of Statement 1
is assuming a sentimental interpretation, then I would claim Statement 1 is
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 and, thus, our choices are no longer restricted to the
Republican field. For all intents and purposes, we can disregard the initial
IF clause because it has no bearing on the statements that follow. To sum
up, one of the following three must be true:
(a) Statment 2 and the initial IF clause of Statement 1 are both
facts, in which case modus ponens works fine;
(b) Statement 2 and the initial IF clause of Statement 1 are both
sentiments, in which case I claim Statement 1 is incorrect and
should be replaced by Statement 1A (and modus ponens again works);
(c) Statement 2 and the initial IF clause of Statement 1 are of mixed
type, in which case modus ponens doesn't apply.
Comments?
--Tony Pak
tjpak@speaker.mit.edugreg@mind.UUCP (Greg Nowak) (07/14/87)
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 ... -- greg Since I now handle mail and news from inside emacs, 90% of my time is spent there ... so I have my .login put me in emacs immediately. ...seismo!princeton!mind!greg MODIFY my BUFFERS! YOW!