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Re: Programming is the Engineering Discipline of the Science that is Mathematics

From: vc <boston103_at_hotmail.com>
Date: 12 Jun 2006 08:06:47 -0700
Message-ID: <1150124807.441420.89810@y43g2000cwc.googlegroups.com>


Bob Badour wrote:
> vc wrote:

[...]
> > My very first message just modestly reminded that probability is not
> > truth functional, that was all.
>
> With all due respect, whether it is truth functional is a philosophical
> matter that applies equally to conditionals in logic. See
> http://plato.stanford.edu/entries/conditionals/
>
> Some philosophers treat both probability and logic as truth functional
> and some don't.

I am afraid you misunderstood the article you are referring to. The article discusses various interpretation of the conditional and speculates on whether the conditional can be regarded as truth-preserving. However, the propositional logic has it quite clear: the implication *is* truth-preserving (as is the entire predicate logic). There is no ambiguity or opinions with respect to predicate logic.

> > The proponents of the probabilistic logics hoped that P(A|B) = P(B->A).
> > Lewis showed that the conditional probability cannot be the
> > probability of implication (the truth functional conditional) thus
> > making truth-functionality impossible though the conditional
> > probbaility either.
>
> Unless one accepts Stalnaker's philosophy instead of Lewis'.

The cornerstone of Stalnaker's ideas was the conjecture that the probability of a conditional is the same as the conditional probability (ca 1968). Lewis showed that it's not the case. I am not aware of any Stalnaker's subsequent work where he would repair the assumptions allowing his hypothesis to hold. His 2005 article referenced in the "Conditionals" does not provide a formal account that would clearly state assumptions, derive results and provided a way to numerically apply his informal thoughts just as PT (or some form of probabilistic logic) does. More importantly, the ordinary PT is not exempt from Lewis's proof so whatever fixes Stalnaker may have had in mind do not apply to PT.

>
> > In order to
> > derive the compound statement probability additional information must
> > be taken into account while with PL the compound statement truth
> > depends only on truth values of its sub-propositions.
>
> Just as relativity depends on the speed of a frame of reference relative
> to the speed of light whereas classical mechanics does not but only
> really holds in some limit of that speed, and just as the cosine law
> depends on the angle between two sides of a triangle whereas the
> pythagorean theorm applies only at one specific angle.
>
> Are you suggesting that relativity is not a generalization of classical
> mechanics to relativistic speeds? Are you suggesting that the cosine law
> is not a generalization of the pythagorean theorem to acute and obtuse
> angles?
>
> You seem to be arguing that the requirement for additional information
> in the general case makes a generalization invalid whereas all other
> generalizations seem to have similar requirements.
>

My very simple point was that PT is not truth functional and therefore cannot qualify as a generalization of propositional logic or any other logic possessing truth-functionality. Your extrapolating this simple statement to the relativity theory vs. classical mechanics, cosine laws, cabbages and kings is truly strange. As I said before, there are profound connections between PT and proopsitional calculus, and PT can be regarded, metaphorically, as a generalization of purely logical reasoning about uncertainty. However, their areas of applicability is quite different and the generalization talk serves nothing but to increase confusion. Since I have nothing more to add except PT lacking truth-functionality and a system of derivation axioms (Abadi/Halpern), this wil be my last comment on propositional logic vs. PT.

>
> With all due respect, your dismissal reeks of evasion. We are discussing
> generalizations and specializations. You apparently argue that
> probability theory is not a generalization of logic, and you allege that
> your challenge demonstrates your argument. It does not.

My challenge merely demonstrates that P(A and B) cannot be derived from P(A) and P(B) alone. I do not know why it is so hard to understand and compare PB non-truth-functionality to the propositional logic truth-functionality -- it's a simple statement of fact. Also, I'd like to remind that the OP made the claim, in so many words, that P(A and B) can be calculated from P(A) and P(B) alone but failed to back up his claim.

> What I find strange is your assumption that deductive logic must treat
> indicative conditionals as truth functional for modus ponens and modus
> tollens or for propositional logic. Since propositional logic does not
> require truth functionality for modus ponens, I don't see how predicate
> logic requires it either. However, I am unfamiliar with any arguments
> either way.

To be precise, modus ponens is a derivation rule (possibly single) in the propositional/predicate logic. In the propositional logic, material implication/conditional tautologies have the same syntactical pattern as modus ponens proofs which can serve as sort of justification for modus ponens truth preservation feature. So the correct statement should be that modus ponens is truth preserving rather than truth functional.

>
>
> > If some other, non-truth functional logic was implied, then the
> > context should have been clearly stated, the usual assumption being
> > that the '[default]logic' = 'propositional logic/predicate calculus'.
>
> I don't see any evidence that Keith assumed any different context
> requiring a statement. Instead, I question your assumption that
> indicative conditionals are necessarily truth functional in deductive logic.

That the propositional/predicate logic conditional is truth preserving is a trivial mathematical fact, not an assumption.

> You did not answer my questions. If you are not confused on the issue of
> generalization, are you suggesting that classical mechanics and
> relativity require merging?

In the sense of truth-preservation and/or formal derivability, PT cannot be called logic generalization by any stretch of imagination. In the sense of being a system of reasoning, yes, PT can be regarded as a propositional logic generalization.

>
> If that was your main point, why didn't you simply state it instead of
> making philosophical arguments regarding truth-functionality and
> generalization?

Truth functionality or its absence is a simple mathematical fact when applied to PT or PL/PC so my argument was purely mathematical.

>
>
> >>>>>What Jaynes did in his derivation of the sum/product rules has got
> >>>>>nothing to do with your mindless playing with formulas. See the
> >>>>>argument from authority in my previous messages.
> >>>>
> >>>>Your argument from authority was flawed. I will reply in the other thread.
> >>>
> >>>The argument from authority was a quote from Jaynes' book , not mine.
> >>
> >>The source of the quote is irrelevant because the flaw itself was a lack
> >>of relevance. Keith never relied on the meaning of any meaningless
> >>value. The fact that the value of x might be indeterminate is
> >>unimportant to the conclusion that x times zero is zero because the
> >>conclusion holds for all x.
> >
> > He relied on the non-existing value, not an existing but
> > unknown/indeterminate value. There is no x*zero simply because x does
> > not exist.
>
> But Keith did qualify that his argument applied in the limit. Would you
> have accepted his argument had he presented it in explicit limit
> notation? P(B|A) is defined in the limit as P(A) approaches 0 after all.

It can be so defined, yes, but there are at least two problems with such definition. (1)P(false) = 0 is usually derived directly from Cox's postulates or from sum/production rules without relying on the limit, so any departure from the usual derivation should be clearly stated along with possible problems such approach may have; (2) more importantly the limit idea is inapplicable in finite domains (e.g. picking a ball from an urn, or some such) where probability values are taken from a finite domain of possibilities.

>
> If you would accept the proof using limit notation, then your argument
> seems more of a quibble over notation to me.

I do not accept it because of (1) and (2). Received on Mon Jun 12 2006 - 10:06:47 CDT

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