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

From: vc <boston103_at_hotmail.com>
Date: 12 Jun 2006 11:58:38 -0700

> vc 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.
>
> Why the sudden switch from truth-functional to truth-preserving?

Because the PL conditional is truth-functional while modus ponens is truth-preserving. You might (or might not) greatly benefit from a trip to the nearest library it seems. There, you you can find an abundant supply of books on propositional logic, derivation, etc.that might put you straight.

> >>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 argument rests on the axiom that conditional statements in
> predicate logic are necessarily truth-functional. I have already shown
> that the axiom is false, which makes your argument unsound.

Could you provide an example of a statement in the propositional logic that would *not* be truth-functional(assuming the standard interpretation) ? Or you'll be as forthcoming with an answer as the OP who claimed that PT was truth-functional?

> > 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.
>
> And as in the case of all statements of fact, the statement can be
> false. In this case, the statement is false. The truth-functionality of
> indicative conditional is not a prerequisite for propositional logic either.

Care to oblige us with an example of such non-truth preserving statement in the propositional logic ?

> 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.
>
> I have followed this thread. I don't recall where he stated that at all.
> Could you perhaps find the relevant quote?

> >
> > That the propositional/predicate logic conditional is truth preserving
> > is a trivial mathematical fact, not an assumption.
>
> Why the sudden switch in terminology? Are you saying that you wasted
> everybody's time with sloppy terminology and now you want us to
> reinterpret everything you wrote previously?

That was a typo for which I apologize. The propositional logic conditional/implication, as rest of the propositional logic connectives, are most certainly truth-functional. The derivation rule of modus ponens is of course truth-preserving.

> >>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.
>
> It's inductive logic and not deductive logic. Big deal. I don't see how
> that makes it any less a generalization. Deduction is a property of the
> special case that does not apply to the general case. Conservation of
> mass is a property that applies to the special case but not the general
> case too. Are you suggesting we should reject quantum and relativistic
> mechanics because they lack this familiar and useful property?
>
>
> > In the sense of being a system of reasoning, yes, PT can be regarded as
> > a propositional logic generalization.
>
> In other words, you agree with Keith and your whole pretense at argument
> has been a waste of time.

Only for people who have no clue what they are talking about. Apparently, you belong to the group.

> >>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.
>
> It is an axiom that is required for neither PT nor PL/PC. Neither is it
> entirely excluded from either. Whether one assumes indicative
> conditionals are truth-functional is a matter of philosophy and not a
> matter of fact.

So how about an example of such non-truth-functional statement in the propositional logic?

>
> Since truth-functionality is equally optional in both the general case
> and the special case, I fail to see any relevance.

Sure you do.

> >>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
>
> Since the probability that a statement is false is not necessarily 0, I
> fail to see how one can derive it. Or are you suggesting that false is
> somehow a meaningful statement on its own?

'False' denotes an impossible, the opposite of a certain(in Jayne's terminology) proposition, not your strawman of 'the probability that a statement is false', P(false) is certainly zero. The fact is derivable both from the sum/product rules as well as from the Cox postulates and is quite obvious intuitively.

>
>
> 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 Keith were talking about the Kolmogorov formulation, you would have a
> point. He has been very explicit about using the Cox formulation.

Who's talking about 'Kolmogorov formulation' ? I meant some finite Bayesian model.
Are you trying to say that Bayesian probability is inapplicable to finite models like population dynamics, ? If yes, how come Jaynes somehow manages to discuss such models (see the sampling chapter) ? If not, then how do you go about finding a limit of a non-real valued function(P(A) takes values from a finite set if the model is finite obviously), with your engineering education and all ?

>
>
> >>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).
>
> I conclude you lack intellectual honesty and are arguing simply for the
> sake of having a pissing contest. Life is too short to waste it on
> idiots like you. Plonk.

Indeed. Received on Mon Jun 12 2006 - 20:58:38 CEST

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