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

Date: Mon, 12 Jun 2006 02:41:26 GMT
Message-ID: <qv4jg.21392\$A26.481017@ursa-nb00s0.nbnet.nb.ca>

vc wrote:

>

```>>vc wrote:
>>
>>>
>>>
>>>>Are you seriously suggesting that true and false are trivial and
>>>>uninteresting? Should we all pack up and go home?
>>>
>>>I am suggesting that a meaningful PT statement reduction to a
>>>propositional logic statement is trivial and uninteresting.
>>
>>Then why did you bring it up in the first place?
```

>
> If you read the whole thread carefully, you'll realize that it was the
> OP who brought the 'reduction' issue, not me:
>
> <OP>
> "
> This is why PT is a /generalization/ of logic. It reduces to
> logic when applied to truth-valued statements. Just as gamma
> reduces to factorial for natural arguments. (Again no quibbles
> about offset by 1 etc).
> "
>
> 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.

```>>   Also,
>>
>>>there is a problem with the conditional probability not being equal
>>>the probability of the conditional (see Lewis's result) which makes
>>>conditional probabily untranslatable to modus ponens in principle.
>>
>>Modus ponens requires a conditional probability of 1 and says nothing
>>about situations with other conditional probabilities which is why the
>>conditional "If A then B" says nothing about B when A is false.
>>
>>For modus ponens, one must start with the premise "If A then B", which
>>is synonymous with P(B|A) = 1. To have a sound argument, one must not
>>only have a valid argument, but the premise "If A then B" must be true.
>>
>>It think your argument might cause more problems for modus tollens than
>>for modus ponens.
```

>
> 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'.

```>>>Going in the opposite direction,  generalization,  PT is not truth
>>>functional ,  that is the probability of a compound statement is not
>>>determined solely by its components probabilities (see my trivial
>>>puzzle).
>>
>>I fail to see what an indeterminate problem demonstrates. If I give you
>>the lengths of two sides of a triangle and ask you for the length of the
>>third side, you cannot answer unless you know the angle where the given
>>sides meet.
```

>
> The problem merely demonstrates that PT does not possess truth
> functionality in the same way the propositional logic does.

I disagree. It demonstrates only that absent P(A|B) and absent P(B|A) one cannot calculate P(A and B) just as absent the angle between two sides of a triangle with known lengths, one cannot calculate the length of the third side.

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.

```>>Such a challenge would neither disprove pythagoras nor disprove the law
>>of cosines. Neither would it disprove that the law of cosines
>>generalizes the pythagorean theorem.
```

>
> That's an irrelevant remark. See above.

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.

```>>   Also, importantly,  it appears impossible to find an axiom
>>
>>>system for any known probabilistic logic that would be sound and
>>>complete (except some special cases). Obviously, lack of such axiom
>>>system makes a formal derivation (a hallmark of any logic) impossible.
>>
>>That precludes it from deductive logic but not from inductive logic.
```

>
> That very well may be the case depending on what you mean by "inductive
> logic", but the original statement appears to be that PT somehow
> subsumes/"generalizes" propositional logic, or any other
> truth-functional/deductive logic, which is a bizzare statement indeed.

Your dismissal seems unsubstantive to me. A system of mechanics that alters the sum of the angles of a triangle depending on the speed of the frame of reference seems rather bizarre from the perspective of classical mechanics and euclidean geometry. However, that doesn't make relativity any less a generalization of classical mechanics nor does it make euclidean geometry any less a special case of the reimannian manifolds underpinning relativity.

It is entirely possible for an inductive logic to generalize deductive logic. Special cases have special properties that do not apply to the general case. The general case often depends on information considered unimportant in the special case. Deductive logic has special properties that do not apply to inductive logic. Probability theory requires conditional probabilities that propositional logic ignores. Big deal.

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.

> 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.

```>>>Apparently,  despite obvious similarities and profound connections,
>>>both had better be used what they are best at and attempts to merge
>>>them do not seem very productive (see abundant literature on
>>>probabilistic logics).
>>
>>Are you suggesting that they need merging?
```

>
> You are confused, the OP does (see his talk about "generalizing"),
> not me.

We obviously disagree regarding who is confused.

```>>Classical mechanics is a
>>special case of relativity. Do they need merging? The pythagorean
>>theorem is a special case of the cosine law. Do they need merging?
>>Deductive logic is a special case of inductive logic. Do they need merging?
```

>
> You are preachig to the choir.

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?

```>>Each of the special cases has properties that do not generalize and each
>>of the generalizations considers factors irrelevant to the special case.
>>
>>The real question is whether inductive logic is appropriate for data
>>management, and I do not think it is.
```

>
> Neither do I.

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

```>>>>>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.
>>>>
>>>
>>>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.

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

Again, the Bayesian conditional probability *exists* only
> if the respective conditioning proposition is true. The very
> derivation of the product and sum rules from the Cox postulates relies
> substantially on B being true in P(A|B). For a rigorous exposition see
> for example Aczel's Lectures on Functional Equations.

Would the rigorous exposition invalidate the proof if the proof were given explicitly in limit notation?

```>>>>>>that should have been P(p1|p2) != P(p1).
>>>>>
>>>>>That's assuming that P(p1|p2) even makes sense.  More general
>>>>>formulation of such independence is just P(p1 and p2) = P(p1)* P(p2).
>>>>
>>>>The formulation is neither more general nor less general. It is, in
>>>>fact, a simple substitution of the equation describing independence:
>>>>
>>>>(1)   P(p1|p2) = P(p1)
>>>
>>>Again,  the substitution is possible only when P(p2) > 0.  (See the
>>>Jaynes book).
>>
>>Okay, the formula for independence only applies when P(p2) > 0. And how,
>>exactly, does that support your assertion that one formulation is more
>>or less general than the other?
```

>
> Independence can be (and is) defined as P(A and B) = P(A)*P(B). It
> works with whatever values of P(A) and P(B) as opposed to the
> conditional probability formulation.

Okay. Fair enough.

```>>>>into the formula for conditional probability:
>>>>
>>>>(2)   P(p1 and p2) = P(p1|p2)*P(p2)
>>>>
>>>>Substitute (1) into (2) gives P(p1 and p2) = P(p1)*P(p2)
```
Received on Sun Jun 11 2006 - 21:41:26 CDT

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