Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 02/14/05 

Date: Mon, 14 Feb 2005 15:07:55 -0500

Wolf Kirchmeir said:
> Allan C Cybulskie wrote:
> > "Wolf Kirchmeir" <wwolfkir@sympatico.ca> wrote in message
> [...]
> >>Actually, your example is a synthetic truth - it's an observation about
> >>the world, which is true, false, or inapplicable, as the case may be.
> >
> >
> > If it's synthetic, it's not in that sense, since once I have the experience
> > the statement IS true, by the definition of visible. Once I've had the
> > experience, the premise is wholly contained in the experience itself, and is
> > derived directly from it. That seems to fit the definition of analytic to
> > me; once I've SEEN something, it follows directly that the object I saw was
> > visible and needs no other supporting evidence.
>
> "Analytic" refers to sentences whose truth values can be determined from
> their logical structure, regardless of content. "Synthetic" refers to
> senetgnecs whos truth value depends on their content.
>
> BTW, what is grammatically a simple sentence may logically still be a
> compound statement -- see below.
>
> >>When used as a premise in an argument, we assume it to be true "for the
> >>sake of argument", as the quaint phrase has it.
> >
> > Um, not in the case I described. If visible simply means "I can see it"
> > then it isn't true "for the sake of argument" to call an object that I am
> > currently seeing "visible". You can debate that it's an object, but that
> > isn't the quality I was talking about
>
> That's a discussion of semantics and morphology, not of logic, and quite
> correct as far as it goes. Ie, in English it's possible to make an
> adjective on the same base as a noun or verb. In your example, we build
> on -vis-: visualise, visible, invisible, vision, etc. We can make a noun
> from an adjective: visible -- visibility.
>
> Anyhow, you seem to be confusing properties of statements with
> properties of the subjects of statements. In logic, the latter is a
> given - hence my allusion to "for the sake of argment."
>
> An "argument" is a series of statements such that the Premise implies
> the Conclusion: [P] --> [C]. That's all. To determine whether the
> argument is valid, you calculate all its possible truth values. If it's
> never the case that [P] is true and [C] is false, then you have a valid
> ragument. When usuing valid argument forms to prove an inference, you
> assume [P] to be true -- there's no point is assuming it's false? You
> may of course have good grounds _outside the argument_ for assuming [P]
> is true, you may even have some proof (ie, another argument) that [P] is
> true.
>
> >>Any statement qualified by "maybe" or its variants is a tautology:
> >>"Maybe Jill likes Jack" is logically the same as "Either Jill likes Jack
> >>or Jill doesn't like Jack", which is always true.
> >
> >
> > I don't accept this. If it is impossible for Jill to like Jack (like she
> > hates him) then the statement is false. "Maybe Jill likes Jack" translates
> > to "It is possible that Jill likes Jack", and that is not always true.
>
> Actually, that is one translation of "maybe", which is a word with
> several meanings in ordinary speech (it also inlcudes "I;'m not sure",
> and "I'm doubt that", which are staments which are true or false
> depending on how honest you are about your beliefs.) But it's still a
> tautology. Logically, a probability statement has the form "A is
> more-or-less true" or "A is more-or-less likely to be true", which
> brings us back to OR - and a tautology. :-) (NB that some logicians
> argue that the first and second forms are different, but that doesn't
> change my point.)
>
> >> SymbolIcally: {A OR
> >>notA}. It's logically true even if there is no relationship between Jack
> >>and Jill at all.
> >
> >
> > Actually, I disagree again. Jill simply cannot like Jack if she has no idea
> > who Jack is, and so it is false since the term really means "It is possible
> > Jill likes Jack".
>
> You are confusing "logically true" with "actually true." They are
> different animals.
>
> Anyhow, if she doesn't know him, she can't dislike Jack, either. In any
> case, I was showing you that tautology's truth depends on its logical
> form, not its content. If you don't believe me, construct the truth
> table for {A OR not-A).
>
> >>"A tautology is a statement that is always true" - well, yes, but as
> >>written it's vague, and subtly misleading - your example meant to
> >>illustrate that you've grasped the concept shows how it misleads. No
> >>simple proposition is a tautology. Only compound statements can be
> >>tautologies.
> >
> >>Thus a better formulation is: "A tautology is a compound statement whose
> >>truth value is T for all possible truth-values of its constituents."
> >
> >
> > That's not the definition, though, since the dictionary definition says
> > "true by virtue of its logical form".
>
> That's what I just said, only more explicitly.
>
> > Can no simple statements be true by
> > the basis of their logical form?
>
> No. Simple statements have no "logical form", ie, their truth value
> cannot be calculated from their constituents. Beware: what is
> grammatically a simple sentence is not necessarily a simple logical
> statement. IOW, watch out for compound subjects or compound objects and
> such. Watch for Aristotelian quantified statements, too.
>
> Eg: "Jack and Jill went up the hill" is logically "Jack went up the hill
> AND Jill went up the hill." It's true only if both propsotrions are
> true, and false otherwise - which conforms very nicely with common sense.
>
> Eg, "All men are mortal" is logically equivalnet to "IF X is a man, THEN
> X is a mortal". "Some men are engineers" is logically equivalent to: "IF
> X is a man, THEN X is an engineer OR X is not an engineer."
>
> >>If a tautology is the premise of a valid argument, the argument will
> >>always have the truth-value pattern T --> T, which has the value T (and
> >>which is also a tautology, BTW).
> >
> >
> > An argument that contains non-tautological premises would not be a
> > tautology, right?
>
> Correct. There are four possible t-value combinations for [P] --> [C].
> like this:
>
> a) T --> T (true; valid and sound)
> b) T --> F (false; invalid and unsound)
> c) F --> T (true; valid but unsound))
> d) F --> F (true; valid but unsound)
>
> Beacuse of a), it's possible to provided "definitve proof" of some
> statements - provided, of course, that [P] is in fact true (which is
> always the rub.)
>
> Because of case c), one cannot infer the truth of [P] from the truth of
> [C]. That inference is one of the most common mistakes in reasoning:
> people often offer a a chain of reasoning from dubious premises, show
> that their reasoning (correctly) leads to an observed truth, and then
> claim that this observed truth "proves" the premises. Many of these
> arguments seem to occur in a theological context, for some reason.
>
> NB that since arguments of interest aren't tautologies, case b) makes an
> argument form invalid, since it means that a true premise may yield a
> false conclusion. (But see "contingent" below.) If an argument form
> includes cases c) and d) as well as a), it is valid. If it is an
> argument about the real world (ie, a theory), then but you need some
> method to determine which case you actually have before you. That's the
> task of experimental science. Theoretical science constructs the valid
> arguments (theories) about the world. Or tries to. Failure is far more
> common than success. That's why people get so excited about valid
> theories, even when it's not obvious how they could be experimentally
> tested, or whether they could be experimentally tested at all (eg,
> string theory.)
>
> A "contingent" argument is one that has both cases a) and b), but is
> structured in such a way that one proposition's truth value produces a)
> if it's true (or false), and b) if it's false (or true.) In that case,
> the argument is valid or invalid depending on the t-value of this one
> proposition - hence we say its validity is _contingent_ on that
> proposition's t-value. Contingent argument forms can be very useful,
> since they allow one to use a single experimental observation to
> distinguish between cases.
>
> Tautologies are useful in abstract contexts, however. If you can prove
> that some statement S tautologically implies some statement S', then you
> can use S and S' interchangeably in other arguments, whichever is more
> convenient. Mathematics is essentially the proving of tautologies. That
> is, a theorem "follows from" some set of axioms if and only if the
> theorem is a tautological restatement of some combination of those
> axioms. Etc.
>
> BTW, the reason many posters in this thread get very het up about
> non-intuitive theorems is that they don't understand the meaning and
> import of the above paragraph.
>
Well, Wolf, I am probably one of those you think gets "het up" about the
"non-intuitive" nature of "theorems", like Cantorian cardinality. Maybe
not. I haven't been the hottest head in the cabbage patch. But, it might
surprise you that everything you've said here makes perfect sense to me.
Well stated. 

-- 
Smiles,
Tony

Relevant Pages

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