Re: What isn't a tautology?

On 20 Jul 2005 07:17:38 -0700, George Dance <georgedance04@xxxxxxxx> said:
> Chris Menzel wrote:
>> On 16 Jul 2005 19:26:47 -0700, George Dance <georgedance04@xxxxxxxx> said:
>> > Chris Menzel wrote:
>> >> On 16 Jul 2005 11:51:09 -0700, George Dance <georgedance04@xxxxxxxx> said:
>> >> > Chris Menzel wrote:
>> >> >> On 16 Jul 2005 09:32:31 -0700, George Dance <georgedance04@xxxxxxxx> said:
>> >> >> > Chris Menzel wrote:
>> >> >
>> >> >> >> Consider, e.g., "A prime number is a positive integer > 1 whose
>> >> >> >> only positive integer divisors are itself and 1", i.e.:
>> >> >> >> (x)(Prime(x) <-> Integer(x) & x > 1 & (y)(Integer(y) & y > 1 &
>> >> >> >> Divides(y,x) -> (y = 1 v y = x))). This definition has a complex
>> >> >> >> logical form in predicate logic, but its propositional form is
>> >> >> >> simply that of an atomic sentence P; it's not a negation,
>> >> >> >> conjunction, disjunction, conditional, or biconditional.
>> >> >> >
>> >> >> > Sorry, but it looks like a biconditional to me:
>> >> > <unsnip>
>> >> > A<->B (A="a is prime"; B="a's only positive integer divisors
>> >> > are a and 1")
>> >> > </us>
>> >> >
>> >> >> Ok, but it's not.
>> >> >
>> >> > Why not?
>> >>
>> >> Because:
>> >>
>> >> >> It's a universally quantified biconditional.
>> >> >
>> >> > And no 'universally quantified biconditionals' are biconditionals?
>> >> > That's a new one to me.
>> >>
>> >> Great, you've learned something today. :-)
>> >
>> > I've learned your opinion: that some biconditionals are not
>> > biconditionals.
>>
>> No, that no quantified biconditionals are biconditionals. Admittedly,
>> the grammatical form of that claim might be misleading: one might think
>> that "quantified biconditional" indicates a subclass of the
>> biconditionals the way that "healthy man" indicates a subclass of the
>> men. But a better linguistic analogy here would be "dead man", or
>> "petrified wood". In such cases, you start with something in the
>> indicated class (a biconditional, a man, a piece of wood), it undergoes
>> some sort of change, and the result is something of a different nature
>> than the original. A quantified biconditional, in particular (and
>> somewhat metaphorically), is the result of taking a formula with the
>> form of a biconditional and turning it into a formula of a different
>> form by prefixing a quantifier.
>
> It's clear, I hope, that what I said looked like a biconditional was
> your definition statement, "A prime number is a positive integer > 1
> whose only positive integer divisors are itself and 1". You indeed
> turned that statement into a 'universally quantified biconditional'

I didn't turn it into a universally quantified biconditional, it *is* a
universally quantified biconditional.

> (a statement about everything) when you formalized it in predicate
> logic, but that's neither here nor there wrt to formalizing the same
> statement in propositional logic.

Its formalization in propositional logic would be: P. Granted, all of
its *instances* would be biconditionals, but they don't do much better
in capturing the content of the original definition; for that you need
the apparatus of quantification.

>> > What I haven't learned, is why it's your opinion.
>>
>> The short answer is that the syntactic type of a (nonatomic) sentence A
>> of predicate logic is determined by the logical operator occurring in A
>> that has the widest scope in A. The operator that has widest scope in
>> the definition of "Prime" above is the universal quantifier. It is a
>> simple exercise to show that every formula is of exactly one type.
>
> Well, that's all well and good; but one normally doesn't use
> propositional logic to formalize sentences of predicate logic, does
> one? What would be the point of doing that?

I haven't the slightest idea. Why would you think I'm interested in
doing it? You wanted to know why I denied that the definition in
question was a biconditional. And the answer is that every statement of
predicate logic (which includes propositional logic) has a unique
logical form, and the form of the definition in question was a universal
quantification, not a biconditional.

>> And just to be clear, this is no more my opinion than that it is my
>> opinion that, say, no biconditional is a conjunction, or that "p v
>> ~p" is a tautology. This is standard material covered in most any
>> thorough text on mathematical logic.
>
> If it is a basic rule of mathematical logic that definition statements
> (like "A prime number is a positive integer > 1 whose only positive
> integer divisors are itself and 1") must be universally quantified
> before being translated into propositional logic (PC) -

Huh, wha? Rule? It's just a fact that definitions like the one in
question *are* universally quantified. The point about their coarse and
inadequate representation in propositional logic was meant only to show
that such definitions are not tautologies.

> and hence all must be translated into PC as non-tautological atomic
> constants - then that rule must apply to tautological statements (like
> "A prime number is a prime number") as well:

That is not a tautology, its a universal quantification; it's of the
form "(x)(Px -> Px)". Granted, the unquantified part has the form of a
tautology, those variable occurrences need to be quantified to capture
the logical form of the example.

> meaning that no tautological statement could be translated into PC as
> a tautology, either.

No. The notion of tautology can be extended to predicate logic in a
natural way. One way to do this, roughly, is to say that a sentence S
of predicate logic is a tautology if there is way of uniformly replacing
subformulas of S with propositional constants in such a way that the
result of the substitution is a tautology of propositional logic. So,
for instance, "(x)(Px v Qx) -> ((x)Rx -> (x)(Px v Qx))" is a tautology
because we can replace "(x)(Px v Qx)" with "p" and "(x)Rx" with "q" and
result is the propositional tautology "p -> (q -> p)".

Chris Menzel

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