Re: What isn't a tautology?

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' (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.

> > 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?

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

.

Relevant Pages

  • Re: What isnt a tautology?
    ... >>> And no 'universally quantified biconditionals' are biconditionals? ... that no quantified biconditionals are biconditionals. ... is why it's your opinion. ... the definition of "Prime" above is the universal quantifier. ...
    (sci.logic)
  • Re: What isnt a tautology?
    ... >> Chris Menzel wrote: ... that no quantified biconditionals are biconditionals. ... >>> form by prefixing a quantifier. ...
    (sci.logic)
  • Re: What isnt a tautology?
    ... that no quantified biconditionals are biconditionals. ... > biconditionals the way that "healthy man" indicates a subclass of the ... > "petrified wood". ...
    (sci.logic)
  • Re: What isnt a tautology?
    ... that no quantified biconditionals are biconditionals. ... >> the grammatical form of that claim might be misleading: ...
    (sci.logic)
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