Re: What isn't a tautology?

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.

> 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.

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.

Chris Menzel

.

Relevant Pages

  • Re: What isnt a tautology?
    ... that some biconditionals are not ... that no quantified biconditionals are biconditionals. ... is why it's your opinion. ... that rule must apply to tautological statements (like "A prime number ...
    (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)