Re: completeness what is it exactly

On Jul 16, 8:13 pm, Jan Burse <janbu...@xxxxxxxxxxx> wrote:
herbzet schrieb:

Would someone please explain to me why propositional logic
does not have variables?  Or does not have free variables?
Or whatever?  I must have missed the memo.

Your propositional formulae at the bottom of this post
sure look to me like they contain free variables.

No, they are zero arity predicate constants.

Bye
Do I now still understand it
(Or am i just getting totally confused)

p-> p

p is a sentence letter
(or equivalent an 0 -place predicate symbol)
(or a zero arity predicate constants
(or a propositional variable)
[ any other name for this concept ;-) ]

Is therefore p->p a sentence?

But if p->p is a sentence then also (p->p)->p is a sentence.

(if what a sentence is is only defined by syntax)



Back to Completeness.

negation completeness says

for every sentence A

|- A or |- ~A


But still neither |- (p->p)->p nor |- ~((p->p)->p)

therefore:
- (p->p) ->p is not a sentence
- propositional logic is not negation complete
or
something else all together????

replacing sentence letter p with Q

where Q stands for any propositional
formula, (then this is a schema, and not
a formula.)

Q -> Q
and
(Q->Q) ->Q

Makes it all just more complicated.
are they both sentences?
only the first one?
neither of them?
(that only the second is a sentence nobody will have that opinion)



Answer to Balthazar and Herbzet
If
Axiom A has sentence letters,

and
Sentences are formula's without sentence letters (Or propositional
variables)

Then
Axiom A is not a sentence.

But it was to test if the definition
Sentences are formula's without sentence letters cut any wood.
(Still think it doesn't)

Defining axioms as sentences feels a bit the wrong way round.
(Completeness means that if formula A is true then formula A is
provable)
so starting with defining axioms as sentences looks at least weird.


So it all goes back to the question

What is a Sentence?
(Other than what is a well formed formula)

still confused
.

Relevant Pages

  • Re: completeness what is it exactly
    ... that has no occurrences of free variables. ... If 'p' is a sentential letter (a 0-place predicate symbol), ... Axiom A has sentence letters, ... Sentences are formula's without sentence letters (Or propositional ...
    (sci.logic)
  • Re: completeness what is it exactly
    ... some authors also call sentence letters 'variables'. ... does not have free variables (i.e., ... 0-place predicate symbols. ... parentheses (not needed if we use Polish notation) ...
    (sci.logic)
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