Re: Semantics of First-Order Languages

On 2 mar, 08:22, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
malc...@xxxxxxxxx wrote:
On 1 mar, 00:24, george <gree...@xxxxxxxxxx> wrote:
On Feb 29, 3:37 am, malc...@xxxxxxxxx wrote:

the language of set theory is just an
instance of a first order logic language, which is precisely the kind
of thing one is trying to give sense.
Not all circles are vicious.
You have to stop SOMEwhere.
This objection is about as sensible as saying
you don't understand what an axiom is because
provability is what axioms are trying to found.

More to the point, nobody is trying to "give sense"
to an first-order language.  If you actually want to do that,
you do it via THE AXIOMS, NOT the semantics.  And
the sense you give is not even specific to THAT language,
in any case, since the exact same sense would be had
by a different language with different names for all the
predicates and functions, as long as they had isomorphic
arities and the axioms were translated isomorphically
into the other language.

I was refering to an interpretation as making a sentence either true
or false.

Right. That's what interpretation does: making a sentence true or false.
But that's also why interpretation is subjective - and relative: there's
no absolute rule why a sentence is interpreted as true or false, instead
of the other way around. For instance, given 2 open longitudinal lines
L1, L2 where the 2 poles are not on them. Is the statement "L1 and L2 are
parallel to each other" true? or false? The answer is it'd depend which
way we'd like to interpret it. If by "parallel" we mean L1 L2 don't intersect
and there would be a 3rd line perpendicular to them, then we've interpreted
the sentence as true. But if by "parallel" we mean L1 and L2 should not
"converge" to a common point, then the sentence would be false.

But there's no way the sentence (which is syntactical) could
be absolutely true, or false.



The problem seems to me that the justification for the syntactic rules
of inference (which I think could be accepted without reference to any
'meaning') seems to lay on the satisfaction relation which is supposed
to give, for any given interpretation, a truth value to every
sentence. I don't understand how this is achieved.

The reason for the difficulty is that we tend to have the "illusion"
that the syntactical rules of inference should be "justified" on the
basis of (first order) truth. The 2 (meta) truths of the matter are:

(1) A formula must necessarily be in its essence a syntactically well-formed
     formula. Whether or not it's semantically meaningful is quite irrelevant
     to this essence.

(2) Similarly, rules of inference are devoid of truth and falsehood.
     If a statement S is a theorem of a formal system then it must be
     absolutely provable. If both S and ~S are not concurrently theorems
     of a system T then T must absolutely be syntactically consistent,
     whether or not we know the fact.

I'd agree with your intuition: "I think [syntactic rules of inference]
could be accepted without reference to any 'meaning'".- Ocultar texto de la cita -

- Mostrar texto de la cita -

I did not mean any sencence should have any absolute true or false
value of truth. What I wonder is how an interpretation might always
give any sentence one (relative) true or false value, being just a
translation into another first order language that maybe should
(again) be interpreted.
.

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