[hist-analytic] The status and relevance of "standard logic"
Danny Frederick
danny.frederick at btinternet.com
Tue Nov 3 14:55:33 EST 2009
Hi Gregory,
Thanks for your contribution. I am not sure if it is intended as a rebuttal
of what I was saying or whether it is intended merely to add some other
points. Let me take it as the former and respond to it on that basis.
You begin: 'I think that a distinction between language and meta-language is
essential to the very intelligibility of the notion of a "formal axiomatic
system".'
You may be right. But I notice that you say 'I think.' And even if it is so,
it is not obvious (certainly not as obvious as elementary arithmetical
truths). Further, I think it is something that it took the paradoxes to
force upon people's attention. Without the paradoxes, anyone making such a
distinction would surely have seemed to be a pedant par excellence! What
will he talk of next? How many angels can dance on a pinhead? Of course, it
needn't have been the paradoxes that motivated the distinction: it could
have been other perplexities into which we fall in using natural language.
But the point is, the distinction is put forward to solve a problem. And any
problem solution is only ever one possibility amongst others. Dialetheists,
for example, find Tarski's solution ad hoc: they think it is better to
accept that some contradictions are true.
You say: 'it is now known how to formulate a first order predicate logic
that does not yield any existential theorems.'
Thanks for the information. Does the system include names (individual
constants)? Either way, it will include a (perhaps unarticulated) theory
about names and designation and meaning - a theory that some people will
dispute - a theory that no one can seriously claim to be as obvious as the
truths of elementary arithmetic.
You say: 'One cannot "say" anything in predicate logic. Saying something
concerns interpretation and is a semantic issue'
I think this is contentious. First, we can treat predicate logic as a pure
formalism, in which case none of its formulae say anything. But even then it
does not follow that the formulae are empty of meaning. They have their
formal meaning: 'P' represents a predicate, 'a' a name, and so on. There's a
semantics of syntax!
Second, it is natural and common to say that there are some things we can
say in first-order predicate logic (such as 'the king of France is bald')
and some things we cannot (stuff involving opaque contexts, for example).
The reason this is natural is that predicate logic is a formal logic, a
formalisation of arguments: the whole point of it is to represent things
that can be said and to show their implications.
Best wishes,
Danny
-----Original Message-----
From: Landini, Gregory [mailto:gregory-landini at uiowa.edu]
Sent: 03 November 2009 14:35
To: 'Danny Frederick'
Subject: RE: The status and relevance of "standard logic"
Hey all:
I think that a distinction between language and meta-language is essential
to the very intelligibility of the notion of a "formal axiomatic system".
It not motivated by any paradox but by efforts to be clear about the meaning
of "well formed formula," "consistency", "semantic completeness", and the
like.
There were, of course, paradoxes that arise from confused uses of words like
"nameability" or "definability," but most of these have been sorted out
thanks to Tarski and demoted from paradoxes to puzzles (whose solutions are
known). For instance, it is now a puzzle whose solution is known (and not a
paradox) that argues (as Koenig and Dixon did) that the non-denumerability
of real numbers is incompatible with the axiom of choice. This was the
puzzle that using choice we can well order the reals and that there would
then be the first unnameable real (which we seem to have just named).
Classical first order predicate logic (quantification theory) with identity
has the theorem: (Ex)(x=x). Hence in the classical semantics for logical
truth, no domain is empty. Therefore, we can add denumerably many individual
constants to the language of first order predicate logic and we could even
allow them to occur in axioms for identity so that we get "c=c" as an axiom.
Hence, for our individual constants c1, c2, ... etc we get the theorems
"(Ex)(x=c1)", "(Ex)(x=c2)" ...etc. This is deemed innocuous since the
semantics of first order logical truth can assign all the constants to the
same member of the domain. This extended first order predicate logic (that
has the constants) is not logically committed to the existence of more than
one entity.
Some don't like this. Russell banned all individual constants (and function
constants) from the language of pure logic. (His theory of definite
descriptions shows how to do this.)
Moreover, it is now known how to formulate a first order predicate logic
that does not yield any existential theorems. (Its semantics embraces an
empty domain.)
One cannot "say" anything in predicate logic. Saying something concerns
interpretation and is a semantic issue.
One can write the following in a first order language:
"-(Ex)( Sy <->y y=x)."
Then one can interpret "Sy" so that it means "y lives at the north pole and
delivers gifts to all good people on Christmas". With that interpretation,
it comes out true. Under the interpretation that assigns "Sy" to "y is
natural satellite of earth" it is false.
"(Ex)(x = Santa Claus)" is a theorem of classical predicate logic with
identity whose language is extended by the addition of the singular term
"Santa Claus" and which allows "Santa Claus = Santa Claus" as an axiom.
But its being logically true just means that in every non-empty domain we
can make some assignment to "Santa Claus" so that it comes out true. Of
course, none of these interpretations make it say that something is Santa
Claus or that "Santa Claus" refers. No interpretation is such that there is
a unique object x in the domain who lives at the North Pole and delivers
gifts to all good people on Christmas and is such that "Santa Claus" is
assigned to x. We must finally admit there is no Santa Claus.
Gregory
Gregory
-----Original Message-----
From: hist-analytic-manager at simplelists.com
[mailto:hist-analytic-manager at simplelists.com] On Behalf Of Danny Frederick
Sent: Tuesday, November 03, 2009 3:30 AM
To: 'hist-analytic'
Subject: RE: The status and relevance of "standard logic"
Hi Roger,
<<There are no "hidden axioms" involved
A type theory is not necessary>>
A distinction between language and metalanguage must be made in order to
avoid semantic paradoxes. This is a theory of linguistic types. Further
theories of linguistic types are also assumed. For example, the following is
a logical truth (in PL + identity):
Ex(x = a).
This amounts to the claim that every name designates. It is not possible to
say (truly) in first-order logic that Santa Claus does not exist. We have to
go to the metalanguage and say that 'Santa Claus' does not designate
anything, or 'Santa Claus' is not a name. Do you think that the claims that
Santa Claus is not a name, and that 'Nothing is identical to Santa Claus' is
ill-formed, are as obvious as parts of elementary mathematics? They seem to
me to be obviously false.
And can a theory of linguistic types be separated from a theory of logical
types? I don't think so.
Thus, there are a many dubious assumptions needed to make the system work.
And they are hidden because they are not explicitly stated as axioms of the
system: they are just assumed by the system builder.
You ask: how can it be that the question of how doubtful something appears
does not bear upon its truth?
Just consider some examples. Basic Law V was not doubtful to anyone before
Russell produced his paradox. All manner of moral and religious truths are
indubitable to fundamentalists. That simultaneity is relative to a
co-ordinate system was highly doubtful when Einstein proposed it (I think it
still is highly doubtful), yet it is nowadays generally accepted as true by
physicists and others.
<<nothing is beyond doubt and we have no absolute warrants of truth>>
This sounds as if it concedes my point. In fact, I think it does. But you
will disagree because you want to order doubts according to HOW doubtful
they are and because you want to link this (purely subjective) order to an
(objective) order of closeness to truth (or perhaps likelihood of truth).
But this cannot be done. I agree we can often say that one thing seems to be
more doubtful to us than another. But this is purely subjective: it has no
bearing on the question of truth (consider the previous list of examples and
others like them).
Cheers.
Danny.
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