[hist-analytic] The status and relevance of "standard logic"

Roger Bishop Jones rbj at rbjones.com
Thu Nov 5 11:31:09 EST 2009


On Tuesday 03 November 2009 09:30:25 Danny Frederick wrote:

> 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.

I can understand your believing this if your information on this matter comes 
primarily from the papers which Tarski wrote in the 1930s, but it is not 
correct.

I believe Tarski proved one definite and very specific result, by formalising 
the liar paradox in first order arithmetic, viz. that arithmetic truth is not 
arithmetically definable.

From this he concluded, without proof, some general and vague principles
along the following lines:

	a) that the semantics of a language L can only be defined in some other
	    language which is strictly more expressive (in some sense) than L

	b) that the relevant kind of expressiveness is that obtained by
            the availabilty of objects of higher type.

Something can be made of the first intuition, but only by making it more 
precise and less general than it appears to be.
I think it doubtful that the second can be cashed in.

The logical system of preference for defining semantics is first order set 
theory.  This is semantically more expressive than any type theory (subject to 
some caveats on what one takes its semantics to be).
However, to define the semantics of a pure first order language, and to 
demonstrate the soundness of the usual deductive system relative to that 
semantics, is particularly simple and can readily be accomplished in PA or 
even weaker systems.
In this is should be noted that PA is, in an appropriate sense relevant to 
intuition (a), much more expressive and proof theoretically stronger than a 
pure first order logic (but does not involve objects of higher type).

I might add here that my allegation is about what logical systems suffice for a 
formal proof of the soundness of first order logic.
Of course, underpinning such a proof there may be all sorts of philosophical 
presumptions, which may be of relevance is assessing the significance of such a 
proof. Of these I say nothing.  I allege only that there exists a valid proof 
in PA, the proof itself making use of no further "assumptions" beyond the 
accepted axioms of PA.

> 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 are back on the "vacuous names" tack here which gave much entertainment to 
Speranza a while back.

There a multiple strategies which enable the expression in first order logic of 
the claim that Santa Claus does not exist, the most famous of course is 
Russell's doctrine that "Santa Claus" should be construed as a description and 
formalised by the method described in his paper "On Denoting".

There is a difficulty in knowing what kind of first order formalisation is 
satisfactory, since we have no generally accepted criteria for when sentences 
in distinct languages express the same proposition.

However, I would suggest that the principle difficulty here is in deciding what 
is the meaning of "Santa Clause does not exist", once that is done finding a 
way of saying the same thing in first order logic will be much less 
problematic.

> 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.

I am inclined to agree with you on this, though I can't see what bearing this
has on the matter at issue (which is I believe, whether first order logic is 
sound).

> And can a theory of linguistic types be separated from a theory of logical
> types? I don't think so.

If you are asking me whether a logical type theory is necessary for 
formalising talk about language then the answer is "no".

> 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.

The considerations you raise speak against appearances being conclusive but 
not against their being relevant.

> <<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.

Which of your points do you take this to be conceding?

> 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).

I don't recall making a claim about objectivity in this.

> 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).

I am puzzled as to the relevance of this to any discussion.
Evidently the appearances lead you to believe various propositions with 
sufficient strength to argue them against others with a determination which is 
suggestive of conviction.

Are you suggesting that my own rather more subtle skepticism should keep me in 
silence?

Roger Jones




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