[FOM] First- Vs Second-Order Logic: Origins of the Distinction?

josef at us.es josef at us.es
Mon May 23 10:47:47 EDT 2016

```Dear Joe Shipman:

I'm happy to elaborate a bit, but I'll keep it short because I fear most
of the arguments are already known. A paper of mine on this topic, which
is about to come out, can be found here:

So, let me briefly try to summarize the main point.

1. Notice something important: I don't say that SOL is not a logical
system, as you seem to suggest -- quite the contrary! I was talking only
about "full" SOL. What I find to be a remnant of logicism is the idea
that full SOL is the _standard_ semantics for SOL. And you're right, I
was also intimating that I find something objectionable there.

2. The question with SO quantification is -- given a domain D of
individuals, what is the totality of predicates over D? (Let's reduce
the question to this, for simplicity, letting relations aside.) Notice
that in practice, when we work with actual mathematical theories, the
predicates and relations we consider are certain concretely given ones,
plus of course anything that can be defined from those. An example is
Hilbert, Foundations of Geometry.

3. There's nothing objectionable in moving from predicates to their
extensions (sets), but it's not obvious that we can define predicates
simply to be the correlates of sets! The situation may not be symmetric.
One must take carefully into account that, in set theory, we work with
arbitrary infinite sets.

In particular, when the domain of individuals is infinite, for any
random collection of individuals, full SOL decrees the existence of a
predicate applying exactly to them (i.e., to that arbitrary collection
of individuals). This is awkward. It runs against the guiding ideas that
logical principles ought to be neutral concerning questions of
existence, and topic neutral.

4. The main point is simply that arbitrary sets are a topic for set
theory to study -- and a topic that is far from clear. It is natural to
expand quantification to predicates and relations. There's no reason to
insist on FOL as the only logical system. So far, so good; but then
comes the 'logicistic' addition, an extra ingredient from pure
mathematics: That natural idea of going second-order is  mixed with the
notion of a full powerset of the domain, taken from set theory -- the
quasi-combinatorial idea of arbitrary set.

5. The main issue, as I see it, is not a technical one. The main issue
concerns the philosophy of logic and the kinds of epistemological
implications people (sometimes) want to extract from logical results.

I propose that we should stop calling the full semantics "standard".
Call it anything else, e.g. the set-theoretic semantics. (This is not
merely a matter of words, since the word "standard" brings with it a
normative connotation.) Henkin semantics is closer to being standard.

Of course one may consider full SOL for certain particular purposes,
e.g. for doing work on model theory, but one should be clear that this
is not a system of pure logic.

Notice however that my position is not Quinean. It seems that Quine was
still too imbued in the ways of thinking of logicism, so that he unduly
identified SOL with its full set-theoretic version. This is a mistake.

Best wishes,

Jose

> ----------------------------------------------------------------------
>
> Message: 1
> Date: Sat, 21 May 2016 19:18:48 -0400
> From: Joseph Shipman <JoeShipman at aol.com>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Re: [FOM] First- Vs Second-Order Logic: Origins of the
> Distinction?
> Message-ID: <897FB754-081D-48A7-B6EE-06339A8A1BCC at aol.com>
> Content-Type: text/plain; charset="utf-8"
>
> I agree that an important motivation for SOL is logicism, but your use of the connotationally loaded descriptors "insist" and "remnant" suggests that you think there is something wrong with this much logicism; please elaborate.
>
> -- JS
>
> Sent from my iPhone
>
>> On May 21, 2016, at 7:57 AM, josef at us.es wrote:
>>
>> Dear Richard:
>>
>> the question how first-order logic became the paradigm logical system was the main topic of a paper of mine, 'The road to modern logic -- an interpretation' (BSL 2001), see http://www.math.ucla.edu/~asl/bsl/07-toc.htm. I argued that simple type theory was taken to be the main logical system by 1930, and I analyzed the main reasons for the move to FOL.
>>
>> Also relevant to this is another old paper, 'Notes on types, sets and logicism, 1930-1950' (Theoria 1997), which I can send you if you'd like. Here the main issue was to analyze reasons for the abandonment of logicism in the 1940s.
>>
>> Concerning second-order logic, let me also indicate my current view, which you may find provocative. I'm convinced that insistence on full SOL is the last remnant of logicism in foundational debates. I have argued for this in several places, a consequence being that we should stop calling "standard" the full powerset semantics.
>>
>> Best wishes,
>>
>> Jose
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