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Abstract

G. Kreisel has suggested that squeezing arguments, originally formulated for the informal concept of first order validity, should be extendable to second order logic, although he points out obvious obstacles. We develop this idea in the light of more recent advances and delineate the diculties across the spectrum of extensions of first order logics by generalised quantifiers and infinitary logics. In particular we argue that if the relevant informal concept is read as informal in the precise sense of being untethered to a particular semantics, then the squeezing argument goes through in the second order case. Consideration of weak forms of Kreisel's squeezing argument leads naturally to reflection principles of set theory.
Squeezing Arguments and Strong Logics
Juliette Kennedy Jouko V¨an¨anen
January 19, 2017
Abstract
G. Kreisel has suggested that squeezing arguments, originally for-
mulated for the informal concept of first order validity, should be
extendable to second order logic, although he points out obvious ob-
stacles. We develop this idea in the light of more recent advances and
delineate the diculties across the spectrum of extensions of first order
logics by generalised quantifiers and infinitary logics. In particular we
argue that if the relevant informal concept is read as informal in the
precise sense of being untethered to a particular semantics, then the
squeezing argument goes through in the second order case. Consid-
eration of weak forms of Kreisel’s squeezing argument leads naturally
to reflection principles of set theory.
1 Introduction
The foundational project is driven by the idea of modeling mathematical
discourse, more or less globally, by giving an adequate formal reconstruction
of it. Adequacy here, as in the phrase “adequate formal system,” is delivered
by the following: the formalism should be sound, complete, and at least
This paper was written while both authors were participants in the Intensive Research
Program - IRP Large Cardinals and Strong Logics at the Centre de Recerca Matem`atica
during the fall semester of 2016, and while the second author was a participant in the
program “Logical Structures in Computation” at the Simons Institute for the Theory of
Computing at UC Berkeley. The authors thank the CRM and the Simons Institute for
their support. The authors presented this paper to the Working Group in the History and
Philosophy of Logic, Mathematics, and Science at Berkeley in November 2016. We thank
the audience for their questions and comments.
1
to the degree possible, eective and syntactically complete. The formalism
should also be meaning-preserving, relative to a given semantics (ideally).
The idea of foundational formalism, as we called it in [17], is that with
such a system in hand one could reasonably claim that the formalism has
“captured” the informal discourse—whichever way one wishes to express this
idea of “capturing.”
At the same time the idea of considering just the informal mathematical
discourse on its own, so to say in situ, has also attracted interest. This is
implicit in so-called “practice-based” philosophy of mathematics—that prac-
tice is situated, after all, in natural language—while other philosophers take
a more direct interest in natural language. M. Glanzberg, for example, in his
[11], argues that the notion of, e.g., consequence at work in natural language
is to be distinguished from a genuinely logical consequence relation:
The success of applying logical methods to natural language has
led some to see the connection between the two as extremely
close. To put the idea somewhat roughly, logic studies various
languages, and the only special feature of the study of natural
language is its focus on the languages humans happen to speak.
This idea, I shall argue, is too much of a good thing.
Glanzberg is not only alerting us to the pitfalls of conceiving of natural
language, at least in its logical aspects, as a kind of thinly disguised formal
language, a matter of cleaning up the relevant definitions, concepts and so on.
For Glanzberg, natural language and formal discourse are instead, with re-
spect to their logical scaolding anyway, two autonomous domains—“though
the processes of identification, abstraction, and idealization can forge some
connections between them.” “Natural language has no logic” is the paper’s
central claim.1
What does it mean to say that natural language has no logic? For
Glanzberg this is to deny the logic in natural language thesis (LNL thesis
henceforth), namely the thesis that:
1Glanzberg construes the term “logic” rather narrowly in the paper. The quote in the
central claim is a deliberate reference to Strawson’s [31]:
Neither Aristotelian nor Russellian rules give the exact logic of any expression
of ordinary language; for ordinary language has no exact logic.
2
A natural language, as a structure with a syntax and a semantics,
thereby determines a logical consequence relation.
This is as opposed to the logics in formal languages thesis (LFL thesis hence-
forth):
Logical consequence relations are determined by formal languages,
with syntactic and semantic structures appropriate to isolate those
relations.. . . Thus, the logics in formal languages thesis holds that
consequence relations are in formal languages, in the sense that
they are definable from them.
The LFL thesis is uncontroversial, if not trivial. The argument against the
LNL thesis is subtle and turns partly on a critique of the model-theoretic
account of logical consequence.2We will not consider Glanzberg’s critique
here, but will do so below, as the issue becomes relevant to Kreisel’s so-called
squeezing arguments, to which we now turn.
2 Squeezing Arguments and their Critics
Squeezing arguments may be thought of as falling on the other side of the
spectrum of belief in the LNL thesis (albeit tacitly). Introduced by Kreisel
in his 1967 “Informal Rigour and Completeness Proofs” [19], and since taken
up by W. Dean [7], H. Field [9], V. Halbach, P. Smith [30] and others, the
arguments go as follows:
Consider an informally defined mathematical concept I. Formally
define two concepts Aand Bsuch that falling under the concept
of Ais a sucient condition for falling under the concept of I,and
falling under the concept of Isuces for falling under the concept
of B.ThusAIB, where the inclusions are understood as
applying to the extensions of the concepts A, B, I .
Now suppose the formal notions Aand Bhave the same exten-
sion. Then by the inclusions AIBthe informal concept I
must coincide, again extensionally, with that of Aand B.
2Glanzberg’s arguments against the LNL thesis generally read “logic” as logical conse-
quence.
3
For the informal concept IKreisel took intuitive validity, denoted Val,
understood as truth in all possible structures. This includes set and class-
sized structures, as well as, in principle at least, structures that have no
set-theoretical definition. Taking formal first order provability, denoted DF,
on the left, and taking truth in all set-theoretical structures,3denoted V,
on the right, Kreisel argued as follows: By soundness, DFVal . By t he
fact that truth in all structures entails truth in all set-theoretical structures,
Val V. Thus
DFVal V.(1)
Invoking the completeness theorem for first order logic Kreisel concludes the
following theorem,ashecallsit,forany first order statement:
Val $Vand Val $DF.
Kreisel’s presentation of the argument has been criticised in the literature.
Smith [30] objects that Kreisel’s somewhat model-theoretic construal of Val
does not obviously capture the pre-theoretic notion in question, validity-in-
virtue-of-form, as Smith prefers to think of Val.
Field’s criticism of the argument in [9] involves the soundness claim,
namely the first inclusion 8(DF!Val ):
In chapter 2 I argued that . . . there is no way to prove the sound-
ness of classical logic within classical set theory (even by a rule-
circular proof): we can only prove a weak surrogate. This is
in large part because we cannot even state a genuine soundness
claim: doing so would require a truth predicate. And a definable
truth predicate, or a stratified truth predicate, is inadequate for
stating the soundness of classical logic, and even less adequate
for proving it.4
Field goes on to prove soundness—or a weak surrogate of soundness—by
means of a formal truth predicate, applying in restricted cases. Taken as a
repair of Kreisel’s argument one might argue that it ignores the methodology
of the paper, which is heavily semantic (see below). Kreisel’s argument does
3A set-theoretical structure is one whose domain, relations and functions are sets in
the usual sense.
4[9], p. 191.
4
not depend on a proof of soundness in classical set theory. Kreisel is asking
us to take soundness for granted, on the basis of historical experience—or
as Kreisel puts it, intuitive notions standing the test of time. Instead, Field
takes DFas “primary”, in Kreisel’s terminology.5
As for Kreisel’s own “proof” of soundness, extending to i(interpreting
as an ith order sentence) for all i, it amounts to arguing that the universal
recognition of the validity of Frege’s rules (DF) at the time, together with
the “facts of actual intellectual experience” accumulated subsequently, should
amount to no more and no less than the acceptance of
8i8(DFi!Val i)
for us. And though a century of logical history has taught the logician
nothing if not to be extremely suspicious of inclusions such as (1)—suspicions
that Kreisel himself airs at the end of the paper—surely what Kreisel has in
mind here is the idea that DFwas formulated ex post facto,thatisprecisely
so as to guarantee soundness. DFstood for Frege and his contemporaries,
Kreisel claims, and stands for the contemporary mathematician too, as a
completely adequate formalisation of Val.
Returning to our discussion of [19]’s critics, Halbach [13] oers a repair of
squeezing arguments in the form of a formal, syntactic substitutional notion
of logical validity, to be substituted in for Kreisel’s somewhat model-theoretic
notion.6Such a concept of validity is, in Halbach’s view, “closer to rough
and less rigorous definitions of validity as they are given in introductory logic
courses”:
I put forward the substitutional analysis as a direct, explicit,
formal, and rigorous analysis of logical consequence. The substi-
tutional definition of logical validity, if correctly spelled out, slots
5From [19], p. 153:
First (e.g. Bourbaki) ‘ultimately’ inference is nothing else but following for-
mal rules, in other words D is primary (though now D must not be regarded
as defined set-theoretically, but combinatorially). This is a specially peculiar
idea, because 99 per cent of the readers, and 90 per cent of the writers of
Bourbaki, don’t have the rules in their heads at all!
6Halbach’s conceptual analysis applies more widely, that is, it is an analysis of the
“natural” concept of logical consequence in terms of substitutional validity ¨uberhaupt, i.e.
not just in connection with squeezing arguments.
5
directly into the place of ‘intuitive validity’ in Kreisel’s squeezing
argument, as will be shown below.7
This is as opposed to the model-theoretic account of consequence, with
its many (in Halbach’s view) drawbacks:
. . . on a substitutional account it is obvious why logical truth im-
plies truth simpliciter and why logical consequence is truth pre-
serving. On the model-theoretic account, valid arguments pre-
serve truth in a given (set-sized) model. But it’s not clear why
it should also preserve simple (‘absolute’) truth or truth in the
elusive ‘intended model’. Truth-preservation is at the heart of
logical validity. Any analysis of logical consequence that doesn’t
capture this feature in a direct way can hardly count as an ade-
quate analysis.8
Under a substitutional account, the connection with set theory is severed, or
such is the claim; and interpretations of logical formulae are now syntactic
objects:
On the model-theoretic account, interpretations are specific sets;
on the substitutional account they are merely syntactic and (un-
der certain natural assumptions) computable functions replacing
expressions.9
We do not address the question here whether Halbach’s is a reasonable
conceptual analysis of the intuitive notion of logical consequence, the notion
of consequence in itself. What seems clear to us is that the informal concept
of consequence at work in natural mathematical languages is often plainly se-
mantic, and moreover model-theoretic. That when the mathematician draws
inferences in natural language, s/he imagines a situation in which the hy-
pothesis is true—i.e. one has a model for the hypothesis in view—then s/he
argues that the conclusion must hold in that model.
Kreisel states the point rather colorfully—“they don’t have the rules in
their heads at all!”—but what he means is that, e.g. group theorists do
7[13], ibid
8[13], ibid
9[13], ibid
6
not derive theorems directly from the group axioms in practice, rather they
employ the semantic method, i.e. they imagine a group and then show that
the group has the property claimed for it. Analysts do not derive formal
theorems from the axioms of real numbers, the real numbers are taken as
astructure which satisfies the axioms, and then theorems are proved about
that structure. In fact this is what mathematicians are trained to do, as
a cursory look at most standard texts demonstrates. For example, Walter
Rudin begins his classic text Principles of Mathematical Analysis [27] with
the existence of the field of reals:
Theorem 1.19. There exists an ordered field Rwhich has the
least-upper-bound property.
The rest of the book is the investigation of this field R. The proof of the ex-
istence of R, i.e. the construction of the reals from the rationals, is relegated
to an appendix.
For another example, Halsey Royden introduces the real numbers in his
Real Analysis [26] with the following:
We thus as su me a s gi ven t he set Rof real numbers, the set Pof
positive real numbers and the functions “+” and “·”onRRto
Rand assume that these satisfy the following axioms, which we
list in three groups.
As for the reasons behind the mathematicians’ semantic mode of thought,
this is in some sense the moral of G¨odel’s speed-up theorem:
Thus, passing to the logic of the next higher order has the eect,
not only of making provable certain propositions that were not
provable before, but also of making it possible to shorten, by
an extraordinary amount, infinitely many of the proofs already
available.10
This can be interpreted as saying that the semantic method, the method
of establishing logical consequence by considering models, enjoys a so-called
“speed-up” over the method of formal proofs. This may explain why the
model-theoretic notion of logical consequence seemed natural to Tarski and
others.
10[12], p. 397
7
The above objections to [19] are certainly appropriate. Kreisel’s notion
of intuitive validity is clearly overly theorised, in the sense that the intuitive
notion considered is not suciently intuitive or pre-theoretic, per Smith; and
under theorised, per Field and Halbach respectively. At the same time though
one might ask, if the intelligibility of Kreisel’s squeezing argument depends
on replacing the pre-theoretic notion of intuitive consequence by some formal,
syntactic counterpart notion (as in [9] or [13]), what is the point of squeezing
arguments at all? Why not simply analyse formal consequence directly, as
logicians have always done?
For it se em s to us tha t the inte re st o f sq ueezi ng a rgume nts l ie s in thei r
being carried through in such a way as to fulfill what was originally claimed
for them in [19], namely to capture an informal, natural language mathemat-
ical notion by “squeezing” it between two formal ones. Of course it can be
argued whether Kreisel himself succeeded in this. The point is that if the aim
was to capture (or “squeeze”) informal notions used in practice—to provide
a conceptual analysis of the informal notion of validity, as it were—then as
we have argued above, Kreisel’s model-theoretic construal of intuitive conse-
quence was the correct and natural one. This is not to question the validity
of the conceptual analyses of logical consequence which have been pursued so
vigorously, especially in the period since Etchemendy’s [8], but rather to ask
whether the concepts emerging from such analyses ought to be slotted in for
the informal notion that appears in Kreisel’s analysis here, i.e. in the origi-
nal squeezing argument. We will return to the issue of “genuine informality”
below.
A final objection may concern what is surely the very unnatural restriction
to the first order case, that is to propositions of the form 1.Theissueis
addressed by Kreisel in the paper, who points to a partial result in this
direction, an analogue of the squeezing argument derivable in the case of
extensions of DFto the !-rule.11
Of interest to us here, and one of the topics of this paper, is the possible
development of squeezing arguments in the direction of infinitary and second
order logics. As it turns out somewhat more is known about completeness
theorems for extensions of first order logic than was known in 1967. Going
beyond the !-rule, which Kreisel mentions, completeness theorems have been
obtained for a number of infinitary logics, as well as for logics intermediate
between first and second order. The second order logic perspective has also
11The completeness of the !-rule is due to Orey [25].
8
been developed a great deal since the publication of Kreisel’s [19].
As a result of this logical work squeezing arguments of the kind Kreisel
seems to be asking for in [19] may now be available. It is an oddity of the
paper that while Kreisel mentions both the completeness of !-logic and infini-
tary logics, he doesn’t mention completeness theorems that would have been
available already during the writing of [19], namely the Henkin Complete-
ness Theorem (1950) [14] for second order logic with the so-called Henkin
semantics, and the completeness theorem for L(Q1), the extension of first
order logic by the quantifier “there are uncountably many” due to Vaught
[36] and published in 1964. We will return to these logics below.
3 Squeezing Arguments and the Logic in Nat-
ural Language Thesis
Before we consider the possibility of expanding squeezing arguments in the
direction of second order and other strong logics, we ask, are the relevant
natural language concepts available for this analysis at all, i.e. even in the
first order case? Keeping Glanzberg’s rejection of the LNL thesis in mind,
can one simply extract what one thinks of as the notion of informal validity
at work in the natural language mathematicians use, and devise a squeezing
argument for that notion? In other words, do squeezing arguments require
the LNL thesis?
Many researchers in the foundations of logic and mathematics may share
a tacit belief in what one might call the “logic in natural mathematical lan-
guage thesis”. At the more formalist end of the spectrum of foundationalist
views which have been pursued traditionally, one might even attribute a be-
lief in the identity of the notion of logical consequence at work in natural
mathematical languages with the notion defined by the relevant formal lan-
guage. This is just the thought that the logical consequence relation defined
by a suitable, maximally adequate formal language is the correct version of
the logical consequence relation at work in natural language mathematics—
what had been meant by the natural language concept all along. Tarski,
though no formalist, seemed to argue for this or a similar view in his conver-
sations on nominalism with Quine and Carnap at Harvard in 1940-41, when
he remarked that “the dierence between logic and mathematics” was that
9
“Mathematics = logic + ”.12
Others, with Glanzberg, might see the notion of consequence at work in
the mathematician’s natural language as exact but fundamentally dierent
from the formal notion. In that case one might ask, what separates formal
entailment from its counterpart in (mathematical) natural language? We
remarked above on Kreisel’s observation that Frege’s rules gained acceptance
among mathematicians over time. This is to say, presumably, that if the
relevant part of the discourse is formalized, then Frege’s rules would be the
mathematician’s obvious choice of logical rules. Formalization also involves
a choice of a semantics, but in contrast to the rules (DF) it is not clear that
a choice of semantics is determined by the informal practice in the second
order case, which is our interest here. The view taken in this paper is that
in the case of intuitive, informal second order validity, a choice of semantics
is entirely irrelevant to the conceptual analysis of the notion of informal
consequence, or so we will argue below. To be informal in the second order
case is to prescind from a choice of semantics.
We first consider the case of strong logics in general.
4 Squeezing Arguments with Completeness
Theorems
Kreisel’s paper is entitled “Informal Rigour and Completeness Proofs,” and
indeed an apparently implicit assumption in [19] is that any time one has
a completeness theorem in hand for a given logic, the associated squeez-
ing argument should go through. More precisely, let Lbe a logic, and let
VLdenote L-validity understood in the standard semantic sense, that is
set-theoretically. Let ValLdenote informal validity (via L), understood in
Kreisel’s sense, that is as referring to truth in all structures. Finally, let
DLdenote derivability in the formal system introduced for L.13 Then one
would expect that a completeness theorem for the logic Ltogether with the
inclusions:
DLValLVL
12Tarski is quoted in Carnap’s notebooks. See Mancosu, [24].
13For many Lit is obvious what DLshould be, but this is not always so.
10
should underwrite the extensional equivalence of the concepts L-provability,
informal L-validity and L-validity construed semantically.
Is this plausible? That is, can new squeezing arguments be obtained
from completeness theorems for strong logics? The following completeness
theorems for strong logics are known: in ZFC,completenesstheoremshave
been obtained for L(Q1), the extension of first order logic by the quantifier
“there are uncountably many”, due to Vaught [36], as was mentioned; for
L!1!, the logic which is otherwise first order, but allowing conjunctions and
disjunctions of countably many formulae, due to Karp [16]; for so-called
cofinality logic, the extension of first order logic by the quantifier denoted
Qcf
xy(x, y), meaning “defines a linear order of cofinality ”fora regular
cardinal, due to Shelah [29]; and for so-called stationary logic, the extension
of first order logic by the quantifier denoted aas(s), meaning “a club of
countable sets ssatisfies (s)”, due to Barwise, Kaufmann and Makkai [3].
The last two require the Axiom of Choice for their completeness theorems.
Going beyond ZFC there is the logic L(Q2), the extension of first order
logic by the quantifier “there are at least @2many” proved complete by C.C.
Chang [6], as pointed out by G. Fuhrken [10], using the continuum hypothesis
CH.
Another interesting case going beyond ZFC is the extension of first or-
der logic by the Magidor-Malitz quantifier [22], defined as follows: M|=
QMM,n
x1...x
n(x1,...,x
n)() 9 XM(|X|^8a1,...,a
n2XM|=
(a1,...a
n)). The completeness theorem for this logic uses the set-theoretical
principle 3, which is stronger than CH, and it is consistent that complete-
ness fails in the absence of 3([1]) .
The squeezing argument for the logic L(Q1), for example, would look
like this: Let DL(Q1)denote the concept of formal provability relative to this
logic. Let VL(Q1)denote the truth of L(Q1)-statements in all set-theoretical
structures. Finally let ValL(Q1)stand for the validity of L(Q1)-statements
relative to all possible structures. Then if the inclusions
DL(Q1)ValL(Q1)VL(Q1)
hold, the squeezing argument relative to the logic L(Q1) must also hold, by
the completeness theorem for L(Q1).
Strengthening the logic escalates one’s set-theoretic commitments, clearly.
The completeness theorem for first order logic is actually equivalent to Weak
onig’s Lemma (WKL), which is also required to prove the completeness
11
both of !-logic and of L!1!. The Axiom of Choice is used for proving com-
pleteness theorems for the cofinality and stationary logics, corresponding
to the generalised quantifiers Qcf
xy(x, y)andaas(s). The CH is used
for proving the completeness theorem for the logic L(Q2), and finally 3is
required for proving the completeness of the extension of first order logic
obtained from the Magidor-Malitz quantifier. A comprehensive study of the
exact nature of these commitments would seem to be in order, but is not our
concern here.
Countenancing such a hierarchy of commitments is acceptable in some
quarters and unacceptable in others—a matter of deciding whether the rele-
vant completeness theorems “speak for themselves,” to quote Kreisel.14 What
about the soundness claim in this advanced setting? In the first order case
we claimed, in the spirit of informal rigour, that DFwas formulated so as to
guarantee soundness—in fact Kreisel’s soundness claim 8i8(DFi!Val i)
extends to all orders, as we saw. There is no obvious reason why Kreisel’s
argument could not be extended to strong and infinitary logics. In the case
of !-logic, the order is somewhat reversed. That is, formal validity (V) is
considered with respect to !-models, in which the positive integer part is
standard. Thus in the case of !-logic the semantics is designed so as to
underwrite the soundness of the omega-rule.
The case of second order logic in this regard is also striking, in that the
Henkin semantics is formulated specifically so as to guarantee not soundness
but completeness. We will now take up the question of whether squeezing
arguments can be obtained in the second order case.
5 Squeezing for Second Order Logic
The mathematician’s informal discourse very naturally includes second order
concepts—quantifying over functions and relations and so forth—so it is rea-
sonable to ask for a squeezing argument for informal second order validity.
14Kreisel used this phrase in discussing the independence of the CH in the paper (p.
140):
The present conference showed beyond a shadow of doubt that several recent
results in logic, particularly the independence results for set theory, have left
logicians bewildered about what to do next: in other words, these results do
not ‘speak for themselves’ (to these logicians).
12
But if a logic has a completeness theorem, then if the proof system of the
logic is eective in the sense that the set of axioms and rules are recursive
and proofs are finite, then the set of valid sentences is recursively enumerable.
By [34, Theorem 1] the set of valid sentences of second order logic is actually
2-complete in the Levy hierarchy. Thus on simple grounds of complexity
no reasonable completeness theorem can exist for second order logic.
Does this mean one shouldn’t pursue a squeezing argument for second
order ?KreiselhimselftooktheviewthatForhigherorderformulaewe
do not have a convincing proof of 82(V2$Val2) though one would
expect one.” We will now argue that a squeezing argument for second order
is available, once one incorporates the concept of validity with respect to
general models.
Before addressing this point, recall that the Henkin semantics is defined
simply so that in the so-called general models, the second order variables of
a given formula are thought of as ranging over a fixed subset of the power-set
of the domain. This subset of the power set may be a proper subset but it has
to satisfy the axioms of second order logic, including the full comprehension
axioms. In case the domain of quantification is actually the full power set,
one refers to the model as “full” or “standard”, and the associated semantics
as full or standard semantics.
Taking the definition of Henkin semantics into account, a squeezing ar-
gument for informal validity of second order would be the following:
Let Ddenote the usual axiom system of second order logic, already given
in [15]. As above, let Val2mean that 2is informally true in all structures,
including class-sized structures and including, in principle at least, structures
that have no set-theoretical definition. Now let V2mean that 2is true in
all set-theoretical structures. The unproblematic implications are:
D2!Val 2!V2.
Note that if the completeness theorem held for second order logic, we
could conclude straightway that Val2$V2and Val2$D2,asbefore.
Now denote by V02the statement that 2is valid with respect to set-
theoretically defined general models (satisfying—as we have assumed—the
full comprehension axioms). Consider the following implications:
D2!Val 2!V02!V2.(2)
Suppose we assume (2). Then by Henkin’s [14] proof of
V02!D2
13
Figure 1: Varieties of validity in second order logic.
together with (2) we would obtain:
Val 2$D2$V02.
The first implication of (2) is clear, modulo the soundness claim, and so
is the last, for trivial reasons. What about the middle implication Val 2!
V02?If2is informally valid in all structures, why is it that general models
should count as such structures? If the second order variables of 2are
thought of as ranging over the full power set of the domain in question, why
is it the case that these second order variables can be regarded as ranging over
the subsets in a general model? Is there a principled way to distinguish non-
standard general models from standard (full) structures, from the “informal
practice” point of view?
To analyse the situation in more detail, let us write Vc|=2if 2is
valid in both class-sized structures and set structures. Let us also write
V0
c|=2if 2is valid in both class-sized general models and set-sized general
models. Finally, let us write Val0
c2if 2is informally true in all structures,
full or general, including class-sized structures. Figure 1 depicts the trivial
implications. The possible implication Val2!V02is the problematic one.
So what distinguishes Val 2from Val02?Weclaimthatontheinfor-
mal level it is impossible to see a dierence between a standard model and
14
a general model. It is true that if we consider a general model in isolation,
from outside, so to speak, it is easy to imagine that something is missing
from the model, in order for it to count as a standard model. For example,
if we consider an infinite general model with a countable set of relations as
the range of second order variables, we know that the model is not standard.
There may be other ways of seeing the non-standardness from the outside.
We may, for example “see” that a general model of arithmetic has an ele-
ment with infinitely many predecessors. The position taken here is that it is
contrary to the idea of informal validity that one should be able to survey
the situation from outside.
One might still think15 that there really is an informal concept of a general
model, encapsulated by the thought: “All the sets I need are there and if
some are missing, they do not change anything”. This would seem to be
dierent from the informal concept of a standard model, encapsulated by
the thought: “All the sets are there and no set, whether I need it or not,
is missing”. If this is the case it is conceivable that for some 2we make
the judgement that it is informally valid only in standard models, not in
all general models. However, while it is crystal clear what the dierence is
between standard models and general models in the technical, logical sense,
it is a dierent matter to see the dierence on the informal level.
We go further and claim that on the informal level, the di erenc e is
not discernable. The reason for this is (essentially) that the general models
“know” all the definable sets and relations (by the Comprehension Axioms)
and they are the ones we refer to in mathematical practice.16
A similar line is articulated in [35], in which the second author has argued
that from the point of view of mathematical practice, when we actually use
second order logic we do not and in fact cannot see a dierence between
ordinary (“full”, or “standard”) models and general models.
I will argue in this paper that if second-order logic is used in
formalizing or axiomatizing mathematics, the choice of semantics
is irrelevant: it cannot meaningfully be asked whether one should
use Henkin semantics or full semantics. This question arises only
if we formalize second-order logic after we have formalized basic
15We are indebted to A. Blass for suggesting this line of thought.
16Note that the definable sets taken on their own are not sucient as they do not satisfy
the Comprehension Axiom. One needs a little “blurring” around the edges, otherwise one
can diagonalise out of the class.
15
mathematical concepts needed for semantics. A choice between
the Henkin second-order logic and the full second-order logic as a
primary formalization of mathematics cannot be made;theyboth
come out the same.17
For example, let us consider Bolzano’s Theorem:
Theorem 1. (Bolzano) Every continuous real function on [0,1] which has
a negative value at 0and a positive value at 1assumes the value 0at some
point of (0,1).
For the proof, by the second order comprehension axiom one can instan-
tiate a universal second order quantifier at X={x|f(x)<0}.ThesetXis
even first order definable, with fas parameter. This is a paradigm example:
we operate on sets definable from existing sets. Of course, principles such
as the Axiom of Choice force us to introduce also non-definable sets, but
they do not exist because “all” sets exists but because we assume—and the
general models are assumed to satisfy—the Axiom of Choice.
We now tur n to t he i ssue of s et v s class -s ized mo del s . Con si de r the weaker
claim that
V2!Vc2(3)
that is, the claim that second order formulas valid relative to set-theoretical
structures are also valid relative to class-sized structures. In other words,
we ask, is it true that if a second order sentence has a class-sized model, it
also has a set-sized model? This cannot be proved from the axioms of von
Neumann-G¨odel-Bernays class theory (NGB), as the following “Zermelian”
argument shows: Let 2be the second order sentence which says that the
universe of the model is an inaccessible cardinal. Let be the least inaccessi-
ble and let Mdenote the cumulative hierarchy up to .ThenhM,P(M)i
is a model of NGB satisfying 2. But no set-sized model, in the sense of
hM,P(M)i,satises2.
Paul Bernays [5] formulated more or less exactly (3), albeit in dual form,
as a reflection principle, and observed that it implies the existence of inac-
cessible cardinals.18 In fact, (3) implies a parameter-free version of so-called
Levy’s Schema [20], which says that every definable closed unbounded class
Cof ordinals contains a regular cardinal. In the original Levy’s Schema the
17[35], p. 505. Emphasis ours.
18We are indebted to A. Blass for pointing this out.
16
definition of Cis allowed to have parameters. Since the class of all cardi-
nals is definable without parameters, we obtain from (3) a proper class of
inaccessible cardinals. Bernays goes on to formulate (3) with second order
parameters and arrives at what became later to be known as indescribable
cardinals.19 L. Tharp [33] showed that the parametrized principle implies
that for every n,theclassof
1
n-indescribable cardinals is a proper class.
This gave immediately a proper class of e.g. weakly compact cardinals. For
an analysis and discussion of the situation we refer to Tait [32, Ch.6].
Thus we cannot expect a proof of V2!Vc2,atleastwithoutad-
ditional axioms. On the other hand, the assumption (3) formulated in a
reasonable class theory (such as NGB) seems plausible. By a result of Scott
[28], it is true in the above hM,P(M)i, assuming that is not only weakly
compact, but even measurable. In fact, it suces to assume that is 1
<!-
indescribable, hence (3) is consistent with V=L, assuming the consistency
of a 1
<!-indescribable cardinal.
What about (3) for sentences in other extensions of first order logic than
second order logic? For first order logic this is an immediate consequence of
the Levy Reflection Principle. For extended logics of the form L(Q)wecan
use translation to first order set theory and get the analogue of (3) as for first
order logic. The same is true for L(QMM,n
), L(Qcf
), and the extension of first
order logic by the H¨artig-quantifier Ixy(x) (y),meaning: the cardinality
of the set of elements xsatisfying (x) is the same as the cardinality of
the set of elements ysatisfying (y). For these powerful logics, unlike for
second order logic, the analogue of the small part of the squeezing argument
represented by (3) is simply provable in ZFC. The situation with stationary
logic is more complicated. We leave the status of (3) open, if 2is taken to
be a formula of so-called stationary logic rather than second order logic. The
Open Question is, whether it is provable in ZFC or not.
Attempts to formulate higher order reflection with higher order parame-
ters leading to larger large cardinals than (!) have failed (see [18]). However,
a dierent approach, due to P. Welch, to a very strong reflection principle
with second order parameters, called the Global Reflection Principle,givesa
proper class of Woodin cardinals [38, 37].
19In our model theoretic context second order parameters would correspond to adding
generalised quantifiers to second order logic.
17
6 owenheim-Skolem Theorems
Kreisel asks for a convincing proof of 82(V2$Val 2), on its face im-
possible as we saw. Short of such a proof, Kreisel then asks a more specific
question, which can be answered. Stating the L¨owenheim-Skolem Theorem
for first order logic in the form 88>!(V!+11$V1), what is the ana-
logue to !for second order formulae?20
First we recall some definitions. Given a logic L,wesaythatLhas
owenheim-Skolem number if is the least cardinal such that for all vo-
cabularies such that the cardinality of is ,ifasentencein the
vocabulary of the logic has a model M,thenithasamodelNof size
.IncaseNcan be taken to be a submodel of Mthen is called the
owenheim-Skolem-Tarski (LS T )numberofthelogic.
Let L2denote second order logic. We can now state Magidor’s result [21],
which answers Kreisel’s question: is the the least supercompact cardinal if
and only if =LST (L2).
In fact there is now a whole range of logics calibrated by large cardi-
nals, in the sense that the assumption of the cardinal is equivalent to or
implies a L¨owenheim-Skolem-Tarski theorem for the logic. For the cases al-
ready mentioned the results are as follows: for cofinality logic, corresponding
to the generalised quantifier Qcf
xy(x, y), the LST number is @1.21 For
stationary logic, corresponding to the quantifier aas(s), the LST number
is consistently @1, assuming the consistency of a supercompact cardinal,
22
but the LST number of stationary logic can also be the first supercompact
cardinal.23
Finally, the interesting case of the H¨artig quantifier: It is now known
that if the LST number LS T (I) of this logic exists, then there is a weakly
inaccessible cardinal and LST (I)isatleasttheleastweaklyinaccessiblecar-
dinal. It is consistent relative to the consistency of a supercompact cardinal
that LST (I) is the first weakly inaccessible, and also consistent that it is the
first supercompact.24
A general approach to strong logics and reflection principles they give rise
20In Kreisel’s notation V1denotes the assertion “1is true is the cumulative hierarchy
up to ”.
21Shelah, [29].
22See Ben-David, [4].
23Magidor, unpublished.
24[23].
18
to is in J. Bagaria et al. [2], where a close connection is established between
LST numbers of strong logics and so-called structural reflection principles in
set theory.
Just as in the completeness theorems, and the ensuing squeezing argu-
ments, obtaining L¨owenheim-Skolem type theorems may require principles
that go beyond Weak K¨onig’s Lemma (WKL), sucient in the case of first
order logic.
7 Squeezing very simple concepts
Consider the concept Wof finite words in a given vocabulary X.Intuitively
we construct a word by placing letters from Xone after another a finite
number of times. What does this mean? We can use a squeezing argument
to shed light on this question. As an analogue of derivability consider the
concept Dof starting from the empty word and then adding one letter from
Xat a time to the end of any word we already have. As an analogue of set
theoretic validity we take the concept Cof being a member of every closed
set, where a set Ais called closed if the empty word is in A,allone-letter
words are in A, and the concatenation ww0of any two words w, w0of Aare
in A.Clearly,
DWC.
The first “” is intuitively obvious because adding one letter to the end of a
word certainly yields another word. The second is less obvious but one can
run an informal induction on the length of the word to see that if Ais closed,
then the word is in A. It is a mathematical fact that
CD,
because Dis one of the sets that Cis the intersection of. Hence
D=W=C,
and the informal concept of a finite word is squeezed between two (exten-
sionally) identical exact concepts. Although everything in this squeezing ar-
gument in on a very elementary level, it is noteworthy that strictly speaking
the inclusion CDis based on an impredicative argument.
19
Similarly, we may consider the concept Fof a finite set. Intuitively we call
asetfinite if we can use some natural number to list the elements of the set.
On the other hand, natural numbers can be identified with finite ordinals.
Thus there is a certain amount of circularity in the concept of finiteness. So
what does “finite” exactly mean? Let us take as Dthe concept of starting
from the empty set and then adding one element at a time to get more sets.
Let Cbe the concept of belonging to every ideal class i.e. to every class
which contains the empty set, all singletons and is closed under unions of
any two elements of the class. Clearly,
DWC.
The first “” is again intuitively obvious because adding one element to
a finite set certainly preserves the set finite. For the second one can use
informal induction on the finite size of the set to see that if Ais an ideal
class, then the set is in A. It is a mathematical fact that
CD,
because Dis one of the classes that Cis the intersection of. Hence
D=W=C,
and the informal concept of a finite set is squeezed between two (extension-
ally) identical exact concepts of class theory.
8 Conclusion
Do squeezing arguments capture the mathematician’s informal discourse,
even as it strays beyond first order talk, quantifying over relations and func-
tions, and making implicit use of infinitary rules? This is dicult enough to
argue for in the first order case. Nevertheless, we hope to have reinforced
Kreisel’s original argument in [19] that squeezing arguments have a general
role in the conceptual analysis of informal mathematical concepts. Moreover,
we have pointed out and given evidence to the claim that the circumstance
that the two sides of the squeeze (extensionally) agree is based in general on
a non-trivial mathematical fact.
In particular, we hope to have shown for strong logics that if we refashion
the relevant informal concepts appropriately (here validity), we can, so to say,
20
filtrate the informal discourse involving those concepts through a hierarchy
of set-theoretic commitments ranging from Weak K¨onig’s Lemma (WKL) up
to 3.
We also s aw th at vari ous str ategi es p resent t he ms el ves i n th e secon d or der
case, that go beyond what Kreisel suggests, if intuitive second order validity
is understood in the right way.
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Preprint
Full-text available
This paper provides a critical overview of Georg Kreisel's method of informal rigour, most famously presented in his 1967 paper `Informal rigour and completeness proofs'. After first considering Kreisel's own characterization in historical context, we then present two schemas under which we claim his various examples of informal rigour can be subsumed. We then present detailed reconstructions of his three original examples: his squeezing argument in favor of the adequacy of the model theoretic analysis of logical validity, his argument for the determinacy of the Continuum Hypothesis, and his refutation of Markov's principle in intuitionistic analysis. We conclude by offering a comparison of Kreisel's understanding of informal rigour with Carnap's method of explication. In an appendix, we also offer briefer reconstructions of Kreisel's attempts to apply informal rigour to the discovery of set theoretic axioms, the distinction between standard and nonstandard models of arithmetic, and the concepts of finitist proof, predicative definability, and intuitionistic validity.
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