The paper on the general interpretation of first-order quantifiers has now been published online by the Review of Symbolic Logic (this direct link requires subscriber/institutional access, or email me for a PDF). The philosophical companion to that paper — “Life on the Range” — is not out yet, but a pre-print can be found here.
Alfred Tarski’s great technical innovation in his Wahrheitsbegriff (1933/36) was the introduction of the auxiliary notion of satisfaction of a first-order formula by a function assigning objects from some model ‘s domain to the variables of the language. Once the notion of satisfaction is in place, one can then define truth of a sentence in a model , by saying that is true just in case it is satisfied by some (equivalently: every) assignment. It is this latter notion of truth that is of real mathematical and philosophical interest, of course, but it cannot be defined directly by recursion because in a first-order language (contrary to the propositional case), sentences are not built up from other sentences by means of constructors such as connectives. Rather, it is formulas in general that are so built, with sentences providing a special case. The characteristic clause of satisfaction is the one for quantified formulas:
- A formula is satisfied by an assignment in if and only if the formula is satisfied, in , by every assignment that that differs from at most in the value .
The detour through satisfaction, while technically interesting, it is just that — a detour. It can also sometimes get pedagogically problematic, as students usually grasp the importance of truth while not always appreciating the necessity of satisfaction.
There is an alternative approach, which does away with the notion of satisfaction, but uses a variant notion of a model. According to this approach, we only allow models that are “rich” in that every object is the denotation of a constant in the language. The truth clause for the universal quantifier is then as follows:
- A sentence is true in if and only if is true in for each constant .
(I believe this definition is fairly standard in some textbooks, but I can’t track down a definition now, any references appreciated). While this definition sidesteps the notion of satisfaction, there are some drawbacks:
- The definition only applies to “rich” models, while models in which some elements are not denoted by constants are also very natural. Moreover, we are forced to consider truth of sentence in a model having a different signature than itself.
- The language changes with the model: for each model , the definition applies only to the expanded language introducing new constants for each member of .
- The definition forces us to consider uncountable languages, in order to assess truth in uncountable models.
- The expanded signature affects the available automorphism of a model. For example, the model has infinitely many automorphisms in the signature with the relation only, but only one if we add the constant .
So here is a way of combining these two approaches in such a way that:
- There is no detour through satisfaction.
- The language is countable and independent of the model in which a sentence is evaluated for truth (although the language will be expanded, just not in a way determined by the model).
Fix a basic language , and let be a countable set of new individual constants. For each let (so that ). We define the notion of truth by induction on -sentences, simultaneously for every . The quantifier case:
- An -sentence is true in the -structure if and only if the -sentence is true in every -expansion of (i.e., if is true in every -structure that differs from only in that it assigns a denotation to ).
So, when is an -sentence, the definition gives us a direct account of the truth or falsity of is a model having the same signature.
On July 24 the system-wide senate of the University of California formally adopted an Open Access Policy (PDF). Under the policy, each faculty member across the ten campuses
grants the University of California a nonexclusive, irrevocable, worldwide license to exercise any and all rights under copyright relating to each of his or her scholarly articles, in any medium, and to authorize others to do the same, for the purpose of making their articles widely and freely available in an open access repository.
Under the policy, faculty will upload a “final version” of their scholarly work to eScholarship, a public repository hosted by the California Digital Library. “Final version” means the post-peer-review version of the manuscript before the publisher’s typesetting and finalizing (although some publishers allow the typeset version to be posted). Faculty have the option to opt out of the policy altogether (for each instance) or delay public access to their work. The University provides an “Addendum” that faculty can use when returning copyright transfer to the publishers in order to alert publishers that the work is subject to a pre-existing license (the license applies by default, even if the Addendum is not returned to the publisher, unless the faculty member explicitly opts out of the policy).
The policy will go into effect on Nov. 1 for UCLA, UCI and UCSF, and a year later at all remaining campuses. More information can be found on the Open Access website and in the Frequently Asked Questions.
Needless to say, this is good all-around.
There is an old paradox, originally due to Montague (JSL 20 (2), 1955, p.140) that only requires the most minimalist set theoretic apparatus, viz., the existence of singletons. It does not seem to be very well known, so here it is.
Let . Intuitively, is the collection of all sets that only belong to well-founded sets. Clearly, is too big to be a set, but that is not the point. The point is that this can be shown in classical predicate logic using only existence of singletons:
We proceed by cases, and derive a contradiction in each.
- ; then only belongs to well-founded sets. Since exists and , the latter must be well-founded, so . But is equivalent to , so , which is impossible if (for then itself is in the intersection ).
- . Then there must be a such that: and . In particular , so there is a which belongs to both and . Now, since :
and since , we have contradicting the choice of .
A characteristic feature of natural languages such as English is that quantifiers can be plugged directly into a predicate’s argument places, thereby dispensing with the whole variable-binding machinery. The example we give students in introductory logic class is the statement of universal love:
Everybody loves everybody
or, as one California motorist efficaciously put it:
Of course, the price to pay for the economy of expression of natural language is potential ambiguity: we do not know, from the license plate above, whether the first quantifier has wide scope over the second or the other way around. The ambiguity is crucial, for instance, when different quantifiers occupy the two argument places:
Everybody loves somebody
is highly ambiguous (especially out of context), another lesson we teach our students. But in the case above, we think, it does not matter, because, as is well known:
Like quantifiers commute.
Or do they? This is another one of those universally known “facts” that are in fact only justified on the basis of a one-sided diet of examples. As pointed out in a previous post, there are non-standard interpretations of and on which the quantifiers do not commute, but we need not go to such lengths to appreciate the point.
Suppose humans are arranged in a countably infinite list where loves if and only if . Let be the quantifier “for co-finitely many,” then is true (there are co-finitely many ‘s that love co-finitely many ‘s), but is false (there aren’t co-finitely many ‘s that are loved by co-finitely many ‘s). So, like quantifiers do not always commute. (This is an adaptation of an example I learned from Thomas Forster.)
I finally have a complete draft of “On the general interpretation of first-order quantifiers,” which can be obtained here. From the abstract:
While second-order quantifiers have long been known to admit non-standard, or “general” interpretations, when properly viewed as predicates of predicates first-order quantifiers also turn out to allow a kind of interpretation that does not presuppose the full power-set of that interpretation’s first-order domain. This paper introduces such “general” interpretations for first-order quantifiers, exploring some of the consequences of the definition especially as regards the unary case, emphasizing the effects of imposing various further constraints that the interpretation is to satisfy.
Comments and other feedback always welcome.
In ” Metaphysics After Carnap: the Ghost Who Walks?,” Huw Price asks us to imagine a philosopher who, in the 1950′s, falls asleep at the wheel while stuck in a traffic jam of the New Jersey turnpike, only to wake up sixty years later and find, much her astonishment, that metaphysics has become respectable again. Back in the days, after all, metaphysics (just like poverty) was supposed to be on its last legs. The fact that both have not only survived but prospered does not make the one any more palatable than the other. The following quote from Price is one for the ages:
What’s haunting the halls of all those college towns – capturing the minds of new generations of the best and brightest – is actually the ghost of a long-discredited discipline. Metaphysics is actually as dead as Carnap left it, but – blinded, in part, by these misinterpretations of Quine – contemporary philosophy has lost the ability to see it for what it is, to distinguish it from live and substantial intellectual pursuits.