# In predicate logic, do we necessarily have to restrict the domain of discourse?

Can we combine formalized statements in predicate logic from widely different domains of discourse (say, one regarding integers, one regarding the fruits in one's garden) by using logical rules/theorems and still predicate logic stays truth-preserving?

Sure, if we use identical free logical variables referring to different “things” (in the case of objects) or having a different sense (in the case of predicates), we may easily get to falsehoods, yet strictly speaking we didn't get them from truths.

Can it be assumed that free variables “carry along” their semantic content faithfully, freed from their original context? And that then the correct combination by predicate logic never gets us from truth to falsehood?

Of course, the resulting formalized statements, then “back-translated” to natural language may sound like written by Eugène Ionesco. For example, “This apple is prime or it is not prime” – yet isn't such a sentence, in the strict sense, true?

So

• is the view that the domain of discourse for predicate logic can be extended to anything that (in a pragmatic way) “exists”, respectable?
• does it have a name?
• which philosophers of logic held it? Who rejected it?
• In general, variables are not restricted. We can do it explicitly using suitable predicate letters, like e.g. ∀x (Natural(x) → x≥0) (usually abbreviated as: ∀x∈N (x≥0). Otherwise, the domain is part of the interpretation of a formula: with different interpretations, the meaning (and truth value) of a formula will change. Oct 5 '17 at 11:15
• The view that the domain of discourse for predicate logic must encompass everything was formulated by Frege and Russell. Today, the "standard" view is due mainly to Tarski, with a precursor in the Algebra of Logic tradition : Boole and Schroder. Oct 5 '17 at 12:45
– user935
Oct 5 '17 at 14:36
• No philosophers reject predicate logic afaik, but many would deny its ability to fully describe the world. As Bradley notes, in metaphysics predication seem necessary yet is illegitimate (since it forces on us a distinction between predicate and predicated that leads to false representations).
– user20253
Oct 5 '17 at 14:53
• @PeterJ I think the question is less about predicate logic generally and more about the question of restrictivism regarding quantification. Oct 5 '17 at 16:08

There seem to be two parts to your question: one concerning whether all quantification is restricted, and another concerning whether free variables can carry contextual information that serves to restrict their admissible valuations. I'll address each in turn.

## Unrestricted Quantification

It seems like your question is regarding the possibility of completely unrestricted quantification. The mainstream view is that such unrestricted quantification over everything is possible. In practice, when doing model theory we often construct models with determinate (restricted) domains to show model existence and consistency of a theory by way of its satisfiability.

This model-theoretic convention, though, doesn't answer the question of whether all quantification must be restricted. The doubter of unrestricted quantification holds a view termed restrictivism. Typically, restrictivists derive motivation for their view from various set-theoretic and property-theoretic paradoxes. They claim that unrestricted quantification over, e.g., sets is impossible because the domain of quantification would be the set of all sets, which leads to paradox in standard set theories. Given the standard move of allowing proper classes to serve as a domain, or opting for a plural logic interpretation of the domain as the "plurality of all sets", this seems like a case of taking the semantics too seriously. That domains are sets is an artifact of model theory but not a necessary postulate.

More sophisticated restrictivists realize that with suitable large cardinal axioms, what was a proper class without such large cardinals becomes a set. They typically complain, then, that the notion of set is indefinitely extensible (an objection popularized by Michael Dummett, who used the notion to push for intuitionistic logic/mathematics). The idea is that anytime you think you have "all" of the sets, you could always admit more by accepting some larger cardinal and extending the height of the cumulative hierarchy. They argue that since there's no non-arbitrary stopping point, and this process can continue on indefinitely, we are never in a position to quantify over a determinate domain of all sets. Such a view, however, seems to tie our abilities to quantify over a domain to our ability to describe that domain uniquely (at least up to isomorphism) -- something the Platonist will surely deny.

The more common response to the paradoxes (though the restrictivist will complain such a response is objectionably ad hoc) is to say that they only show us that some predicates (like "set of all sets") do not have sets as extensions -- not that unrestricted quantification over sets is impossible. (This is more or less the idea behind Zermelo's approach in formulating the axiom of separation. Such predicates are rules out since you must always "separate" a set from a set -- ensuring the extensions of any predicates used in instances of the axioms are sets.)

Additionally, restrictivism is often claimed to suffer from a paradox of self-defeat akin to the flaw afflicting the verificationist theory of meaning. There seems to be no way to state the view without rendering it false. Consider:

(R) Every quantified statement is restricted.

(Typically they'd claim this holds of necessity, but I've eliminated any modal notions for simplicity and because the additional complications don't help the restrictivist.) Now there is a dilemma:

1. The use of "every" in (R) is restricted, and so (R) does not capture the restrictivist's view -- it's compatible with instances of unrestricted quantification that fall outside the scope of the restricted "every".
2. The use of "every" in (R) is unrestricted, and so (R) is false since it provides its own counterexample.

How to coherently state restrictivism is a subject of controversy, with many dismissing the view as incoherent. The primary source for this topic is Rayo and Uzquiano's (eds.) Absolute Generality.

## Contextually Restricted Free Variables

The second question is largely separate from the first. There are a few uses of such contextually restricted free variables, off the top of my head these divide into (at least) two major categories: Dynamic Semantics and (some) presentations of Quantified Modal Logic. I'll start with dynamic semantics.

## Dynamic Semantics

In dynamic semantics the central idea is that context serves to restrict the admissible variable assignments. It introduces quantification over variable assignment functions. Atomic formulae are of the form P(x) and are interpreted as the set of all assignments which assign to x an object that is P. Existential quantification acts to "reset" the set of admissible assignments to be those that differ from the original set of assignments by a renaming of the relevant variable/constant according to the (usually newly, otherwise the renaming is the trivial identity) introduced variable bound by the existential quantifier. This is the approach of Dynamic Predicate Logic, which takes these "actions" on assignments to provide the meaning of quantified statements (hence "dynamic").

A different formulation of dynamic semantics is given by Discourse Representation Theory, tracing back to work of Lauri Kartunnen from the 60's (1968?) but finding more complete development in the early 80's in the work of Irene Heim and the work of Hans Kamp (Kamp's work and its descendants are what is usually meant in modern references to "discourse representation theory"). The main innovation of DRT was to introduce the notion of "discourse referents" to treat phenomena like anaphoric reference. Anaphora occurs when an expression (typically a pronoun) depends for its reference on some previously introduced individual -- a referent derived from the preceding discourse. Consider:

A man walked into the bar. He asked for a drink.

The idea is that the phrase "a man" introduced some particular but underspecified individual into the discourse and that "he" refers to this individual. This is treated via "discourse representation structures" (DRSs).

The first sentence is given the following DRS:

[x: Man(x), Walked-into-bar(x)]

Here, the variable "x" denotes a discourse referent introduced by "a man". Note that this is a particularly individual and so not simply what would be conveyed by an existential statement.

The second sentence has the following DRS:

[x: Ordered-a-drink(x)]

(With the understanding that the gendered pronoun supplies additional information marking x as male. I've put x in bold to indicated that it's unresolved and must be identified with a previously introduced discourse referent.)

These two DRSs are joined via the operation of "merge" to produce the DRS for the whole discourse (the two sentences):

[x, x: Man(x), Walked-into-bar(x), Ordered-a-drink(x), x = x]

The discourse referent "x" is often glossed as a free variable whose value is supplied by context (i.e., context restricts the admissible assignments).

## Quantified Modal Logic

Finally, some presentation of quantified modal logic -- notably Fitting and Mendelsohn's First-Order Modal Logic -- use contextually restricted free variables, under the name of parameters, to deal with names in modal discourse. A parameter differs from an ordinary free variable in that it is never allowed to be bound by a quantifier in the course of a proof -- its value is always supplied via contextual restriction on admissible assignments.

This is similar to the use of "dummy constants" in natural deduction systems for predicate logic. When you move from an existential claiming that "Something is P" to the claim that P(a), you are required to use a "fresh constant" -- a constant not previously introduced in your proof (without having been already discharged). The value of a here isn't really constant at all, it denotes some arbitrary thing that is P and is indifferent as to which P you choose. P then serves to restrict the assignments to a and in some systems free variables with such restrictions are used in place of dummy constants.

## Summary

So, to conclude, it is possible to assume that free variables "carry contextual information", even across a change in context, but the details of how this works will depend on the particular framework. Certainly it is a common phenomenon, even if not usually explicitly treated in introductory discussions of predicate logic.

As to unrestricted quantification, the majority view is that it is possible and coherent. The view does, however, have its detractors. It tends to be treated as almost a matter of dogma that unrestricted quantification is coherent and unproblematic, with the burden being placed on detractors to argue against it and formulate a coherent alternative.

• I am not sure I understand, the bigger problem with unrestricted domains is that there are no predicates to speak of that would apply to everything. SEP article you linked says the same:"In typical contexts, the use of the quantifier “everything” is tacitly restricted to a domain of contextually salient objects... But the fact that unrestricted quantification is relatively uncommon is no reason to doubt it is attainable in certain contexts". Even "indefinitely extensible" classes in math would not contain candy as elements. The main problem seems to be not incoherence but pointlessness. Oct 5 '17 at 18:50
• @Conifold I see, I haven't encountered that objection. Most defenders of unrestricted quantification as a possibility are happy to admit that most quantificational utterances contain (at least tacit) restrictions. The "usefulness" would typically be in formulating metaphysical theses. For instance, nominalism might be stated as the claim that "Everything is concrete"/"There are no abstract objects". For that to capture the nominalist's view the quantifiers would need to be unrestricted. I'm not sure if there are non-metaphysical claims that require unrestricted quantifiers.... Oct 5 '17 at 19:26
• Maybe empiricism of a sort "All things are knowable through the senses" (roughly)? I took OP's question about whether unrestricted quantification was respectable to be more a matter of whether it's coherent or tenable, but I admit to not knowing what to make of the parenthetical suggesting a notion of "existing in a pragmatic way". Oct 5 '17 at 19:26
• @wolf-revo-cats Oh, that's a bit different. You have in mind a sort of Carnapian set-up where different metaphysical claims might exist relative to different "frameworks". Neither the nominalist nor the platonist will accept such a set-up, since the nominalist holds that no matter how you loosen up restrictions on quantification, you'll never find abstract objects in your domain. The platonist holds that we can restrict to avoid talking about abstract objects, but then we're not talking about all there is.... Oct 7 '17 at 2:30
• Some have argued for a non-ontological notion of quantification whereby you can quantify over things without committing yourself to their existence -- it's a sort of modern Meinongian position. The logic to capture that would be free logic which allows an "inner domain" and an "outer domain", where the nominalist could claim it's only the entities in the inner domain that really exist. But the key would be to keep the "existence loaded" quantifiers separate from the "free lunch" quantifiers. Oct 7 '17 at 2:35

I am going to assume extra context provided by OP’s earlier related question, Does fictional discourse pose special difficulties for logic?

Classical devices

Even the classical set theory has a simple device for restricting the range of quantifiers without changing the domain of discourse, ∃x∈S P(x) restricts the domain to S. If the use of set theory is objectionable it can be simulated within the predicate logic itself, let x∈ S be denoted S(x), then the above formula is equivalent to ∃x (S(x) ∧ P(x)), which blocks attempts to infer existence outside of S. Similarly, ∀x∈S P(x) is simulated by ∀x (S(x) → P(x)). If x is not in S the antecedent of the implication is false, so it is trivially true, meaning that whatever happens ouside of S has no effect on the truth of the sentence. Thus, we can have a “unified” universe of discourse but de facto restrict it to subsets/subclasses when statements are made.

Of course, in conversational language such restrictions are often left implicit, but this is a problem not so much for logic or semantics as for linguistic pragmatics, which studies speech acts and how speakers infer implicit restrictions from the context, like prior conversation, environment, shared background, etc. In programming languages the requisite background is made explicit by splitting variables into data types, and modularization, restricting the use of a variable to within the procedure where it is defined, and its subprocedures, as in object-oriented programming.

Fictionalization

Sometimes a more radical change of domain is called for, where S is not saliently described as either a set or a class, for example with fictional discourses. The role of x∈ S is then taken over by modal operators of fictionalization introduced by Lewis in Truth in Fiction which apply to sentences rather than to variables, so the Platonist make believe that fictional objects populate some kinds of sets is not needed. But the mechanism of blocking an unwelcome mixing of discourses is the same in spirit:

But if we prefix the operator "In the fiction f" to some of the original premisses and not to others, or if we take some but not all of the premisses as tacitly prefixed, then in general neither the original conclusion nor the prefixed conclusion "In the fiction f” will follow. The premiss that Holmes lived at 221B Baker Street was true only if taken as prefixed. The premiss that the only building at 221B Baker Street was a bank, on the other hand, was true only if taken as unprefixed, for in the stories there was no bank there but rather a rooming house.

Taking the premisses as we naturally would in the ways that make them true, nothing follows: neither the unprefixed conclusion that Holmes lived in a bank nor the prefixed conclusion that in the stories he lived in a bank. Taking both premisses as unprefixed, the unprefixed conclusion follows but the first premiss is false. Taking both premisses as prefixed, the prefixed conclusion follows but the second premiss is false”.

So this apple is not prime, but according to Ionesco it is prime, no contradiction. Rosen adapted Lewis’s framework to modal semantics of possible worlds for those unwilling to accept the explosion in ontology it requires. All possible worlds and objects in them are reduced to fictions, see his Modal Fiction.

Meinongian universe

Finally, there is an option “opposite” to fictionalism, which adopts the most prolific ontology imaginable, that of Meinong, with golden mountains and round squares admitted on a par with desks and chairs. A coherent formal version of it was worked out by Parsons in A Prolegomenon to Meinongian Semantics. In a way, the mechanics is not that different from fictionalism or Russell’s paraphrase:

Individuals are not sets of properties. However, corresponding to each individual is a unique set of properties-the set of properties that the individual has. I call this set the "correlate" of the individual… Likewise, Meinongian objects are not sets of properties either, but I'll talk as if they were. More literally, sets of properties (i.e., subsets of P) will "represent" objects in this theory, much as certain sets of sets represent numbers for logicists.

[…] There are incomplete objects. For example, the gold mountain is not determinate with respect to any properties except for its goldness and its mountainhood… There are supposed to be objects that don't exist. These will simply be sets of properties that aren't the correlates of any individuals… There are impossible objects, for example, the round square. This object may be represented by {roundness, squareness}. The round square happens to be incomplete, but there are complete impossible objects too.

[…] I intend 'there are' in the broadest sense, the sense of 'some'. In particular, my quantifiers are to range over all objects… It is clear from M4 that Meinong would not regard such a quantifier as existentially loaded.

This is perhaps the closest option to the unrestricted uberdomain asked for in the OP. When restriction of context is needed one can simply use the classical restriction device described above, there are no qualms to be had in Meinongianism over what “sets” may contain. Pace Kant, there is also an existence predicate in this formalization, and blocking incoherence with impossible objects is simple: their properties may be inconsistent, but the existence predicate is false on them. "There are" and "there exist" come apart here.

Considering that Meinongian “objects” are essentially identified with collections of properties, and to us those pretty much manifest as indefinite descriptions, i.e. conjunctions of descriptive sentences, we get something very much like fictionalism in disguise. So primality is not among this apple's properties, but Ionesco-primality is. There are also apples that combine primality and non-primality, they just happen not to exist.

• +1 thanks for linking to OP's other question, I've been meaning to get to that one. Taught half of a class on fictional discourse, should have some good material for OP once I get my computer sorted and find those lecture notes. Oct 5 '17 at 21:00