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The notion of domain of discourse (also: domain of discourse, universe of discourse, universal set, or universe) is a fixture of mathematical logic which is sometimes claimed to be necessary to the logic of quantified predicate logic expressions. For example, the Stanford Encyclopedia of Philosophy says:

The interpretation of the language of pure quantificational logic requires one to specify both a domain for the variables to range over and an extension for each non-logical predicate of the language. — Stanford Encyclopedia of Philosophy

Reasoning in a natural language, we don't necessarily need to specify any domain. Very often, we just make blunt assertions. For example:

God exists.

Unicorns do not exist.

We don't bother to specify that things exist or don't exist in some particular domain. We could well say:

God does no exist in the real world.

But, in effect, usually, we don't, because we just say that God does not exist.

In the following expression, why would we need to specify any domain?

¬∀y(∃x(y ∈ ℕ ∧ x ∈ ℕ ∧ y = x²)

So why in mathematical logic (predicate logic), and presumably therefore in mathematics, would it be necessary to specify a domain?

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  • It is the way to define the formal concept of "interpretation" of a FOL language, i.e. the way to give meaning to formulas. – Mauro ALLEGRANZA Oct 22 '20 at 8:55
  • The usual "domain of discourse" of natural language is the "real world" of our experience. But this is not so simple... What about an historical book dealing with Napoleon ? and a fiction book dealing with Peter Pan ? – Mauro ALLEGRANZA Oct 22 '20 at 8:57
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    Because predicates are interpreted as functions that need to be well-defined on all inputs for the calculus to be sound with the standard model theory semantics. The mismatch with natural language is well-known, and is one of the arguments for alternatives to the predicate calculus that allow plural expressions with vague and varying domains, see e.g. Ben-Yami, Logic and Natural Language, ch.6:"the way quantification functions in the calculus shows that its semantics is fundamentally different from that of natural language". – Conifold Oct 22 '20 at 9:08
  • In your example, you have restricted the domain of quantification to natural numbers. For Socrates as value of x, the clause Socrates ∈ ℕ is false. But predicate logic is general; we want to express the validity (i.e. universal truth) of formula like ∀x(x=x) and ∀xPx → Pa. – Mauro ALLEGRANZA Oct 22 '20 at 9:20
  • You're missing the point that "requires one to specify" is in the context of "interpretation" i.e. providing the "standard semantics" via model theory... which defines what it means for a formula to be true in a model The FOL formulas themselves don't have such a restriction. So if you take "God exists" simply as "∃g" this means nothing basically other than assuming a non-empty universal set. – Fizz Mar 5 at 10:41
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The domain is just a way of making clear the entire set of things over which you are quantifying. In ordinary discourse by default this is simply everything that exists in the actual universe, so as you point out, it tends to go without saying. But we might wish to speak about a fictional domain. If you were to say, for example, my favourite Star Wars character is Han Solo, then it would be clear that you are quantifying over things in the Star Wars universe.

Mathematicians like to be precise about what they are quantifying over so that functions and operators have well-defined inputs. Your mathematical example shows why a domain is needed. As it stands, it does not express what you want.

¬(∀y)(∃x)(y ∈ ℕ ∧ x ∈ ℕ ∧ y = x²)

By not specifying the domain that x and y range over, this sentence is trivially true, just because there is at least one thing that is not a natural number. Also, because you have not specified any domain, the square function will fail for many values of x. What is the square of x if x is an elephant? Granted that if x is an elephant, x ∈ ℕ will be false, but we still need a value for y = x² or the conjunction cannot be evaluated. (In programming languages with lazy evaluation, this problem can be ignored, but not with predicate logic.)

Instead, you could allow the quantification to range over everything, thus:

¬(∀y)(∃x)((y ∈ ℕ ∧ x ∈ ℕ) → y = x²)         where → is material implication

But this still requires you to be able to square x for any value of x. A better option is to restrict the domain of quantification:

¬(∀y.y ∈ ℕ)(∃x.x ∈ ℕ)(y = x²)

This approach is more accurate and neater because it avoids having to put material implications everywhere.

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  • So predicate logic expressions don't require the specification of any domain? – Speakpigeon Oct 22 '20 at 17:08
  • They require a domain, but the domain by default is simply the set of everything that exists. – Bumble Oct 22 '20 at 17:26
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Short Answer

Necessary? Strictly speaking, no, but it does aid in eliminating ambiguity and other problems. The first-order predicate calculus (FOPC) is a formal system of symbols, and any set of symbols derives meaning from context. In natural language, context is often understood, but in formal systems, explicitly stating the domain of discourse helps avoid contradictions and infinite regress because self-reference can and does lead to both. In natural language, domain of discourse is often restricted in an implicit manner through what is known as the cooperative principle which lead to certain maxims argued by Grice and others, explicit statements of rules, or other methods of encouraging cooperation such as the linguistic prescriptivism. In a formal language, it avoids paradox. From WP:

In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with those of the universe itself. But more usually we confine ourselves to a less spacious field. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life, or of men under some other condition or relation. Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse. — George Boole, The Laws of Thought. 1854/2003. p. 42.3

Long Answer

The domain of discourse is important in language generally because a proper understanding of linguistics lays bare the notion that meaning is derived partially from context in multiple ways: grammatically, semantically, lexically, etc. For instance, is the word 'lead' a verb or a noun? In the sentence 'His will is being done', does 'His' refer to the Christian God? Ever hear the statement 'Depends on what the meaning of is is'? A domain of discourse helps to avoid misinterpretation.

Paul Grice was a British philosopher who recognized that in natural language, there are domains of discourse, so to speak, that are implicit in communication between people that go above and beyond deixis, and help faciliate the transfer of meaning. Personal deixis is easily understood as the implicit antecedents in the use of pronouns. For example:

He came down the chimney with care, and left gifts for the little boys and girls.

Do you know who 'he' is in this sentence, and if so how? I suspect most children born in the Western world would guess Santa Claus if asked. But how do they know? Context. It could be that he is an altruistic burglar unless specified explicitly because pronouns function much like variables in that they have reference, a distinction from meaning (sense) proper recognized by Gottlob Frege.

In fact, Grice's maxims have the effect of narrowing a domain of discourse substantially, and with other psychological, psycholinguistic, and linguistic mechanisms encourage joint attention. Let's take your example:

God does no exist in the real world.

Why is the 'real world' not usually specified? Because usually, our conversations are about the real world, and we only specify briefly otherwise to shift the domain of discourse. Here are the maxims from WP:

Maxim of quantity

Make your contribution as informative as is required (for the current purposes of the exchange).
Do not make your contribution more informative than is required.

Maxim of relation (or relevance)

Be relevant.

In other words, it would be too informative if we included the phrase in the real world every time we were referencing something that might not be real and would lead to a lack of economy and a degree absurdity.

Santa Claus, the myth, not the royal inspiration from history, came down the chimney, the real brick chimney, to give the gifts, bundled and hidden objects that the children desired, to Robert Smith of 221B Baker Street yesterday, December 25th, 1891 in the year of Our Lord, the Judeo-Christian God of the Anglican faith.

In normal conversation, explicitly stating domains of discourse would violate some of Grice's maxims. Likewise, in formal systems like FOPC, a domain of discourse avoids the principle of explosion and other nastiness. The most famous mathematical logical example of the problems related to not restricting domains of discourse is probably Russell's paradox. Both ZFC and BNG restrict the domain of discourse of the nature of a collection to avoid the paradox because of self-reference. FOPC is a type of mathematical logic. The use of an explicit domain of discourse in ZFC for instance helps prevent contradiction, infinite loops, and avoids ambiguity.

Let's take f(x)=x. which is the identity property. Do you need to define x as a natural? an integer? No. But, it's a different story in f(ab)=ba (the commutative property). Properly speaking there exists f for all a and b such that f(ab)=ba when a and b are in N (naturals). Commutativity doesn't hold true for groups or matrices generally, though there are Abelian groups, for instance.

So, a domain of discourse is just a way of ensuring that someone using the formal mathematical system is aware of restrictions and definitions. with the f(ab)=ba example, the domain of discourse whether N or G(S,*) determines whether or not the statement is true. An even more elementary use of a domain of discourse is the restriction on division. There exists f(a,b)=a/b for all a in R and b in R-{0}. The domain of discourse for b is restricted to exclude 0 because division by zero in standard arithmetic is undefined. A domain of discourse is usually just a restriction on the domain.

Another elementary example of a domain of discourse would be calculating area. In normal geometric usage lengths are confined to real numbers greater than zero. So, the area of a rectangle would have the following domain of discourse: There exists A(l,w) for all l,w in R>0 such that A(l,w)=lw=wl. In Euclidian geometry, it is meaningless to talk about an area of 0 or a negative area, so we restrict in FOPC notation the variables l,w to positive reals.

Now, is it possible to have a mathematical system with negative areas? Sure, but doing so would a non-standard formal mathematical system. So, domains of discourse, in a sense are always there implicitly even if they are often invoked explicitly, and when paradox, contradiction, etc. arises, generally, it can be eliminated within the system by defining/redefining the domain of discourse explicitly.

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  • So it is not necessary in predicate logic, but necessary in mathematical logic, and so essentially a flaw in mathematical logic? – Speakpigeon Oct 22 '20 at 14:12
  • @Speakpigeon I just moved my comments to my answer for ease of reading. – J D Oct 22 '20 at 14:39
  • I'm sure domains are useful but apparently essentially a matter of convenience for mathematicians for presenting their results. My question was about (mathematical) predicate logic, not mathematics in general. – Speakpigeon Oct 22 '20 at 16:53
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    @Speakpigeon I think it's a fair characterization to say that it is very convenient to specify a domain of discourse much in the same way using an symbolic notation to represent ideas is; it cuts through the hanky panky. I would suggest to you that you need to consider that mathematical logic is the use of predicate logic in mathematics in general. What the foundationalism program really accomplished is to show that mathematics and logic are essentially two aspects of a single formal system... – J D Oct 22 '20 at 21:48
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    Anyway, just to recap, ∀y(∃x(y = x²)) is not a true statement in all cases, and without specifying a domain of discourse, could lead to errors. ∃y(∃x(x∈Q, y=2, y = x²)), in fact, leads to contradiction. – J D Oct 22 '20 at 22:18
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Not sure if this helps with your confusion/skepticism, but natural language is full of under-determined sentences, e.g.

Today is Tuesday.

That statement is true if where you're at right now (time and space) is a Tuesday and false otherwise. Words like 'today' are known as indexicals in linguistics and the philosophy of language. Think of that formalized as a variable statement that says "x is Tuesday"; it's true for some values of x and false for others. Further, if you lived in world in which no day of the week is called "Tuesday" it would always be false.

In first-order logic, the point of fixing a domain of interpretation for a first order language is precisely so that you can say for every sentence if it's true or not.

Statements like "All S are P" (i.e. more formally, "For all x, if x is [in] S, then x is [in] P") are basically true in some models and false in others. To be able to assign such statements a truth value requires providing an interpretation of the (non-logical constants symbols S and P and of some interpretation of the quantification in terms of what domain it ranges over. E.g. if the domain is the "real world" and the interpretation of S is "x is a man" and that of P is "x is mortal" then the sentence is true in that model. But if the model is (a fictional) one in which all men are immortal, then the same sentence is false for the same S and P.

When someone says (in natural language)

God exists.

they mean a whole lot of things; that sentence has a lot implicature. They don't want to be understood as stating for example that a dog named God exists! In an unrestricted quantification ∃g can well be satisfied by any dog (in the real world, assuming that's the domain of discourse) or anything else in the domain of discourse. Normally when someone says "God exists" they mean that entity has a lot of properties (being a dog usually not included), i.e. they mean g is omnipotent, omniscient, it has created the world and what not:

the sentence ‘God exists’ does not carry its meaning with it, independent of context and background beliefs (Phillips 2004, 67). For this reason, implicaturism argues that what varies with one’s religious beliefs and practices is what language additionally conveys.

  • Phillips, D. Z. Religion and Friendly Fire: Examining Assumptions in Contemporary Philosophy of Religion. Aldershot: Ashgate, 2004.

As for

In the following expression, why would we need to specify any domain?

¬∀y(∃x(y ∈ ℕ ∧ x ∈ ℕ ∧ y = x²)

You're confusing two (other) things here: the domain in which interpretation is provided for a (first order) sentence and syntactic use of ∈ in a first-order language that includes the (ZF[C]) axioms of set theory! ZFC's (nine) axioms basically define (syntactically) the properties of that sole binary predicate denoted by ∈

ZFC is an axiom system formulated in first-order logic with equality and with only one binary relation symbol ∈ for membership. [Axioms omitted]

If you want an analogy here, if you want "God exists", as a first-order language formula, to (syntactically) mean more than the mere triviality of some element existing in the domain of discourse/interpretation, then you then need to specify what axioms God fulfills!

As for that "math expression" you gave, y ∈ ℕ in that expression is surely meant for it to be interpreted in the context of set theory, i.e. considered as added to axioms if you want to prove something follows from that logic formula in ZF. But you also need to define what x² and ℕ means, which the axioms of ZF alone don't do for you. Note also that you can define natural numbers (albeit not completely) via Peano's axioms, which don't actually need to include ZF, and in that context there's no need to write ∈ or ℕ.

[the successor,] the addition[,] and multiplication operations are directly included in the signature of Peano arithmetic, and axioms are included that relate the three operations to each other.

And the PA axioms (which have 3 predicates) look like

enter image description here

The don't carry any ∈ in them (which isn't among the 3 PA predicates), and you'll note that the quantification is thus not "restricted" to anything.

But normally a "math statement" is by default interpreted in the language of ZF[C], plus whatever might be "well understood from context", e.g. what ℕ means, unless otherwise specified.

If you don't want to take my word on that, here's a mathematician saying it

the theory [i.e. axioms and whatever follows from them] ZFC specifies that there is one relation symbol ∈, which we usually read as "element of". A model of ZFC interprets the symbol ∈ as a binary relation, so in this sense it defines its meaning.

[...] most mathematicians seem to be a bit unclear about this distinction, as they never really have to distinguish the language they use from its [intended] meaning.

-- Andrej Bauer

And another concurs (you many want to read all their comments as they are insightful by giving some more examples of this practice, including arithmetic, but I'm not going to copy all of that here).

It is very common, when talking about a first-order theory, to conflate the symbols in the theory with their intended interpretations. [...]

even though the elements of an arbitrary model of ZFC might not "really" be sets, or the interpretation of the ∈ symbol may not really be set membership, we often speak as if they are. The key observation is that, if someone "lived inside" the model, and only had access to the ∈ relation, that person would have no way to tell that the things they see are not sets. One way of making this observation precise is the following [Mostowski Collapsing] lemma, which is proved from "outside" a model of set theory. [...] This lemma says that if we look from the outside at a model that looks even vaguely like a (well-founded) model of ZFC, we can replace it with an isomorphic model whose elements are actually sets and whose binary relation is actually set membership.

-- Carl Mummert

So yeah, you're confusing syntax and semantics here, but you may excused for doing so. That "math formula" you gave is surely meant to have an interpretation in a set universe though, but it's not what it intrinsically carries with it, except by social/math convention(s) and possibly the [con]text given before it in the work where it is found. (You could, e.g. define/intend '∈ ℕ' to mean [be interpreted as] "is beautiful", which of course nobody could guess unless you were explicit about that.)

(There's also that technical sense-- Mostowski Collapsing lemma --in which it is somewhat appropriate to forget about the distinction between language and intended model of ZFC in a lot of "run of the mill" math.)

When a mathematician writes ℕ you may think you know what it means because you were probably taught that symbol with that meaning in school. But that doesn't mean the first-order sentence you've shown carries that meaning for ℕ... except by implicature! This is almost the same as when someone says something about God actually, except that in the ℕ case there's less ambiguity about what properties we expect from it. (There surely are less obvious terms in math, e.g. when a mathematician speaks of a ring or a group, he [usually] doesn't mean it in the sense of common day usage of those words. And there are also such terms that have multiple definitions in math itself, e.g. "filter".)

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  • Very interesting answer but beside the point. I'm sure we are all well aware that no formal expression, including therefore logical expressions, can be understood if we don't first understand the vocabulary involved. My point relative to the notion of domain as used in mathematics is that instead of specifying a domain, for example ℕ, we can insert a premise stating the domain, as I did in my example. I also don't mean to say that this would be better, only that we don't actually need the notion of domain in the sense that it is not necessary. Possibly convenient, but not necessary. – Speakpigeon Apr 8 at 9:56
  • In re "we can insert a premise stating the domain, as I did in my example". If by that you mean that you wrote "y ∈ ℕ", sure, but you still need to say how that symbol is to be interpreted. Your pragmatics argument that one needs to "understand the vocabulary involved" is exactly that--providing some specific semantics, albeit not very explicitly. – Fizz Apr 9 at 1:44
  • "you still need to say how that symbol is to be interpreted" Sure, but this isn't relevant my question. Further, Your point doesn't concern logic. It is about language It only affects the expression and comprehension of our verbal arguments. Not my question. – Speakpigeon Apr 9 at 10:15
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The real world is the domain when we use natural language and don't specify, though!

"There does not exist an x with P(x)" or "There exists a y with P(y)" is not something we can prove to be true or false in predicate logic.

We can prove something if it true in all models. That's literally how If I choose a model where P(x) is empty (let's say P(x) is "x is a unicorn", and my model is the real world), then the formula is true in that domain. If instead the model is that P(x) is "x is a clock inside a crocodile" and the model is a Peter Pan story, then the formula is false. So, just in predicate logic, it's not a formula that is provable to be true or false.

https://en.wikipedia.org/wiki/First-order_logic#Validity,_satisfiability,_and_logical_consequence

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