Have you heard the phrase “he’s an everyman” or “he’s a nobody” or “he’s a somebody”. What does this mean?
English has quantifiers. “Every dog is a mammal”, “some dog is brown”, “no dog is a prime number.” The standard semantics (Frege) treats these as higher-order functions: “every dog” denotes a function from properties to truth-values, true on input F iff every dog has F.
What if quantifiers corresponded to objects, the same way names do? “Pope Leo is a mammal” has a subject “Pope Leo” and a predicate “is mammal”. What if “every dog is a mammal” had the same structure — a subject “every-dog” and the same predicate “is mammal”?
This is maybe the worst semantics of quantifiers. Nobody endorses it, I invented it in the shower.
It’s a complete non-starter. To see what kind of object every-dog is, you can examine it’s properties. For any predicate φ, every-dog satisfies φ iff every dog is φ. For example, every-dog is a mammal, weighs less than 800 tonnes, etc. But every-dog lacks the property of being four-legged, and lacks the property of not-four-legged, since dogs vary on this. Every-dog violates excluded middle — it’s properties are gappy.
Some-dog is the dual: it has every property instantiated by at least one dog. So it’s simultaneously brown-all-over and white-all-over, male and female, three months old and twelve years old. Some-dog violates non-contradiction — it’s properties are glutty.
No-dog has every property that no dog has. It’s a prime number. It’s the Eiffel Tower.
Despite being a terrible semantics of quantifiers, this corresponds (if you squint) to the isomorphism between a finite-dimensional vector space and its double dual. Think of the domain of objects D as a vector space, and predicates as linear functionals on D — elements of D*. Then quantifiers live in D**, functionals on predicates. There’s a canonical map D → D** given by x ↦ (f ↦ f(x)): each object corresponds to the quantifier “evaluate at x”, i.e. the proper name quantifier. In finite dimensions this map is an isomorphism — every quantifier is a name in disguise.
fwiw, i think this sort of picture gets somewhat better at what happens in your mind when you grasp a sentence involving “a dog” or “the dog” than the canonical proposal by russell involving quantifiers. this probably deserves a long essay but i’ll try to communicate the idea quickly:
as the first sentence of a novel, i claim that “porthos the dog crossed the street” and “the dog crossed the street” basically do the same thing in your head (except in the former case the object tracker/representer that is created has the name “porthos” attached to it[1])
there are many languages in which the 4 sentences “a/the dog crossed a/the street” are[2] literally the same string, namely “dog crossed street” in word-by-word translation. this is related to the thoughts being very similar
more speculative: roughly i think that “a” tells you to create a new object tracker/representer and “the” tells you to add to an old one. except that it’s fine to start a novel with “the dog was tired” even though the reader doesn’t already have an object tracker initialized beforehand, but i’d guess in that case you can still as if add to an old tracker (like, you imagine already being familiar with that dog)
roughly i think russell’s view of determiners (at least insofar as it is trying to get at what it is to grasp a sentence involving a determiner, which it needn’t be trying to get at) confuses a [frame/model]-internal proposition which doesn’t really have a quantifier with a statement capturing the adequacy of applying the frame/model which has a quantifier. like, in my view: we really think of “the round square” as some sort of ordinary object inside a frame/model, but then that frame/model turns out not to hang together or refer, and that can be seen from “there is a (unique) round square” being false. (this is related to wittgenstein’s hinge vs free belief distinction.)
this view has an easier time making sense of “a dog has four legs” usually communicating that dogs have four legs
this view coheres better with how “a/the dog” sits in the syntax tree of a sentence
(i haven’t thought this through carefully. an interesting challenge here is to spell this picture out much better and to [explain why]/[ascertain whether] this sort of thought-syntax “works” when doing eg mathematical thinking)
and also except for properties created by other associations like “porthos” being a name used for male dogs and maybe being used by such and such a person and reminding you of a particular porthos you knew and whatever
Is “The dog” in “The dog crossed the street” perhaps similar to “this dog”? “This” is often conceptualized logically as a constant/name. I guess “Porthos the dog” could also be analyzed as a name. Though one would have to ensure that “Porthos the dog is a dog” is a logical truth. Usually “a dog” would be formalized as “some dog”, “there is at least one dog such that”, which is usually plausible but not the intended meaning of “A dog has four legs” (in that intended meaning “A dog has three legs” would be false).
One underappreciated fact (which in my opinion clearly invalidates Russell’s analysis of the definite article) is that several languages use the definite article in front of names. So e.g. “The porthos crossed the street” rather than “porthos crossed the street.” Even English does that sometimes, e.g. “the Eiffel Tower”, “the NSA” (but not “the NASA”). Which suggests there is no fundamental logical difference between proper nouns (or “proper names”, as philosophers like to call them) and common nouns. Which suggests that names (parthos) are not logical constants, but predicates, just like common nouns (dog).
What you are talking about are ′ generic objects’ .
They (re)appear in language/linguistics, logic and computer science. E.g. in computer science they are closely related to ‘parametric polymorphisms’/ ′ generic data types’, in logic with Girard’s System F.
There is a large literature on them. Eg semantics of generic objects is occasionally used to semanticize anaphora, see e.g. here
The most interesting bit to me is that they are related to a dual connective to universal/existential quantifiers that are defined by elimination.
Concretely—if I have to prove \exists \phi(x), the logical introduction rule says that I have to supply an instance t such that \phi(t). I can then conclude \exist \phi(t) by the \exists-introduction rule. In a standard natural deduction style type theory by logical harmony there is a corresponding elimination rule that would eliminate these \exists \phi(x).
However, we could also posit a new quantifier that is defined by it’s elimination rule. So in that case \forall x \phi(x) would produce a c such that \phi(c) iff \forall x \phi(x), in other words a generic object c.
It’s a complete non-starter. To see what kind of object every-dog is, you can examine it’s properties. For any predicate φ, every-dog satisfies φ iff every dog is φ. For example, every-dog is a mammal, weighs less than 800 tonnes, etc. But every-dog lacks the property of being four-legged, and lacks the property of not-four-legged, since dogs vary on this. Every-dog violates excluded middle — it’s properties are gappy.
exactly. so this generic c would be your ‘every-dog’. Since we’re working in type theory we are not forced to use excluded-middle so these objects can happily exist without contradiction. =)
Some-dog is the dual: it has every property instantiated by at least one dog. So it’s simultaneously brown-all-over and white-all-over, male and female, three months old and twelve years old. Some-dog violates non-contradiction — it’s properties are glutty.
No-dog has every property that no dog has. It’s a prime number. It’s the Eiffel Tower.
No-dog is not so interesting for non-typed settings. However, if we type or use various universes we can do much richer things.
rk. More generally in full (Martin-Lof) type theory
We can extend from \exists and \forall to all \Sigma and \Pi -types [and Universe types]. This would give constructions of arbitrary generic objects and universes satisfying any collection of properties...
Despite being a terrible semantics of quantifiers, this corresponds (if you squint) to the isomorphism between a finite-dimensional vector space and its double dual. Think of the domain of objects D as a vector space, and predicates as linear functionals on D — elements of D*. Then quantifiers live in D**, functionals on predicates. There’s a canonical map D → D** given by x ↦ (f ↦ f(x)): each object corresponds to the quantifier “evaluate at x”, i.e. the proper name quantifier. In finite dimensions this map is an isomorphism — every quantifier is a name in disguise.
Yes you are definitely on to something here. There are some connections with double-dual constructions. I think the definite treatment is still left to be worked out though.
Quantifiers as objects
Have you heard the phrase “he’s an everyman” or “he’s a nobody” or “he’s a somebody”. What does this mean?
English has quantifiers. “Every dog is a mammal”, “some dog is brown”, “no dog is a prime number.” The standard semantics (Frege) treats these as higher-order functions: “every dog” denotes a function from properties to truth-values, true on input F iff every dog has F.
What if quantifiers corresponded to objects, the same way names do? “Pope Leo is a mammal” has a subject “Pope Leo” and a predicate “is mammal”. What if “every dog is a mammal” had the same structure — a subject “every-dog” and the same predicate “is mammal”?
This is maybe the worst semantics of quantifiers. Nobody endorses it, I invented it in the shower.
It’s a complete non-starter. To see what kind of object every-dog is, you can examine it’s properties. For any predicate φ, every-dog satisfies φ iff every dog is φ. For example, every-dog is a mammal, weighs less than 800 tonnes, etc. But every-dog lacks the property of being four-legged, and lacks the property of not-four-legged, since dogs vary on this. Every-dog violates excluded middle — it’s properties are gappy.
Some-dog is the dual: it has every property instantiated by at least one dog. So it’s simultaneously brown-all-over and white-all-over, male and female, three months old and twelve years old. Some-dog violates non-contradiction — it’s properties are glutty.
No-dog has every property that no dog has. It’s a prime number. It’s the Eiffel Tower.
Despite being a terrible semantics of quantifiers, this corresponds (if you squint) to the isomorphism between a finite-dimensional vector space and its double dual. Think of the domain of objects D as a vector space, and predicates as linear functionals on D — elements of D*. Then quantifiers live in D**, functionals on predicates. There’s a canonical map D → D** given by x ↦ (f ↦ f(x)): each object corresponds to the quantifier “evaluate at x”, i.e. the proper name quantifier. In finite dimensions this map is an isomorphism — every quantifier is a name in disguise.
fwiw, i think this sort of picture gets somewhat better at what happens in your mind when you grasp a sentence involving “a dog” or “the dog” than the canonical proposal by russell involving quantifiers. this probably deserves a long essay but i’ll try to communicate the idea quickly:
as the first sentence of a novel, i claim that “porthos the dog crossed the street” and “the dog crossed the street” basically do the same thing in your head (except in the former case the object tracker/representer that is created has the name “porthos” attached to it [1] )
there are many languages in which the 4 sentences “a/the dog crossed a/the street” are [2] literally the same string, namely “dog crossed street” in word-by-word translation. this is related to the thoughts being very similar
more speculative: roughly i think that “a” tells you to create a new object tracker/representer and “the” tells you to add to an old one. except that it’s fine to start a novel with “the dog was tired” even though the reader doesn’t already have an object tracker initialized beforehand, but i’d guess in that case you can still as if add to an old tracker (like, you imagine already being familiar with that dog)
roughly i think russell’s view of determiners (at least insofar as it is trying to get at what it is to grasp a sentence involving a determiner, which it needn’t be trying to get at) confuses a [frame/model]-internal proposition which doesn’t really have a quantifier with a statement capturing the adequacy of applying the frame/model which has a quantifier. like, in my view: we really think of “the round square” as some sort of ordinary object inside a frame/model, but then that frame/model turns out not to hang together or refer, and that can be seen from “there is a (unique) round square” being false. (this is related to wittgenstein’s hinge vs free belief distinction.)
this view has an easier time making sense of “a dog has four legs” usually communicating that dogs have four legs
this view coheres better with how “a/the dog” sits in the syntax tree of a sentence
(i haven’t thought this through carefully. an interesting challenge here is to spell this picture out much better and to [explain why]/[ascertain whether] this sort of thought-syntax “works” when doing eg mathematical thinking)
and also except for properties created by other associations like “porthos” being a name used for male dogs and maybe being used by such and such a person and reminding you of a particular porthos you knew and whatever
or i mean: can naturally be
Is “The dog” in “The dog crossed the street” perhaps similar to “this dog”? “This” is often conceptualized logically as a constant/name. I guess “Porthos the dog” could also be analyzed as a name. Though one would have to ensure that “Porthos the dog is a dog” is a logical truth. Usually “a dog” would be formalized as “some dog”, “there is at least one dog such that”, which is usually plausible but not the intended meaning of “A dog has four legs” (in that intended meaning “A dog has three legs” would be false).
One underappreciated fact (which in my opinion clearly invalidates Russell’s analysis of the definite article) is that several languages use the definite article in front of names. So e.g. “The porthos crossed the street” rather than “porthos crossed the street.” Even English does that sometimes, e.g. “the Eiffel Tower”, “the NSA” (but not “the NASA”). Which suggests there is no fundamental logical difference between proper nouns (or “proper names”, as philosophers like to call them) and common nouns. Which suggests that names (parthos) are not logical constants, but predicates, just like common nouns (dog).
What you are talking about are ′ generic objects’ .
They (re)appear in language/linguistics, logic and computer science. E.g. in computer science they are closely related to ‘parametric polymorphisms’/ ′ generic data types’, in logic with Girard’s System F.
There is a large literature on them. Eg semantics of generic objects is occasionally used to semanticize anaphora, see e.g. here
The most interesting bit to me is that they are related to a dual connective to universal/existential quantifiers that are defined by elimination.
Concretely—if I have to prove \exists \phi(x), the logical introduction rule says that I have to supply an instance t such that \phi(t). I can then conclude \exist \phi(t) by the \exists-introduction rule. In a standard natural deduction style type theory by logical harmony there is a corresponding elimination rule that would eliminate these \exists \phi(x).
However, we could also posit a new quantifier that is defined by it’s elimination rule. So in that case \forall x \phi(x) would produce a c such that \phi(c) iff \forall x \phi(x), in other words a generic object c.
exactly. so this generic c would be your ‘every-dog’. Since we’re working in type theory we are not forced to use excluded-middle so these objects can happily exist without contradiction. =)
No-dog is not so interesting for non-typed settings. However, if we type or use various universes we can do much richer things.
rk. More generally in full (Martin-Lof) type theory
We can extend from \exists and \forall to all \Sigma and \Pi -types [and Universe types]. This would give constructions of arbitrary generic objects and universes satisfying any collection of properties...
Yes you are definitely on to something here. There are some connections with double-dual constructions. I think the definite treatment is still left to be worked out though.
By the way you might also be interested in Aristotle & Boethius’s original work on the topic.
The math level skyrocketed at the end and I still have no idea how it relates to “he’s a nobody”, but it was a fun read.