Extension, Intension, Comprehension – Revisited

There are traditional examples that move us quickly away from the idea that our language is just extensional.  And there are some that put into doubt that our language is only intensional, with no distinctions between any concepts that are necessarily co-extensional.  These examples suggest that predicates have, or may have, three distinct semantic values: extension, intension, and comprehension.^[1]

What the examples leave largely open is the character of relations between these three levels or modes of meaning.  Seemingly best understood is the relation between extension and intension.  I shall explore that first, and then explore how the same sort of relationship may obtain between intension and comprehension.  There will be a connection with paraconsistent logic.

1.      Traditional examples

From Medieval philosophy we have a notable theory of distinctions.  There is a distinction de re between featherless biped and rational animal: Diogenes the Cynic displayed a plucked chicken as real instance of the one and not the other.  Today we say: these two concepts do not have the same extension.  But there is only a distinction of reason between woman and daughter: there are no real entities instantiating the one but not the other, but there could be, as are represented in pictures, stories, and myths.  Today we say:  these concepts have the same extension but not the same intension.  

Are there examples that push us still further?  We can symbolize has a property, as x̂ (∃F)Fx, and is identical with something, as x̂(∃y)(y = x).  Let’s refer to these concepts as being and existence.  Then we can puzzle over them:  there surely could not be, or even be fantasized to be, an instance of one that would not be an instance of the other.^[2] As the Medievals would say, not even God could create something that has one but not the other.  If there is a distinction nevertheless, it is what Duns Scotus called a formal distinction. Today we would say, if anything, that the two concepts do not have the same comprehension, and that if so, our language is hyper-intentional.

2.      How extension is related to intension

Terminology. I will use “property” only for what can be an intension of a predicate in a given language.  Only different terms are to be used for extensions and comprehensions.^[3]  

There are well-known ways in which properties (intensions of predicates) are represented in semantic analyses of modal logics.  

Let’s abstract from the details. Properties, in this sense, form an algebra A_I.  As a working hypothesis I will take this to be a Boolean algebra.  It has a top, T, and a bottom, f. In a given world, our world say, any property x has an extension |x|, which is a set of entities in the world.  Necessarily, T‘s extension includes everything and f‘s extension is empty. A distinction of reason is then a distinction between properties that have the same extension.

The function | | is a homomorphism:  if x, y are properties and x  ≤ y then |x| ⊆ |y|, ⎯ |x| = | ⎯ x|, and |x . y|  = |x| ∩ |y|. As a result the sets, which are the extensions of properties, form a Boolean algebra, A_E.    

To formulate the distinction of reason we can define an equivalence relation, extensional equivalence, on the properties:  (x ≡ y) iff |x| = |y|.  Then A_E is isomorphic to the quotient algebra A_I modulo  ≡ : the elements of this are the equivalence classes  of elements of A_I,  [x] = {y: x ≡ y}, with [x] ≤  [y]  iff x  ≤ y,  ⎯[x] = [⎯x],  and |x|  ∧ |y| = |x  ∧ y|. 

But there is another nice way of thinking about this relation between properties and their extensions.  Think about the properties that | | maps into the empty set.  Let us call these the ignorable properties from a extensionalist point of view. 

properties x and y are co-extensional  iff they differ only by an extensionally ignorable part, that is, there is some extensionally ignorable property z such that x v z = y v z.

The distinction of reason pertains then to exceptions to co-extensionality.  For example, woman = (woman who has parents) v (woman who has no parents), and that is the join of daughter with an extensionally ignorable property.^[4]

Now the ignorable properties form an ideal in the Boolean algebra B_I of properties: if x ≤ y and y is ignorable then so is x, and moreover, (x v y) is ignorable iff both x and y are ignorable.  This is not coincidental: for any equivalence relation E on a Boolean algebra there is an ideal J such that x E y iff for some z in J, x v z = y v z.

Are co-extensional and extensionally equivalent the same relation?  In other words, if x and y have the same extension must there then be an ignorable property z such that x v z = y v z?

The answer is yes:  z is the symmetric difference between x and y, that is, the join of (x ⎯ y) and (y ⎯ x).  If either of those had a non-empty extension then |x| and |y| would not be the same.

I would like to explore this idea of ‘ignorables’ to get at the relationship between comprehension and intension.

3.      Positing the same relationship for intension to comprehension

The idea is that the above abstract form of the relationship between extension and intension obtains also for the relationship between intension and comprehension.  That is, the comprehensions of concepts form an algebra A_C, and A_I is (isomorphic to) the quotient algebra formed by reducing A_C by an appropriate equivalence relation.

In view of the above, the way to identify that appropriate equivalence relation is to specify an ideal in A_C: the ideal of intensionally ignorable comprehensions.  

If x is in A_C then it has an intension ||x|| in A_I, and x is intentionally ignorable exactly if ||x|| = the absurd property f.  Define x and y to be intensionally equivalent (x ⇔ y) exactly when x v z = y v z for some ignorable comprehension z.  And I submit that A_I is (is isomorphic to) A_C modulo ⇔.

Let’s see how this plays out with the example of being and existence.  A quick check shows that

(∃y)(y = x)   v  [(∃y)(y = x) & ~(∃F)Fx]  v  [(∃F)Fx & ~(∃y)(y = x)]

 is logically equivalent to 

[(∃F)Fx]  v  [(∃y)(y = x) & ~(∃F)Fx]  v  [(∃F)Fx & ~(∃y)(y = x)]:

that is 

existence or [existence and non–being] or [being and non–existence]

= being or [existence and non-being] or [being and non-existence]

Therefore to complete the example we must declare that [existence and non-being] as well as [being and non-existence] are intensionally ignorable comprehensions.  The mapping || || sends these into the absurd property .  

In this case the three comprehensions being, existence, and being cum existence have the same intension.  Necessarily, any real things that have being exist, and any that exist  have being.   So the intension is the summum genus among properties, T.

4.      Pertinence of paraconsistent logic

The example of being and existence already shows that the logic pertaining to comprehension cannot be classical, where the definitions of those properties are tautologically equivalent.  

Medieval discussions of such concepts as being, (‘transcendentals’), included also examples such as finite or infinite.  While distinct from being, that property it is clearly not distinguishable from being by any real or possible instances.  From a classical logic point of view, finite or infinite is just an ‘excluded middle’, hence tautological, and there is no logical leeway.

To do justice to the formal distinction between being and ‘excluded middles’, therefore, the logic pertaining to comprehension must allow for ‘excluded middles’ that do not imply each other.   

The most modest logic of this sort is FDE, which corresponds in algebra to De Morgan lattices: distributive lattices equipped with an involution.  The involution ⎯ is like a Boolean complement:

x = ⎯ ⎯ x

if x ≤ y then ⎯y ≤ ⎯x, 

and from these two, given distributivity, the De Morgan laws follow:

⎯(x v y) = (⎯x  ∧  ⎯y)

⎯(x  ∧ y) = (⎯x v ⎯y)

But (x v ⎯x) may not be the top, (x  ∧ ⎯x) may not be the bottom, and it can happen that x = ⎯x. 

As models for the logic of comprehension I propose the De Morgan lattices with top (Θ), bottom  ⊥ , and such that ⎯⊥= Θ. The function || || assigns an intension to each comprehension; ||  ⊥ || = f, and ||Θ|| = T.

Boolean algebras are De Morgan lattices, with the characteristic that their involution ⎯  is such that for each element x, (x v ⎯ x) = the top element.  Happily the connection between ideals, homomorphisms, and equivalence relations holds as well for De Morgan lattices.^[5]

To explore how comprehensions fare in this landscape, let us take a simple example of a De Morgan lattice for a model.

The 8-element De Morgan lattice DM8 (aka Mighty Mo) looks quite like the three atom Boolean algebra B3, but the involution is different:

Suppose we take as our ideal of intensionally ignorable elements the set of elements marked with ⎯, which consists of ⎯ 0 and everything below that.  To represent the two classical tautologous ‘excluded middles’ of finite or infinite and round or not round let us choose +2  for finite  and +1 for round.  Then we see that:

finite or infinite = (+2 v ⎯2) = +2

round or not round = (+1 v ⎯1) = +1

+2 v ⎯0 = +1 v ⎯0    ( = +3), 

so the two ‘excluded middles’ differ by the ignorable element ⎯ 0, hence are intensionally equivalent though distinct.  And yes, they are also equivalent to the top +0 and +3, their intension is the property T, the summum genus.

5.      Identifying the intensionally ignorable comprehensions

There is a first-blush troubling question about the insistence that A_C must be a De Morgan lattice, and not Boolean. Above I had advanced as working hypothesis that A_I is Boolean.  But now we have A_I as a quotient algebra, namely A_C modulo ⇔, which implies that A_I is a De Morgan lattice as well.   That does not rule out that A_I is Boolean.  But is it?

I will take this question as the clue to how to identify the intensionally ignorable comprehensions.  

First, to have a clear example, let’s suppose again that A_C is Mighty Mo, DM8.  We chose an ideal of ignorables, namely the ideal generated by ⎯0.  Then we already saw that with that choice +0, +1, +2, +3 are intensionally equivalent.  A quick check shows that ⎯1 v ⎯0 = ⎯2 v ⎯0 = ⎯0 v ⎯0 = ⎯3 v ⎯ 0 = ⎯3.  So ⎯0, ⎯1, ⎯2, and  ⎯3 are all equivalent as well.  Therefore in this case AI has just two elements, a top and a bottom (‘the True’ and ‘the False’).  That is the two element Boolean algebra. 

Can this be the case in general? What candidates do we have for intensionally ignorable comprehensions?  Clearly the elements (x  ∧ ⎯x).  They are mapped to the absurd property: ||x  ∧ ⎯x|| =  f.  So let us choose for the ideal of ignorables an ideal that includes {z: for some x in A_C, z = x  ∧ ⎯x}.^[6]

But then in A_I, whose members are exactly the properties ||x|| with x in A_C, 

||x  ∧ ⎯x|| =  f =   ||x||  ∧ ⎯||x||  

hence by De Morgan’s laws, ⎯||x|| v || x|| =   ⎯  f = T, the summum genus.     

Therefore A_I is a Boolean algebra.

6.      Coda

I have not so far mentioned comprehensions that are not ignorable although they strike us at once  as self-contradictory.   At first blush 

            brother of someone with no siblings, 

sister of someone with no siblings

are different concepts, despite their apparent self-contradictory-ness.  For one is a concept of a male and the other is a concept of a female.  

What we can say about it is only this, I think:  they do not have the same intention, and are not intensionally equivalent, but each is the meet of something with the intentionally ignorable comprehension being a sibling of someone who has no siblings.

I doubt that this is the end of the matter.  The logic of comprehension must be, with respect to self-contradictions at least as liberal as FDE … yes, but who knows what else lurks in these deep-black logical waters, yet to be appreciated?

INDEX TO SYMBOLS 

≤ , v,  ∧ ,  ⎯ : partial order, join, meet, involution in an algebra

the absurd property:  f,   the summum genus (top property): T

equivalence class of property x: [x]

assignment of extensions to properties:  | |

assignment of intensions (properties) to comprehensions:  ||  ||

extensional equivalence of properties:   ≡

intensional equivalence of comprehensions:   ⇔

top of a De Morgan lattice: Θ

bottom of a De Morgan lattice: ⊥

NOTES

---

^[1] Terminology varies.  Alonzo Church’s review of C. I. Lewis’ The Modes of Meaning begins with “As different mtodes, or kinds, of meaning of terms the author distinguishes the denotation of a term, the comprehension,  the connotation or intension, the signification, the analytic meaning.”  (JSL 9 (1944): 28-29).  Lewis’ terminology was not standard, and as Church shows, not clear; though influential, his work did not standartize usage in this respect.  

^[2] More familiar is the example of a distinction between triangle and trilateral, discussed by Leibniz among others.  His oft quoted passage on the matter: “[T]hings that are conceptually distinct, that is, things that are formally but not really distinct, are distinguished solely by the mind. Thus, in the plane, Triangle and Trilateral do not differ in fact but only in concept, and therefore in reality they are the same, but not formally. Trilateral as such mentions sides; Triangle, angles.”

^[3] I am tempted to adopt “concept” for the comprehension of a predicate.  But its associations to the mental might become a constant worry.

^[4] To help my intuition and imagination I keep in mind reduction modulo sets of measure 0 as paradigm example.

^[5] Thm. 5 on p. 27 of Birkhoff  1967 Lattice Theory 3^rd edition

^[6] Minimally, the ideal generated by that set.  But we may want additions to the ignorables, like the meet of being and non-existence.