Coexstensive Properties and Being

If two properties are coextensive, they are said to be the same as one another. For example, the property of being water, and the property of being H2O - these are generally conceded to be the same property, even though we can refer to that property in multiple ways.

Another example: The property of being trialateral and the property of being a figure containing exactly three angles.

We can logically express coextensive properties the following way:

(X) an object has the property of being trilateral if and only if that object has the property of containing three angles

First question: Does (X) necessarily mean that being an object with 3 interior angles is the same as being trilateral

Second Question: If the answer to the first question is yes, then can there ever be exceptions to the assertion that when two properties are necessarily coexstensive they refer to the same thing? For instance is the property of being the only even prime number the same as the property of being the square root of four? Both properties are attributed only, and in every conceivable mathematical instance, to the number TWO, and both properties are reducible to the character of TWO-ness, so are they, therefore, the same property - or are the 2 properties somehow ontologically distinct?

I’d say it doesn’t follow. Take water and H20. I think water is coextensive with H20 but H20 is not coextensive with water. After all, when I’m talking about ‘water’, I am generally talking about what is in the glass in front of me. That will necessarily contain some ions. For example, water is conductive but H20 is not conductive. This is a very important distinction.

Interesting, i hadn’t thought of that at all . . . but the property of containing ions is not a necessary property of water, whereas being composed of H2O is (you can deionize water and it will still be water, but you can’t alter the H2O formula without it becoming something other than water). . . could we still preserve the coextensivity of being water and being H2O with one another if we add that both may in some instances have the additional, different property of being accompanied by ions (as well as a broad array of minerals and impurities)?

Here is the thing: can H20 contain different things without ceasing to be H20? Water can contain ions, or organic molecules, or whatever impurity you want, and still be water. H20 is water. But H30+ is a hydronium ion. Water contain both H20 and H30+. And that is just the tip of the iceberg . . .

Okay - however, how could we apply that to something more precise and conceptually pure, like being the only even prime number and being the square root of four, which both equally qualify as necessary properties of the number two? are they ontologically distinct properties (in the same way that you are saying being water and being H2O are ontologically distinct properties) even if they both necessarily equal the number two, or is one the same thing as the other?

The problem with purely defined categories is that you can invoke all manner of nonsense in their execution.

But given your example, I would say that being the only even prime number is perfectly coextensive with the number two. However, being the square root of four is not, since the square root of four can be either two or negative two. Think of it as a language game. ‘The only even prime number’ is a synonym for ‘two’, whereas ‘the square root of four’ is more ambiguous in its meaning.

they are not the same thing… the square root of 4 is not necessarily the only even prime number…


edit Xun got it

So it IS a language game - i suspected that much - conceptually pure categories really are the problem - i think you’re both right.

Within a language game, it is a language game. Outside of that is another matter: this is where Imp and I disagree.

Should two issues be separated: identity and meaning? ‘7 + 5’ and ‘12’ are numerically identical, but have different meanings. For example, being divisible by 3 is part of the meaning of ‘12’ but not ‘7’ or ‘5’. H2O and water are identical substances, but their meanings are also different.


you can separate identity from meaning, but the question then would be is there any meaning without identity, and, if not, is meaning then “reducible”, so to speak, to that with which it is identical? the same object can always have more than one set of meanings, but the question i’d want to then address would be can there be any meanings prior to, or apart from, those objects - do we derive meanings from the objects (is it strictly the interplay of sensuality and conciousness?), or are there transcendent meanings beyond sensuality, so to speak, which we intuit and then attribute to objects? I mean, is it ever possible for properties of an object to have any being, or substantive ontological status, independent of the objects those properties are HAD by? that is, does the property of being trilateral have any meaning in a universe in which there are no objects with three internal angles?

  • i’m sort of fumbling here - i don’t know if what i’m asking makes sense, or even addresses the question i’m actually thinking of . . . maybe i need more coffee.


What are you citing as the dynamic “outside of” the language game, upon which you and Imp disagree?

Imp is free to correct me, but my understanding of his position is that truth only exists within deductive logic, built from tautologies. However, since deductive logic is an artificial construct, a language game, it doesn’t really have anything to do with reality. Despite this, and this is where I am confused on his position, “the limits of language is the limits of his world” (something demonstrated via deduction) so reality itself becomes a self-deconstructing language game. Despite being detached from deductive logic . . .

I’m more of a pragmatist, so I define truth as a function of utility. Imp and I talked about it a little here.

I do too, with the acknowledgment that to say it’s a function of utility is to grossly simplify the consequences of a bigger existential truth - which is that, in the end, truth is just another word.

saying it’s a function of what works (which it in large part in fact is) is to make being correct contingent upon a perhaps individually subjective judgement as to the definition of the phrase “what works”.

Sure. That is, admittedly, one of the problems with pragmatism. Rorty tried to address that, but in so doing (IMHO) he kinda destroyed what makes pragmatism work . . . so it is admittedly a tricky area. But I think as long as clear goals are defined, it holds. Now, how and why those goals are defined is another issue. That’s why I think that pragmatism works best as an adjunct philosophy. It needs to be mixed with some other narrative. In so doing, I think both are deepened and strengthened. Otherwise, it collapses into Rortian post-modernism or something akin to utilitarianism, neither of which are terribly appeasing.

Well stated. I agree.


Thanks! That clarifies a great deal on your position.


So is language itself NOT a part of reality?