Thursday, October 7, 2010

The Meaningfulness of Abstract Objects as Existents

It is alleged by the noncognitivist that talk of abstract objects as existing things is literally meaningless. What does it mean to say, for example, that the law of non-contradiction exists, or that the number 7 exists? Can such statements be defended from the noncogntivist's attack?

Notice that one needn't hold the belief that abstract objects actually exist in order to consistently maintain that it's meaningful to talk as if they exist. The nominalist might say that the number 7 does not exist, but that "the number 7 exists" is a meaningful, coherent proposition.

One may advance an argument based on the indispensability of abstract object-talk:

1. If a proposition is meaningful, each of its referents must possibly exist. (Premise)

For example, if I were to say, "unicorns are magical horses with a horn," I am expressing a proposition in which each of the words in the sentence at least possibly exists, even though this particular entity does not, in fact, exist.

2. Some propositions include abstract objects as referents. (Premise)

"The Prime Minister is not a prime number," includes an abstract object - prime number (e.g. 7).

3. Some of the propositions in (2) are meaningful. (Premise)

"The Prime Minister is not a prime number" expresses not only a truth, but a necessary truth. Prime Ministers just aren't the type of entities that can be prime numbers. Yet, a proposition can only be true if it is meaningful. From this it follows that:

4. Therefore, abstract objects are meaningful referents. (From 1 - 3)

As stated before, the meaningfulness of abstract objects does not necessarily entail that abstract objects exist. However, one may very well advance an additional argument that states that indispensable truths must exist. It is self-refuting to reject the laws of logic, for example, which entails their indispensability, and it's their indispensability that entails their existence as abstract objects. (I'm thinking of logic as abstract for fairly obvious reasons. The law of non-contradiction doesn't do anything; it doesn't stand in any causal relations, so it cannot be concrete.)

Logic, then, is necessary at all times and all places. This is a viable starting point for an argument of natural theology. Logic is either the concept of a mind, or else it is mind-independent. If it is conceptual, then it cannot be the concept of just any mind (you and I possibly fail to exist at various times), but must be the concept of a necessary mind, God.


  1. If it were true that meaningful sentences presuppose the possibility of their referents, then we might have a good basis for the ontological argument's possibility premise. For we'd just argue that the premise 'Possibly, God exists' is meaningful; and hence it is possibly true.

    But, this might also work to a disadvantage:

    It is both the case that George is a perfect being and John is a perfect being, and George is not identical to John.

    if we admit its meaningfulness, then we admit the two possibility of two compossible perfect beings. Mind you, the Trinity may cover our butts in this respect. But then we could just propose that we add two more names where none are identical to each other.

    But consider this:

    2. Squared circles do not exist.

    If meaningful, then squared circles possibly exist. But, obviously they do not. However, I wouldn't be so quick to deny it as meaningful either; it may just a logical contradiction. If it were not meaningful, some say, it would be as if I said gibberish.

  2. Those are some interesting points to consider, Mickey. The Trinity states that there is one God (being) that is three persons. Presumably, John and George are supposed to be the same being, but different persons. If they were different beings, then at least one of them could not be perfect. If "John is a perfect being" and "George is a perfect being" are both meaningful, then one might say that "John" and "George" are just words that correspond to the same referent.

    On the other hand, I'm not sure ontological arguments can be used to establish the existence of any persons, anyway. You allude to the following:

    1. There is possibly a Trinity.

    1*. There is possibly a Quadrinity.

    Both (1) and (1*) entail the possible existence of a perfect being (and therefore, the actual existence of a perfect being, given their conjunction with S5). However, both (1) and (1*) cannot be true. The Trinitarian could argue that non-Trinitarian views of God are incoherent, but I'm not sure that's a viable route to take. The alternative I suggest is that the ontological argument can only be used to establish the existence of a perfect being, the number of persons notwithstanding.

    The last point I thought was especially interesting, since it serves as a potential counter-example to my claim that meaningfulness suggests possibility. "Squared circles do not exist" is actually synonymous with, "there is no object that is both a square and a circle." This revision makes it clear that "square" and "circle" are distinct referents, both of which are meaningful.

  3. My larger point about the possibility premise is this. Presume the premise 'God exists' is meaningful; and hence it is possible. Likewise, consider (1)

    (1) George, Paul, Scott and Matt are are perfect beings and no one (from these four names)is identical to the other.

    If meaningful, (1) would suggest its possibility. But no Christian of the monotheistic/necessary being sort would ever grant (1). Yet, it seems just as meaningful as 'God exists', or perhaps 'A perfect being exists'. Perhaps not all are true, but are all meaningful?

    consider a new point: You want to affirm that 'God exists' is meaningful. Hence, we call it possibly true. But for any proposition p, if p is a sentence, and a meaningful one at that, then so is not-p. This is a point of logic too: if X is a sentence, then so is not-X.

    This would suggest that 'god does not exist' is meaningful; and hence it is possibly true. Can a necessitarian theist entertain this? I'm unsure.

    To the last point, Even if square' and 'circle' have meaningful referents themselves seems to be besides the point. what is being denied is a referent which is both these things. Is such a referent meaningful? I dunno. But suppose we treat it as a single property: That object has the property of being-a-squared-circle. What then?

    but these questions aside, I'm unsure your response was faithful to my statement. I could be wrong but 'squared' in 'squared circle' is functioning as an adjective for 'circle'. In conventional predicate logic, we cannot so easily separate nouns and their adjectives. For instance, consider this:

    A large female does not exist.

    Now, if we interpret this to mean:

    Nothing is both large and a female

    then it's no longer clear that the adjective is meant to describe the referents size AS A FEMALE. We clearly have the denial of something being large as well as a female, but it's unclear that this is the same thing. For 'large' was meant to modify 'female', not the referent itself.

    I gotta run. As always, i enjoy our talks.

  4. Keep in mind the original claim:

    1. If a proposition is meaningful, each of its referents must possibly exist. (Premise)

    "Each" can be modified to "each individual." "Squared circle" is a conjunction of two separate meaningful and possible referents. I will have to think about this some more, though.

    "God does not exist" can be considered meaningful because the referents, "God" and "~exist" are both individually meaningful and possible. The claim isn't that whole statements are possible if each of its individual referents is meaningful and possible.

  5. Unless you want to consider '~exist' to be a property, I'm a little baffled how it is a referent or has reference. The referent in this case is just God, and his existence is denied.

    Consider: impossibly, God exists.

    This is meaningful. Yet, its meaningfulness, on your argument, would entail its falsity. why? Well, it's only meaningful if its referent, God, possibly exists. Yet, that's precisely what it denies! That's problematic, I think.

    We basically throw out statements which deny existents as impossible. I'm not ready to do that.