Tuesday, September 15, 2009

The Gaps in Gappy Propositions

Some of you are familiar with The Gappy Proposition View (GPV). According to (GPV), the meaning of a name is the thing to which it refers and if a name such as 'Vulcan' has no referent, then when it is used in a sentence, such as 'Vulcan is a planet', that sentence expresses a gappy proposition. We might represent the proposition expressed by 'Vulcan is a planet' as follows {____, planethood}. On (GPV) we assign truth values to propositions as follows: A simple sentence of the form 'N is F' expresses a truth iff N has a referent and the referent of N has the property expressed by 'is F'. A simple sentence is false otherwise. Truth values for non-simple sentences are determined in the standard way.

(GPV) seems to have some ontological commitments. For example, (GPV) is committed to propositions. (GPV) along with the thesis that there are genuinely empty names is committed to gappy propositions. Finally, it seems that (GPV) along with the thesis that there are genuinely empty names is committed to gaps. Gaps, of course, are the unfilled positions in gappy propositions. But, now it seems that there is a problem for the gappy proposition view.

Consider the following sentence:

S1 'Vulcan exists.'

Assuming that 'Vulcan' is an empty name, then according to (GPV), (S1) expresses the following proposition:

P1 {_____, exists}

But, now let's introduce a new name. Let's introduce 'Gappy' as a name for the gap in proposition P1. Now consider the following sentence:

S2 'Gappy exists.'

Since the meaning of a name is the thing to which it refers (that is, since a name contributes its referent (if any) to the proposition it expresses) it should follow that S2 expresses the following proposition:

P2 {____, exists}

But, P1 is clearly identical to P2. There is a problem because S1 clearly does not express the same proposition as S2. For one thing, S1 expresses a falsehood whereas S2 expresses a truth. (In fact, given the truth conditions outlined above it seems that S2 cannot express a truth. This is a further problem for (GPV)).

9 Comments:

Blogger Joshua said...

Now, I have a response to my argument. Perhaps the sentence 'Gappy exists' expresses a proposition that has a gap that fills the gap attached to the property of existence. So, first consider the gappy proposition {____, exists}. Now, fill the gap in that proposition with a gap (namely gappy). The resulting proposition is the proposition that Gappy exists.

One weird thing about this view is that it seems that Gappy is filling itself in the proposition that Gappy exists. Does this spell any trouble? Can a gap fill itself? Are there any mereological worries? Those are some of my questions right now.

3:55 PM  
Blogger Andrew said...

This is a REALLY fun puzzle...

One option would be to deny the existence of gaps. Treat them like holes. This seems bad...cuz holes exist. But a way to flesh this puzzle out into a paper would be to anticipate that kind of objection and explain why people should believe in things like holes (and therefore gaps).

It seems the best option would be to deny that the propositions expressed are the same.

I don't have a great idea as to how to do this, but here's an initial stab...Proponents of the gappy view don't think 'Vulcan' means gap. They don't think it refers to gap either, right?

However...'Gappy' does refer to gap.

So...'Gappy' must contribute something to the proposition expressed that 'Vulcan' doesn't.

Although, this is why I love this puzzle...I'm not sure what the heck it would be.

7:47 PM  
Blogger Greg Fowler said...

I have to agree with Andy: This truly is a REALLY fun puzzle.

It reminds me of the puzzle, which we've talked about before, for Millian Russellians who accept that definite descriptions have semantic contents (and which Salmon, in his paper 'On Designating', credits Russell for discovering). For anyone unfamiliar with the puzzle, it goes like this. For any expression E, let '*E*' refer to the semantic content of E. Now assume that definite descriptions have contents and consider the propositions expressed by the following two sentences:

a. The center of mass of the Solar System is a point.
b. *The center of mass of the Solar System* is a point.

Just as the propositions expressed by Joshua's S1 and S2 are distinct, so too are the propositions expressed by (a) and (b). After all, (a) is true and (b) is false.

One of the things I find really interesting about Joshua's puzzle is that at least some of the responses to the puzzle involving the propositions expressed by (a) and (b) are not, so far as I can see, available in the case of Joshua's puzzle. For instance, one might hold that whereas *the* is an (indirect) constituent of the proposition expressed by (a), it is not a constituent of the proposition expressed by (b) at all. I don't think any similar solution to Joshua's puzzle is forthcoming, however.

I wonder, then, whether providing an alternative solution to Joshua's puzzle might be used to motivate a different response to the puzzle involving definite descriptions. That said, I'm not sure I have anything useful to say (yet) about how to solve Joshua's puzzle.

BTW, Joshua, I think you should e-mail David about this puzzle, especially if you're thinking about writing up a paper on it.

7:35 PM  
Blogger Joshua said...

Andrew,

Thanks for the comment. I think you and I were having roughly the same idea. My thought was that what Gappy contributes to the proposition is a gap and there is a difference between there being an empty gap in a proposition and there being a gap that is filled by a gap in a proposition. But, I wonder if there are any metaphysical reasons to deny the claim that gaps can fill gaps. Or, more importantly if there are any problems with the claim that a gap can fill itself.

Greg,

I was thinking of the Russell argument when I came up with this argument. I also was wondering if there are interesting relationships between solutions to the two puzzles. I'm glad you pointed out that the direct vs. indirect constituent distinction is not helpful.

8:47 AM  
Blogger avwake said...

Hey Joshua,

I'm wondering about the first option Andy suggested.

How important is the "gap" talk to(GPV)? Suppose I endorse (GPV) and I'm with you until you say this:

"it seems that (GPV) along with the thesis that there are genuinely empty names is committed to gaps."

(GPV) and the thesis that there are empty names commits us to gappy propositions, but not gaps. A gappy proposition is a proposition that is incomplete because it is missing a constituent. Isn't that good enough? Do I have to go on and endorse the existence of gaps?

The holes analogy is different, you might think, because there's plenty we want to say using "holes" talk and its hard to give paraphrases - whatever those are - for it. Is the same true of the "gap" talk?

6:05 AM  
Blogger Joshua said...

Andrew,

I think that Andrew's first suggestion doesn't work because we need to say things like this: The proposition that Venus is bigger than Venus has more gaps then the proposition that Venus exists. It is not enough to quantify over missing constituents instead of gaps because both propositions have an infinite number of missing constituents. The proposition that Venus is bigger than Venus is missing the number 2 and 3 and ... Similarly, the proposition that Venus exists is missing the same constituents.

I just don't see a paraphrase strategy that will work out. But, I guess there may be one that I'm just not thinking of (I have a hard time with paraphrases anyway)

10:32 AM  
Blogger Chris Tillman said...

One could perhaps avoid the reification of gaps by talking about different ways of being an incomplete proposition. If we permit talk of saturation in argument places of properties or relations, we can talk about propositions that result from the failure of one or more argument places to be saturated. This will still be mereologically squicky since presumably 'is identical to' would encode what 'Vulcan is identical to Vulcan' encodes, except that the latter encodes a proposition and the former does not. But people with this sort of view better get used to mereological squickyness anyway (so say me and Greg!)

Another option is to reify gaps but take gaps to *represent* nothing. After all, propositions are supposed to be representational entities. Then we get an easy solution though we have to deny the negative thesis of Millianism.

Presently I prefer the former option.

2:42 PM  
Blogger Joshua said...

Chris,

What is Mereological Squickyness? Is it the denial of extensionality of proper parthood?

I agree with your first comment. Since I responded to Andrew I began to think about the view that propositions are structured trees that have individuals and properties are the nodes. We might be able to retranslate our talk about gaps into talk about branches of the tree that have no individual or property at the end.

I'm not sure I understand the second proposal. How would it help to say that gaps are not representational and what millian thesis would you be denying?

6:58 AM  
Blogger Chris Tillman said...

Mereological Squickyness is any mereological view Lewis dislikes. I'm fine with being squicky.

This is what I had in mind for the second proposal. Hold that there are gaps and that empty names contribute them to propositions. Now, strictly speaking, you're denying that the contribution of a proper name is its referent, if any, or nothing. But maybe that's okay. Now hold that the gap contributed by 'Vulcan' represents nothing. But hold that the name for that gap, 'Gappy', is something that represents a gap (namely, that gap). This seems to handle things but I still prefer the other option.

12:54 PM  

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