A paradox regarding representation

I recently developed a novel paradox involving a variety of representational states and activities, and I am wondering if readers might have any thoughts about my ideas here.  To illustrate the paradox, I first prove that there are certain contingently true propositions that no one can occurrently believe.  Then, I use these conclusions as lemmas in a further proof to derive a contradiction, thus giving us a paradox.  Finally, I show how the general ideas behind the paradox regarding occurrent belief can be extended to a wide range of other types of representational states and activities.

To begin, let W  be the proposition, <Some narwhal weighs more than 1.5 tonnes>, and let R  be the proposition, <No one at any time truly occurrently believes a proposition that entails but is not entailed by W>.

As a point of clarification, I am not saying that R  is the modal proposition, <It is impossible that someone truly occurrently believes a proposition that entails but is not entailed by W>.  Rather, I take it that for any possible world ν, the proposition is true at ν if and only if no one ever truly occurrently believes at ν a proposition that entails but is not entailed by W.  Thus, R  is true at some worlds and false at others.  Moreover, I take it that (W & R) is likewise true at some worlds and false at others.  Specifically, it is true at any world at which W  is true and at which no one ever truly occurrently believes a proposition that entails but is not entailed by W.  As I now show, however, it is not possible that someone truly occurrently believes that (W & R).

Suppose, for the purpose of reductio ad absurdum, that it is possible that someone truly occurrently believes that (W & R).  Thus, there is a possible world µ, and a subject S, such that S truly occurrently believes at µ some proposition that entails but is not entailed by W; for, (W & R) entails but is not entailed by W.  So:

(a)     someone truly occurrently believes at µ a proposition that entails but is not entailed by W.

Furthermore, given our initial assumption, we know that S truly occurrently believes at µ that (W & R).  Thus, both of these claims also hold (since S’s belief is a true belief):

(b)    W  is true at µ; and

(c)     R  is true at µ.

However, given (a), it is clearly false at µ that no one at any time truly occurrently believes a proposition that entails but is not entailed by W, which is to say:

(d)    ~ R is true at µ (since R  is the proposition, <No one at any time truly occurrently believes a proposition that entails but is not entailed by W>).

And given (c) and (d), it follows that (R & ~R) is true at µ.  So, given our initial assumption that it is possible that someone truly occurrently believes that (W & R), it follows that it is possible that (R & ~R).  But since it is not possible that (R & ~R), we know that our initial assumption is false.  That is:

Lemma 1:    It is not possible that someone truly occurrently believes that (W & R).

Now, again let W  be the proposition, <Some narwhal weighs more than 1.5 tonnes>.  And let N  be the proposition, <No one at any time truly occurrently believes a proposition that entails ~but is not entailed by ~W>.  Then, consider how we can run an argument that is similar to the one provided for Lemma 1, but where the new argument shows that it is impossible that someone truly occurrently believes that (~W & N).

Suppose, for the purpose of reductio ad absurdum, that it is possible that someone truly occurrently believes that (~W & N).  Given this, it follows that there is a possible world µ′, and a subject S, such that S truly occurrently believes at µ′ some proposition that entails ~but is not entailed by ~W; for, (~W & N) entails ~but is not entailed by ~W.  So:

(a′)    someone truly occurrently believes at µ′ a proposition that entails ~but is not entailed by ~W.

Furthermore, given our initial assumption, we know that S truly occurrently believes at µ′ that (~W & N).  Thus, both of these claims also hold (since S’s belief is a true belief):

(b′)   ~W  is true at µ′; and

(c′)    N  is true at µ′.

However, given (a′), it is clearly false at µ′ that no one at any time truly occurrently believes a proposition that entails ~but is not entailed by ~W, which is to say:

(d′)   ~is true at µ′ (since N  is the proposition, <No one at any time truly occurrently believes a proposition that entails ~but is not entailed by ~W>).

And given (c′) and (d′), it follows that (N & ~N) is true at µ′.  So, given our initial assumption that it is possible that someone truly occurrently believes that (~W & N), it follows that it is possible that (N & ~N).  But since it is not possible that (N & ~N), we know that our initial assumption is false.  That is:

Lemma 2:    It is not possible that someone truly occurrently believes that (~W & N).

With Lemmas 1 and 2 in mind, we can now provide an argument to derive a contradiction, as follows.  Suppose that at some possible world µ*, only one proposition is ever occurrently believed that entails W  but is not entailed by W.  Namely, Omar occurrently believes at µ* that (W & R).  Similarly, only one proposition is ever occurrently believed at µ* that entails ~but is not entailed by ~W.  Namely, Sarah occurrently believes at µ* that (~W & N).

First, consider Omar’s occurrent belief at µ* that (W & R).  We know that R  is true at µ*.  For, R  is the proposition, <No one at any time truly occurrently believes a proposition that entails but is not entailed by W>.  This is true at µ*, since the only proposition ever occurrently believed at µ* that entails but is not entailed by W is the proposition (W & R); and given this, along with Lemma 1, it follows that there is no proposition ever truly occurrently believed at µ* that entails but is not entailed by W.  So, in sum:

(i)       Omar occurrently believes at µ* that (W & R);

(ii)      Omar does not truly occurrently believe at µ* that (W & R); and

(iii)     R  is true at µ*.

Given (i) – (iii), it follows that W  is false at µ*.  So:

(iv)     ~W  is true at µ*.

Next, consider Sarah’s occurrent belief at µ* that (~W & N).  We know that N  is true at µ*.  For, N  is the proposition, <No one at any time truly occurrently believes a proposition that entails ~but is not entailed by ~W>.  This is true at µ*, since the only proposition ever occurrently believed at µ* that entails ~but is not entailed by ~W is the proposition (~W & N); and given this, along with Lemma 2, it follows that there is no proposition ever truly occurrently believed at µ* that entails ~but is not entailed by ~W.  So, in sum:

(i′)      Sarah occurrently believes at µ* that (~W & N);

(ii′)     Sarah does not truly occurrently believe at µ* that (~W & N); and

(iii′)    N  is true at µ*.

Given (i′) – (iii′), it follows that ~W  is false at µ*.  So:

(iv′)    W  is true at µ*.

Finally, given (iv) and (iv′), it follows that (W & ~W) is true at µ*.  So, it is possible that (W & ~W).  Moreover, one can replace W  with any contingent proposition and run the same type of argument.  Therefore, for any contingent proposition p, one can employ the above line of reasoning to derive the conclusion that it is possible that (p & ~p), thereby giving us the paradox.

Lastly, note that one can likewise create such paradoxes by replacing the notion of occurrent belief with other types of representational states or activities.  For instance, we can run a similar argument to the one just provided by replacing occurrently believing that p with any of the following alternative representational states or activities: writing a sentence that expresses p, uttering a sentence that expresses p, asserting that p, hoping that p, and imagining that p.

Any thoughts from readers will be much appreciated.

6 Comments

  1. Eric Thomson

    My guess is that you will be much more likely to get feedback if you give a summary of the paradox and the gist of the argument (i.e., something like an abstract) before diving in with the W’s, R’s, and N’s. What is the crux of the argument?

    • Hi Eric, thanks for the comment! Yes, I agree that it would have been helpful to provide a clearer summary of the crux of the argument. Part of the issue relates to the fact that I accidentally ran into the paradox while working on a somewhat different topic. And yet, while I couldn’t find any flaws with the steps in my reasoning, the proofs lead to the claim that for any contingent proposition p, it is possible that (p & ~p). Given that I am not inclined to accept this conclusion, I figured that there must be some assumption that I’m making that is incorrect. But I couldn’t figure out what that assumption might be. I mention this because I would want to frame the crux of the paradox in light of any assumption that I was making that might seem contentious. And given that I couldn’t find any such assumption, the only way I could think to summarize the paradox was to merely describe it as a collection of seemingly plausible claims that together result in a contradiction. That said, however, I think that I can now say a bit more about what might be the crux of the paradox. These new thoughts on the crux of the paradox actually developed in light of the comment that Wo made about Prior’s paper. So, perhaps my reply to Wo (which I will give below) will provide a better summary of what might be going on in the paradox.

  2. Your paradox reminds me of cases discussed in the later sections of Arthur Prior’s “On a family of paradoxes”. My hunch is that it’s essentially the same problem, and should have the same response.

    • Hi Wo, thanks so much for the helpful reference. I hadn’t read Prior’s paper until you suggested it, and I certainly agree with you that it’s relevant. I found this quote by Prior to be particularly applicable to my post:

      “So far as I can see, we must just accept the fact that thinking, fearing, etc., because they are attitudes in which we put ourselves in relation to the real world, must from time to time be oddly blocked by factors in that world, and we must just let logic teach us where these blockages will be encountered. . . . If it is a fact that no fact is being assented to in Room 7 at 6, then this fact (that no fact is being assented to, etc.) cannot be being assented to in Room 7 at 6.”

      In essence, I take it that the proofs in my post for Lemmas 1 and 2 are proofs of particular examples related to what Prior is talking about. For instance, given Lemma 1, any world at which (W & R) is true is a world at which (W & R) cannot be occurrently believed. Similarly, given Lemma 2, any world at which (~W & N) is true is a world at which (~W & N) cannot be occurrently believed.

      However, I find the specific examples that I offer interesting in light of the second part of my post. Namely, if we imagine a case in which the only relevant occurrent beliefs are Omar’s occurrent belief that (W & R), and Sarah’s occurrent belief that (~W & N), then it seems that we are led to a contradiction (as shown in the post). Now, someone following Prior might respond as follows. We know that, in the case being considered, both R and N are true (as shown in the post). And since it must be the case that either W or ~W is true, it follows that either (W & R) or (~W & N) is true. Hence, someone following Prior might say that if W is true, then this blocks Omar from being able to occurrently believe that (W & R); for, if Omar were to occurrently believe that (W & R) while W is true, then Omar would truly occurrently believe it (which we know is not possible given Lemma 1). Similarly, if ~W is true, then this blocks Sarah from being able to occurrently believe that (~W & N). Assuming that this is the best way to avoid the contradiction, I still find the result somewhat surprising. For, given that W is an arbitrary contingent proposition that I happened to choose, it suggests that any contingent truth, regardless of whether the truth is about anyone’s particular mental state, is able to block a person’s ability to have certain mental states. This seems to differ from Prior’s example in which the given truth is about people’s mental states; i.e., the “fact that no fact is being assented to in Room 7 at 6”. In the case with Omar, by contrast, the truth that would block Omar’s belief is something quite different; i.e., it is the truth that some narwhal weighs more than 1.5 tonnes. So, in sum, I think that the paradox in my post accords with much of what Prior says. It’s just that the cases that I consider seem to lead to some surprising results when carried forward; e.g., the result that any contingent truth can block a person’s ability to have certain mental states. Does this seem right to you? Thanks again!

  3. Call me naive, but aren’t paradoxes non-extensional? So how does your paradox count as something which contains information about ‘representation’ or ‘the brain’?

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