Intentionality and Formalizability


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Following philosophers like Tim Crane and Uriah Kriegel, let’s call the Problem of Intentionality the problem of motivating the rejection of one of the three propositions in the following inconsistent triad:

1. We think about non-existents
2. One can bear relations only to existents
3. Thinking about is a relation

Part of my interest in the Problem of Intentionality is that a big chunk of the Unicorn Argument involves an acceptance of 1 & 2 and a rejection of 3.

I’ve gotten grief from philosophers like Chase Wrenn and Eric Steinhart about whether the Unicorn can be stated in a formal calculus. Such grief can equally be directed at the Problem of Intentionality. We can motivate such grief by formulating what I’ll call the Steinhart Principle:

Steinhart Principle: A set of propositions exhibits logical properties (e.g., validity, inconsistency) only if there is at least one calculus in which the propositions are jointly formalizable.

I have a worry about the applicability of the Steinhart Principle to either the Unicorn or Intentionality that I would like to raise in terms of what I’ll call the Mandik Principle:

Mandik Principle: The adoption of a formalism is philosophically fruitful only if doing so doesn’t beg (pro or con) the question at hand.

Consider, then, the following challenge: State the Problem of Intentionality in a way that simultaneously respects both the Steinhart Principle and the Mandik Principle.

Can this challenge be met? I haven’t made up my mind one way or another, but here are some reasons for doubting that the challenge can be met.

Consider that meeting this challenge would involve formulating the three propositions in a way that doesn’t require one to assign a particular truth-value to any of them. Now consider proposition #1. It is very difficult to see how to proceed with its formalization without also taking a stand on the truth of 1, 2, or 3. For example…

Suppose that we formulate 1 as
($x)($y)(Px & ~Ey & Txy)
where “($x)” is the existential quantifier, “Px” is “x is a person”, “Ex” is “x exists”, and “Txy” is “x thinks about y”.

Lots of problems arise aside from the fact that one may be squeamish about an existence predicate. In particular, formulating 1 in terms of the two-place “Txy” presumes the truth of proposition 3.

On the other hand, we might try to formulate 1 as
($x)($y)[Px & Tx & ~($z)(Uz)]
where “Ux” is “x is a unicorn” and “Tx” is a predicate we construct by presuming a language of thought and an apparatus of thought-quotation giving us “x is thinking ‘($z)(Uz)’”.

On this formulation lots of problems arise aside from the fact that we are quantifying into the opaque context of thought quotation. In particular, it looks like such a formulation in terms of a one-place thinks predicate presumes the falsity of 3.

Let’s suppose for the sake of conversation that there is no formalization of the Problem of Intentionality that satisfies the Mandik Principle. What, then, is the most appropriate response to the Problem of Intentionality? Rejecting it as a non-problem seems itself to beg genuine philosophical questions.

5 Responses to “Intentionality and Formalizability”

  1. R Brown says:

    Pete, interesting point…how about if we can formalize it one way and not the other, wouldn’t that be some evidence for thinking that the question being begged is being begged in the right way?

  2. I’m more sympathetic to the intuitions behind the Unicorn than it might seem; I’m not sympathetic to the argument itself. The problem with the Unicorn is not a problem with the limits of formalization. It’s far more mundane. Namely, that there are very good reasons to say that the Unicorn is either unsound or invalid. And I think there may be ways to get around all these problems without making Big Scary Noises about the Failure of Logic, the coming Apocalypse, and the End of all Reason.

    Consider Statement 1: “We think about non-existents”. Or “We think about things that don’t exist”. Historically, three strategies have been proposed to deal with this.

    Strategy 1: Deny that we ever think about non-existents. This probably goes all the way back to Parmenides (“Thought and being are the same”). In the Middle Ages, for all x, if you think about x, then x is an essence. Some essences exist, some don’t. Many thinkers, such as Leibniz, would say that for all x, if you think about x, then x is an idea in the divine mind. Every consistent set of properties is instantiated by a divine idea. But God has blessed only some of these with actuality. According to Meinong, for all x, if you think about x, then x subsists. Every consistent set of properties is instantiated by some subsistent thing. Only some exist. Latter day Meinongians like Zalta have worked this out in excruciating detail. The modalists (e.g. Lewis) say that for all x, if you think about x, then x is possible. Every consistent set of properties is instantiated by some possible object. One of the morals of these stories is that (for all x)(for all y)(if x thinks about y, then (there exists some property P)(x has P)). Historically, this seems by far to be the most popular strategy. And it’s pretty hard to beat (though it may be beatable). On this strategy, the Unicorn is unsound. Premise P4 in the paper is false.

    Strategy 2: We think about what transcends existence by way of approximation. I think about the proper class V of all sets; but strictly speaking, that is impossible. V is not the value of any variable. When I think I’m thinking about V, I’m only thinking about some set that’s a member of V. I’m only thinking about V[k] where k is some ordinal. But as I think about V[k] for larger and larger k, the objects of my thoughts more and more closely resemble V. Thus V comes into view more and more clearly. This strategy is used routinely in math. Likewise, I think about God. But, as we all know, God is beyond all comprehension. So we reach out towards God by thinking about things that are more and more like God. This strategy was used routinely in theology (the via positiva). And on this strategy, it remains true that (for all x)(for all y)(if x thinks about y, then (there exists some property P)(x has P)). I don’t see how this helps the Unicorn.

    Strategy 3: Give a Russellian analysis of “We think about non-existents”. A Russellian analysis would likely look like this: (there exists mind M)(there exists idea X)((X is an idea of M) and (there does not exist T)(X represents T)). On this strategy, the Unicorn is invalid (given the formulation in the paper, there’s no way to pass from P4 to P5).

    But there are two other promising strategies:

    Strategy 4: Maybe statement 1 should be reformulated like this: We think about things that lack intrinsic properties. That is, (there exists mind M)(there exists idea X)((M has X) & (there exists Y)((X represents Y) &(there does not exist any intrinsic property P)(Y has P))). In other words, we can think about bare particulars. There is some hope for this strategy. It would require minor revision to the Unicorn.

    Strategy 5: Maybe statement 1 should be reformulated like this: We have ideas for which the representation relation is undefined. This parallels statements like: We have calculi that have only partial interpretations; we have theories that have only partial models. This is perfectly plausible and avoids the absurdity of saying that there exist non-existent things that we think about. I believe something like this is the Way Out.

    Also, I took my road bike out for a great rides yesterday and today.

  3. Chase Wrenn says:

    Just adding $0.02 to Eric’s contribution:

    I’d favor treating 1 as a semantic ascent: Some of our thoughts employ empty terms. That might evaporate the Problem of Intentionality, but I’d see such an evaporation as progress.

  4. Pete Mandik says:


    Thanks for the detailed reply. Strategy 5 also strikes me as the most appealing (of the five. It’s not the greastest thing in the world, though). Strategies 1-4 are for people hell-bent on treating representing as a relation and 5 at least backs off on that a bit.

    My current concern, and what I wanted to convey with this post, is that the various reasons one might have for holding positions such as strategies 1-5 are non-formalized, philosophical reasons. They are the sorts of reasons one might bring to bear in arguing for one formalization or another, not reasons stated in one formalization or another. They may very well consitute good reasons in spite of not occurring in a formalization. And so may the Unicorn.

    Consider, for example, strategy 1, which argues for the undsoundness of the Unicorn based on substantive metaphysical considerations. What’s doing the heavy lifting there isn’t simply the acceptance of a logical calculus–it’s the acceptance of a particular ontology. It involves, among other things, a denial of thoughts about necessarily uninstantiated properties, something that I explicitly address in the paper.

    Anyways, I do appreciate your suggestions and do think they merit further exploration. My main concern here, though, is to defend against any suggestions along the lines of “anyone who knows a little quantified modal logic can immediately shoot down the Unicorn”.

  5. You and the unicorn you rode in on want to avoid all these:

    (1) The Unicorn is absurd. Pete says there exist non-existent objects. Ha ha.

    (2) The Unicorn treats existence as a property or tries to quantify over non-existent objects. It has the same status as Anselm’s argument.

    (3) The Unicorn is unsound. Whenever we appear to think about non-existents, we’re thinking about essences, things that subsist, or non-actual possibilia, or whatever. This approach has always been and still remains very popular.

    (4) The Unicorn is invalid. This is the Russellian way of handling the negation in “We think about things that don’t exist”.

    Saying “We have ideas for which the representation relation is undefined” gets you out of all those traps. Of course, the challenge will still be to find some x for which the representation relation is undefined.

    How about mythical beasts, golden mountains, present kings of France? Many will say that the representation relation is defined for those — they’re non-actual possibilia. Given such things, the power of the Unicorn seems to vanish.

    How about things that existed in the past or will exist in the future? Ah, but the eternalists & other 4-dimensionalists will say that such things exist.

    How about mathematical entities? Well, platonists or other math realists will just say that math terms represent real objects.

    How about things like round squares and other inconsistent objects? Maybe, but it won’t interest anyone that there are problems representing such things.

    Can you find any idea at all for which the representation relation is undefined? Perhaps ideas like “truth” and “existence” are such ideas (other candidates are still God and V). But they aren’t likely to deliver the punch in the Unicorn.

    If you can’t, you’ll have to conditionalize the argument to an ontology: If the following theory of existence is true, then the Unicorn is true. Example: If strict actualism is true, then the Unicorn can be formulated like this…. But then your phil mind depends entirely on some controversial theory of metaphysics.

    So you might try to handle it like this: For any theory of existence, there is some term in the theory for which the representation (or reference) relation is undefined. In set theory, the term “set” has no referent. There is no set of all sets. There is a class (a proper class) of all sets. And thus in class theory, the term “class” has no referent. For there is no class of all classes. Such are formulations of the Unicorn that would be of interest to logicians; but probably not in philosophy of mind.