Is desire closed under logical equivalence?

Suppose some person P desires a statement S to hold. Also, S is logically equivalent to S'. Does this mean that P desires S' also? Basically, is desire closed under logical equivalence?

• You need to distinguish between P desiring S' and P knowing that he desires S'. Apr 18, 2023 at 18:13
• "Logically equivalent" implies an abstraction. But desire is concrete, so, desires might not be "equivalent". Example: (1) "100% is water". A logical equivalent is (2) "0% is not water". But, while (1) is ok to wash my hands or drink, (2) is not ok (pure oil meets the condition). Apr 18, 2023 at 18:37
• @RodolfoAP No, pure oil is 0%-water, not 0% not-water.
– J.G.
Apr 20, 2023 at 6:47
• @J.G. False dichotomy. A negative affirmation does not imply that any other complement is positive. Pure oil can be "0% not water", because the rest (100%) can also be not water. Apr 20, 2023 at 10:04
• @RodolfoAP If you're using the numbers as bounds, not exact values, you've misidentified what's logically in/equivalent to what. It goes without saying, of course, 100% oil isn't 100% water. What you've actually shown is at-most-100%-water is equivalent to at-least-0%-not-water.
– J.G.
Apr 20, 2023 at 13:02

No. Firstly, a person may not know that S and S' are logically equivalent, in which case the desires may fail to agree quite straightforwardly. If you amended the question to: is desire closed under known logical equivalence, it is still unlikely to hold.

For one thing, considerations of hyperintensionality apply. Propositions may be necessarily equivalent but still mean different things and have different truth conditions. In such cases, a person might desire a proposition to be true but not some other that is equivalent to it.

For another, humans have a tendency to treat something differently depending on exactly how it is described. Suppose I enter a competition and I am uncertain of whether I shall win. It may be that I desire to have a 90% chance of winning, but not that I desire to have a 10% chance of losing.

More generally, desires can themselves be inconsistent. I may desire A and also desire B, even though I know it is quite impossible to have both. I desire to marry Jane and I desire to marry Betty. I know I cannot do both, but that does not make the desires go away. If my desires are contradictory, and if we construe logical equivalence along classical lines, then all contradictions are logically equivalent to one another, so closure would have the consequence that I desire all contradictions. Perhaps it is even more complex than that, since I may desire to marry Jane and also to marry Betty even though I have no desire to marry both.

• I don't think your last example shows that desire is not closed under logical equivalence. Rather it shows that it is not closed under logical implication: I desire to marry Jane. Marrying Jane implies not marrying Betty. But I have no desire to not marry Betty. Apr 19, 2023 at 7:17
• True. Though we might say that being married to Jane, together with being married to Betty is logically equivalent to being married to Jane and Betty. I desire being married to Jane and also being married to Betty but I do not desire to be married to both. Apr 19, 2023 at 9:30
• No, I don't think that you desire "being married to Jane and also being married to Betty". You desire being married to Jane and you desire being married to Betty. Apr 19, 2023 at 9:33
• `It may be that I desire to have a 90% chance of winning, but not that I desire to have a 10% chance of losing.` Just a nitpick, you would not want to have a 90% chance of winning, but a >= 90% chance of winning. And this is perfectly fine if you invert it (i.e., desiring a <= 10% of loosing would be perfectly fine as far as I can tell).
– AnoE
Apr 19, 2023 at 12:37
• @Squala Maybe in this case, but you could construct an example where the implication holds in both directions. Desires are just not subject to rational constraints. I've known mathematicians say they wished some theorem were false. One might for example desire that Gödel's incompleteness theorems were false. But the object of that desire is itself necessarily false. And as I say in the post, under classical logic all contradictions are logically equivalent, so under closure this would be equivalent to desiring all contradictions. I may desire some contradictions to be true and not others. Apr 19, 2023 at 14:42

In the strictest sense: no.

There's three questions here, with varying degrees of justification required:

1. Can I desire S without desiring S'?
2. Can I desire S and desire not-S'?
3. Can I desire S and desire not-S' while knowing that S logically leads to S'?

1 seems very reasonable. I'm assuming desiring something is an active process whereby an individual wants S. I may be unaware of S' or just haven't thought about S' or I want S because it is inherently good and the other stuff that leads from S is inconsequential to me.

2 seems similarly reasonable though to a lesser degree. If I am bad at reasoning, or if the equivalence depends on a complex relationship then maybe I am just unaware that S and S' are logically equivalent.

3 is a bit tougher. There's no a priori reason to assume that people's desires are logically consistent. So, it's not strictly impossible the way logical contradictions are. However, it's really tense. I'd say maybe Orwellian doublethink is an example of 3, but that's only possible under extreme conditions.

BUT at its core, your question seems to be about whether desire is closed in an absolute logical sense, and the answer there would be no. There's no logical contradiction in any of those constructions.

So, how strict are we about defining "logical equivalence" here? Whether the answer is "yes" or "no" depends entirely on that.

If what we mean is just "S' holds in all circumstances S does, and vice versa", then empirically the answer is probably "no": wishful thinking, ignorance, and bias can all turn the same facts into different conclusions. That is, P may or may not be aware that S and S' are logically equivalent (for instance, they may never have considered S' enough to form an opinion), and if they're not, then they can quite easily desire one without the other.

You can still salvage logical equivalence somewhat by postulating that most things we think of as S and S' are not actually logically equivalent, just very similar: think "it'll be picnic weather tomorrow" vs "it'll be warm and sunny tomorrow".

Let A a 10-page long logical formula (or the corresponding sentence in English if you prefer) which is logically true but verifying it is beyond your or anyone else's capabilities.

You, presumably, desire it to be true that you find a dollar on the street tomorrow.

You, presumably, do not desire it to be true that if A holds, then you find a dollar on the street tomorrow, and if A does not hold, then your parents will get shot in the head tomorrow. This is because you do not know whether or not A is true.

(Sorry for the gruesome example, but the point is to make it very clear that you really do not desire this to be true.)

But the two sentences, "I will find a dollar on the street tomorrow" and "If A, then I will find a dollar on the street tomorrow, and if not A, then my parents will get shot in the head tomorrow" are logically equivalent.

Therefore, what you desire is not closed under logical equivalence.

A better question would be if it is closed under known logical equivalence.

To supplement Bumble's answer which invokes hyperintensionality which describes a property of logic, let's introduce a property to describe people: irrationality.

Is desire closed under logical equivalence?

Because human beings, while rational creatures (can use logic), often have irrational beliefs (often fail to do so in specific circumstances creating contradictions). According to the substantivist position of practical rationality, page 59 of the Oxford Handbook of Rationality (GB):

An agent can be open to have a desire whether or not the agent can rationally reach this desire from the beliefs and desires that he or she has.

Let us say that a person desires to be a marathon runner. Now, to be a marathon runner, one must train the body to get from short distances of a mile or 2, until one can run 26.2 miles. Thus, one can be a marathon runner if and only if one trains to run distances over 25 miles. So, a marathon runner is someone who trains to run distances over 25 miles by metaphysical necessity. Yet, a person who desires to be a marathon runner may also desire to avoid the costs to run over 25 miles. That is, they desire not to run distances over 25 miles because they desire to not run distances over a mile. (I suspect this is a great number of people who desire to attain a goal and desire to avoid costs generally!) So:

P desires S (Person desires to be a marathon runner.)
S entails S' (A marathon runner is someone who trains to run more than 25 miles.)
P desires not S' (Person desires not to run more than 25 miles.)

Thus, it's is fair to rationally criticize person P for desiring S and not desiring S'. According to substantivism, person P should desire S' since it can be logically seen as being entailed by the desire for S.

It turns out (shock!) that Homo economicus is a fictional entity, and that actual people fall short of being rational engines. As such, alternative descriptions of rationality include bounded rationality (SEP) and instrumental rationality (SEP). Human beings simply often fail to see and heed the logics of rationality. From a naturalistic perspective of epistemological matters, one strong reason for that are cognitive biases.

• This answer also shows that desire is not closed under logical implication. It does not say anything about logical equivalence. Apr 19, 2023 at 7:28
• @Squala I would suggest that in the claim "A marathon runner is someone who trains to run more than 25 miles" the copula functions as natural language version of logical equivalence. The purpose and function of a real definition is to assert a factual claim of equivalence about two distinct things. In this case there are "marathon runners" and "runners who run more than 25 miles", and that one is the first IFF one is the second. It's that IFF that makes the real definition an assertion of logical equivalence, no?
– J D
Apr 19, 2023 at 12:53
• Sure, but this is something else then "someone who runs several miles", i.e. more than 1 mile Apr 19, 2023 at 12:56
• @Squala Ah, and the game of the fuzzy logic of language categories begins! :D Point granted!
– J D
Apr 19, 2023 at 12:58
• To be honest it is a bit confusing because you use 26.2 miles, over 25 miles, 2 miles and over 1 mile in your answer Apr 19, 2023 at 12:58

Several answers here thoroughly examine that desire is not closed under logical implication. But it is not only possible to show that desire is not closed under logical equivalence, it is in fact possible to hold directly contradictory desires simultaneously.

There is presently in the next room a Twix bar which I have every right to eat if I choose to. I simultaneously want to eat the Twix and do not want to eat the Twix.

I directly want S and not-S at the same time.

Both desires can be supported at least to a degree rationally. I want S, to eat the Twix, because I am slightly hungry and I know it will taste good. I want not-S, to not eat the Twix, because I am trying to lose weight.

The reason I hold both desires is because they carry different outcomes, but that does not change the fact that I directly hold contradictory desires at the same time without any question about understanding this is a contradiction.

If it is possible to hold contradictory desires simultaneously, then desire is clearly not closed under logical equivalence and in fact does not obey first order logic at all.

• +1 Welcome! Critical hit on this roll.
– J D
Apr 21, 2023 at 2:20
• @Jason Ordendorff thank you for the edit. That is an improvement. Apr 21, 2023 at 18:35

Some of the answers show that desire is not closed under logical implication. However, this does not show that desire is not closed under logical equivalence. In this answer I will describe how I think a rational person could think about 'desire' (written from the first person even though I do not claim to be rational).

The world can develop itself in many ways. For each of the ways the world can develop itself, I will have a certain amount of satisfaction. For a statement X, saying 'I desire X' means that, from my estimate, the occurrence of X is positively correlated with my satisfaction. I.e., given X my estimated satisfaction will be higher than given not-X.

Now this model of desire is necessarily closed under all logical equivalences that I know of. It is not closed under logical implications. In the example in Bumble's answer, my safisfaction would be high in case I am married to Betty or Jane. Since marrying to Jane correlates positively with being married to Betty or Jane, I desire to be married to Jane. Similarly I desire to be married to Betty.

In the example of J D, my satisfaction may be very high if I can run a marathon, but very low if I run several miles but not a marathon. Then I desire to run a marathon while simultaneously not desiring to run several miles. This shows that this pattern is not a sign of irrationality at all.

Most of my desires are for counterfactual states of affairs: "I wish I had a sandwich." / "I wish I had not sent that email." / "I wish Jude liked me better." These correspond to statements S that are flatly false: "I have a sandwich", etc. Well, all false statements are logically equivalent, but I do not equally desire all counterfactuals.

I think the question assumes an unworkable view of desires as selecting from fully-fleshed-out, plausible hypothetical worlds. Reflecting on my own desires, I suspect they are as senseless and fragmentary as dreams until I have taken the mental effort to trim and square them and have them make some sort of sense. I'm not sure it could be otherwise. What are desires for? If a desire is ever to serve as the impetus for thinking or planning, the desire has to come first, and logic after.

Two cents.

Assuming the statements are not identical ie S =/= S', then equivalence necessarily means modulo some differences between the two statements that would make them equivalent under some conditions, ie S ~ S' (modulo C).

But desiring S might depend on the extra conditions that make S different from S' and not on the other conditions that make it equivalent.

So, in general, no, it is not necessary that desiring S means desiring S' as well (assuming all this logic/desire makes sense).

• The reason there's a term "logically equivalent" is that there are different ways of stating things, and it can be far from obvious that they are equivalent. Apr 22, 2023 at 0:02
• That is, certainly C can be empty. Alternatively, I think S and S' can both be read as implicitly including all of the subject's taken-for-granted desiderata, plenty to draw a substantial C from. Apr 22, 2023 at 0:03
• @JasonOrendorff C can be empty but then S and S' would be identical. In any case C may not be empty and desire may depend on C as well, that is my point. Apr 22, 2023 at 5:39
• @JasonOrendorff for example equivalence between S and S' would mean some terms T in S can be translated into terms T' in S', but this translation means conditions C under which T "can be seen as" T', which may not be the case. Apr 22, 2023 at 5:43
• Let me revise my position. "Logical equivalence" is when the only context is the rules of logic. For example, suppose S is "I have a pen and some paper" and S' is "I have some paper and a pen". I mean these to be logically equivalent, but not identical. How do you see them? Apr 27, 2023 at 14:26