On Philosophy

September 30, 2007

The Method Of Socrates

Filed under: Metaphilosophy — Peter @ 12:00 am

In my previous two posts I have discussed methods of answering the question “what is justice?” embodied in the proposals of Thrasymachus and Glaucon. Which brings me to the third (technically fourth, if we count the opening appeal to authority) method for answering the question, the method that Socrates uses to produce his answer, and the one that Plato endorses. The methods of Thrasymachus and Glaucon, while not identical to each other, share a number of similarities, including some dependence on empirical reasoning. The method of Socrates is, however, completely different from those that have come before it, Socrates’ approach would, in theory, allow us to uncover what justice is with almost no knowledge of the world in general.

Socrates begins his investigating by considering justice as it appears in an ideal city. Although he recognizes that we are after an understanding of justice on the personal level, in order to decide whether it is good or bad for us, he proposes that “there is a justice of one man, we say, and, I suppose, also of an entire city … Is not the city larger than the man? … Then, perhaps, there would be more justice in the larger object and more easy to apprehend.” (368e) He goes on to construct an ideal city, one that is best suited to our needs (and desires), in great detail. When this is done he then returns, as promised, to look for justice where it appears in this city. He states, “I think our city, if it has been rightly founded, is good in the full sense of the word. … Clearly then it will be wise, brave, sober, and just.” (427e) And given this he goes on to identify wisdom, courage, and sobriety with various aspects of the city. Since the city seems exhausted by these virtues he concludes that justice must be a kind of structural feature, that which leads each virtue to be expressed properly and in harmony with the others. This then is the account of justice at the level of the city that is provided by the method of Socrates. And this account is extended to describe justice at the level of individuals by arguing that each person is divided in much the way the city is, and thus that the just person is one who displays an inner harmony with each aspect of their soul playing its proper role.

Upon a critical examination this method seems questionable, at best. Major leaps in reasoning are made without any justification and none of the other participants in the dialogue call Socrates on them, in fact there is remarkably little disagreement in this entire section of the Republic, as compared to book I. Specifically there are four major questions the method raises: 1) Why assume that justice will be found in the best city? If we don’t know what justice is at the beginning of our investigation then it is quite possible that it is not a virtue but a vice, and thus will be absent in the best city. Now we might suppose that the fact that justice is a virtue has been established in the previous discussion, and that justifies the assumption. But this doesn’t solve the problem for three reasons. First it was established only in discussion with Thrasymachus, which shows only that Thrasymachus’ beliefs commit him to thinking justice is a virtue. Secondly if we suppose that justice is a virtue to answer the question “what is justice?” we are begging the question because we have already made an assumption about what justice is. And thirdly, and most significantly, Socrates himself says “For if I don’t know what the just is, I shall hardly know whether it is a virtue…” (354c) 2) Why would considering an ideal city make us better able to recognize justice in it? Granted if justice had a size then perhaps we could enlarge it by considering a larger example, but it is hard to see how justice will become more or less apparent no matter what we investigate. 3) Why should we conclude that the city is wise, brave, sober, and just if we agree that it is good? Certainly these are traditional virtues, but surely Socrates can’t be appealing to a kind of “common sense” knowledge about what goodness implies, because again it would be to beg the question to an extent. 4) What justifies the identification of wisdom, courage, and sobriety with the parts of the city that Socrates claims they are to be identified with? Certainly the other people present don’t disagree with Socrates’ claims about them, but each the nature of those virtues seems as much open to debate as the nature of justice, so there is no good reason to treat our intuitions about them as correct.

To really understand Socrates’ method and how it is supposed to lead us to the correct understanding of justice we must answer those questions. To do that we must take a quick look at Plato’s epistemology. The foundation of Plato’s theory of knowledge is that the best and truest knowledge involves grasping the unchanging forms, which we can have direct intellectual access to. To grasp the forms we must turn our mind way from “the world of becoming”, in Plato’s terms, in which it is hard to see the forms. Instead we must consider idealizations, and from them abstract even further, until we are dealing with the ideas themselves and can proceed in our thinking on the basis of them alone. As Plato explains it through an analogy with geometry: “they further make use of the visible forms and talk about them, though they are not thinking of them but of those things of which they are a likeness, pursuing their inquiry for the sake of the square as such and the diagonal as such, and not for the sake of the image of it which they draw.” (510c) With this in mind we can answer the four questions raised earlier, and thus provide a logical underpinning for the method of Socrates. Question 1) we can now answer by denying that the virtue of justice was being presupposed at all. The ideal city being envisioned is like the drawings of the geometer, it is to lift our mind above the world and to the idea of the ideal city and thus to the idea of justice. And so if justice was not in fact in the ideal city we would be immediately aware of that fact, and could then proceed to construct the worst possible city in order to find it. Question 2) then has already been answered, we become better judges about justice because our intellect is raised above the world of becoming and into the world of forms where justice actually is. Questions 3) and 4) can be answered in a similar way, it is supposed that by dealing with the ideal city the forms are made apparent to us. Thus we can rely on our judgments about what virtues are present in the good city and what they correspond to in it because our judgments are now motivated by the forms themselves.

But while this does provide a solid foundation for the method, a rationale for why it works, it isn’t a very acceptable one to modern minds. The existence of a world of forms that provides the objects of knowledge and direct intellectual access to it seems questionable to most. So before we evaluate the methods let us see if we can separate the method of Socrates from Plato’s epistemology, in order to put it on a more even footing with that of Thrasymachus and Glaucon. A task I will take up next time.

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September 29, 2007

The Method Of Glaucon

Filed under: Metaphilosophy — Peter @ 12:00 am

Previously I discussed how Thrasymachus investigates what justice is, using a method I described as based on an ordinary language conception of philosophy. Today I am going to discusses the next method through which justice is investigated, that of Glaucon. Not only does Glaucon argue to one of the same conclusions as Thrasymachus, that the unjust are better off than the just, but he uses a method that is surprisingly similar as well. And Glaucon’s argument also seems to have its roots in some kind of ordinary language philosophy, although it proceeds with a different style of empirical investigation. In addition Glaucon, unlike Thrasymachus, is completely uncommitted to the idea that injustice is better, and thus he does not abandon his method midway simply to press for his conclusion.

Glaucon’s investigation of the nature of justice is rather short, most of his speech is concerned with elaborating on the idea that the unjust man is better off than the just man. Thus we can repeat it in its entirety.

“By nature, they say, to commit injustice is a good and to suffer it is an evil, but that the excess of evil in being wronged is greater than the excess of good in doing wrong. So that when men do wrong and are wronged by one another and taste of both, those who lack the power to avoid the one and take the other determine that it is for their profit to make a compact with one another neither to commit nor to suffer injustice; and this is the beginning of legislation and of covenants between men, and that they name the commandment of the law and the lawful and just, and that is the genesis and essential nature of justice-a compromise between the best, to do what is wrong with impunity, and the worst, which is to be wronged and be impotent to get one’s revenge.” (359a)

I would say that the logical order of this investigation is reversed, but it doesn’t really matter, given that it is so concise. The first step, I would maintain, is that which Glaucon puts last, namely the affirmation: “that they name the commandment of the law and the lawful and just”. The reasoning he has just carried out can only be part of a valid method for investigating justice if that which is named just actually is just. And so Glaucon’s method contains an implicit commitment to some form of ordinary language philosophy; if that which is called just is not necessarily just then Glaucon has shown nothing about justice by this argument.

And, as with Thrasymachus’ method, the next step is to empirically investigate just what kinds of things are called just, under the assumption that they share some common features, and that the fact that they are all designated by the same term is not a coincidence. Thrasymachus’ empirical “investigation” searched for similarity in the properties that the laws called just all have, which he concluded was being advantageous to the ruling parties, but which might be better said to be desired by the ruling parties. And this in analogous to the way we might try to discover what water is. By empirically investigating it we find that all water has the property of being composed primarily by a certain molecule and so conclude that that molecule simply is water, just as Thrasymachus concludes that the advantage of the stronger simply is justice. Glaucon, however, proceeds with what we might call a functional or evolutionary empirical investigation. Often when we come upon things created by people there are no immediately obvious properties that they have in common. Consider musical instruments. The things we call musical instruments have little in common with each other. Some are made of metal and some of wood. Some make sound by vibrating strings and some make sound by having air blown through them. Some are large and some are small. Thus we cannot put musical instruments under our investigative microscope and come up with something that they all have in common by investigating them individually. But when investigated in the context of society they do have something in common, namely that they are all used to make music.

Like musical instruments, laws, and thus justice according to the ordinary language approach, are human constructions. And so we might suppose that, like music instruments, they have nothing intrinsic in common, but rather a common purpose for existing. Glaucon attempts to uncover this purpose by considering why laws exist in the first place. To do that he considers what a world without laws would be like (in theory something that can be discovered empirically), and hypothesizes that without any regulation people would do whatever was best for themselves, which would often bring them in conflict with other people (again, an empirical claim about psychology). If we were ever to find ourselves in such a state we would attempt to get out of it by establishing laws, so that overall we are better off both as individuals and as a society. The function and nature of justice then, according to Glaucon’s method, is to regulate individual behavior so that everyone is better off by virtue of avoiding being harmed by others, even though such laws restrict how some would like to pursue their own desires.

Now some (Santas 2004) would consider this a kind of contractarian theory about justice. This does accurately reflect the content of Glaucon’s theory about justice, that it is a way of regulating individuals to overcome their natural tendencies for the benefit of all. However, I do not see any evidence that Glaucon proceeds to this conclusion in the same way that contractualists do, nor does his claim include many of the common contractualist ideas. For example, there is no need to understand Glaucon’s argument as implying that there really ever was a state of nature, Glaucon can be read as explaining the motivations behind laws, not their actual historical origin. Nor do we need assume that there is an explicit social contract or that anyone need agree to it. It could be that justice emerged without anyone intending it to, that groups of people regulated themselves in different ways, and those that regulated themselves in ways we now describe as just thrived and came to dominate.

Unfortunately, as with Thrasymachus, Plato does not really address the method of Glaucon’s investigation. In fact he doesn’t directly address Glaucon’s claims at all, he simply proceeds with his own method for investigating justice. Which leaves us in an interesting position. Clearly Plato will come up with some conception of justice, but just because Plato makes a claim about justice doesn’t mean that Glaucon’s claim is wrong. If they conflict surely one of them must be wrong, and although we might argue that they say much the same thing when it comes to society as a whole they differ greatly about whether it is better for the individual to be just or unjust. To decide which of them is right the method of one or the other (or possibly both) must be shown to be lacking, but this simply isn’t done. And so in a sense the nature of justice is actually left unresolved in the Republic, even though Plato clearly feels that his version is superior.

September 28, 2007

The Method Of Thrasymachus

Filed under: Metaphilosophy — Peter @ 12:00 am

In the Republic Thrasymachus puts forwards what might be the first argument based on an ordinary language conception of philosophy (and is thus a kind of empirical argument well). Obviously this is the exact opposite of the philosophical method used in the majority of the Republic. Most of the Republic proceeds on the assumption that the majority of speakers could easily be wrong in their use of words and, furthermore, that the correct understanding of concepts such as justice can be achieved through philosophical insight stemming from the rational reflection on the concepts achieved when we leave behind our existing biases and prejudices. This method is itself justified by Plato’s metaphysical and epistemic doctrines, since he believes that the mind has direct access to the world of forms, and thus can perceive general truths, such as those about justice, directly. The method of Plato and the method of Thrasymachus couldn’t be less alike, and yet Plato never criticizes Thrasymachus’ method, and in fact Thrasymachus himself is unable to completely adhere to it. And so an opportunity for the two methods to go head to head is missed.

But before I say any more about the method of Thrasymachus let me first discuss the argument he invokes to defend his claim that justice is the advantage of the strongest. First Thrasymachus says who the stronger are (since Socrates has just previously claimed that he cannot understand Thrasymachus specifically on this point). He states that the strong are the ruling party, the person or persons who make the laws. Thrasymachus then goes on to observe that the rulers always create laws that are to their advantage. “a democracy democratic laws and a tyranny autocratic and the others likewise”. (338e) And finally he notes that everywhere this is called justice. This argument can only be considered valid if we accept the key premise behind ordinary language philosophy, that the meaning of words is determined by how the majority of people use those words. And thus if the majority of people use the word “justice” to refer to rules designed for the advantage of the stronger then justice is the advantage of the stronger. Naturally such a method drags with it an empirical investigation as well, which Thrasymachus sums up in his second claim, namely that all (or at least most) people do in fact use the word to designate this thing. Thrasymachus’ empirical claim then is that all ruling parties do in fact make laws to their advantage, a claim which must be falsified or confirmed by actually going out there and looking at what ruling parties do.

Socrates is of course unhappy with this conclusion and so he challenges Thrasymachus. But he does not challenge his method, he does not dispute the claim that justice is whatever the majority of people happen to call just. And he very easily could have. Knowing Socrates he probably would have argued by analogy, and perhaps reasoned that we wouldn’t consider counterfeit money genuine even if the majority of people were fooled by it, and thus that we shouldn’t consider Thrasymachus’ “counterfeit justice” real justice, even if citizens believe it to be such. But Socrates does not do this. Instead he simply points out that surely some rulers error and enact laws not to their advantage. And thus what people call just cannot always be the advantage of the strong. Obviously this criticism could in turn be rejected by an ordinary language philosopher who had refined their method. They might argue that the meaning of a word is what it most often is used to refer to, and thus that as long as the majority of rulers enact laws to their advantage most of the time that Thrasymachus’ conclusion holds. But, on the other hand, Thrasymachus’ claim may not in fact hold, many, probably most, government have a pattern of enacting laws that are not completely to their advantage. And so let us simply accept that there is something wrong with Thrasymachus’ epistemic generalization.

Thrasymachus’ friend Cleitophon is the first to respond to this criticism. He attempts to avoid such objections by reinterpreting Thrasymachus’ original claim, saying that “by the advantage of the superior he meant what the superior supposed to be for his advantage.” (340b) And this could be improved further by adopting a new claim, similar to Thrasymachus’ original, that justice is what the stronger desires. This avoids the possible confusions arising from making a distinction between what is to the stronger’s advantage and what they think is to their advantage. And it has the virtue of allowing for ruling parties, however rare, who may be guided by principles and not purely by their selfish desires.

However Thrasymachus rejects this improvement. Instead he pursues a much stranger course, redefining what a ruler is so that people only count as rulers when they are in fact making laws that actually are to their advantage. In fact he would extend this to all professions, stating that “do you call one who is mistaken about the sick a physician in respect to his mistake or one who goes wrong in his calculation a calculator when he goes wrong and in respect of this error? Yet that is what we say literally – we say that the physician has erred and the calculator and the schoolmaster.” (340d) Here Thrasymachus is abandoning his method, or at least one of the assumptions that underpins it. Previously we were to accept as just whatever most people call just. And yet now Thrasymachus would have us believe that most people can be in error when they call someone a ruler, if they are ruling in a way that is not to their advantage. Socrates does not follow up on this, but clearly we must admit that, given this, Thrasymachus’ argument falls down because of internal contradictions: either he is wrong now or he was wrong earlier, and so we need consider it no further.

This leaves us with two pressing questions: why did Thrasymachus abandon his method and why did Socrates completely overlook the methodological issues? One of my colleagues has suggested that Thrasymachus makes methodological mistakes because his method is not the motivation for his claims. Rather Thrasymachus is committed to the idea that justice is to the advantage of the stronger (perhaps he was channeling Nietzsche) and was simply using whatever tools would allow him to get to that conclusion. And thus Thrasymachus is moved to abandon his method rather than his claim. Perhaps that is part of it, but the Thrasymachus of the Republic is probably at least a bit fictional, surely Plato didn’t have to make him reason in such an illogical way (i.e. prioritizing the conclusion over the premises, which is methodologically backwards because we should always be less certain of the conclusion than we are of the premises, else we should be arguing in the other direction). Maybe Plato simply isn’t sensitive to such methodological issues. Although Thrasymachus uses a different methodology to arrive at his claims perhaps this is an accident, a single argument which has appeared for this theory of justice, but which was never examined from a methodological standpoint because it was the only one of its kind. Or it may be because Plato (and Socrates) simply don’t have a way to deal with the argument on the methodological level, which may also explain why Cleitophon’s modified (and more resilient) claim receives no attention whatsoever. Although Plato has the firm belief that his method works and can lead us to the truth he simply may not have a way to justify it from a methodologically neutral position, or argue against alternate methods. Thus the issue remains untouched even though everything else, from metaphysics to epistemology, seems to receive some time in the Republic.

September 27, 2007

Epistemic Possibility

Filed under: Logic — Peter @ 12:00 am

Last time I discussed the logic of necessity and possibility under basically two interpretations. First we considered the possibility that □φ should be interpreted as claiming that φ was a tautology, or, in other words, that it can be proved without appeal to any additional axioms. This is a very simple way to interpret possibility and is also, in my opinion, very intuitive. The other way discussed to interpret the logic of necessity was in terms of sets of possible worlds. For our almost purely logical purposes it was sufficient to understand a possible world as a set of facts. □φ was then determined simply by considering whether φ held in each possible world belonging to the set of possible worlds.

Although these two approaches to possibility are conceptually very different they have in common a central idea, namely that statements involving claims of necessity or possibility are to have their truth values definitively determined in an objective and purely formal way on the basis of the totality of all facts. This is similar to the way the truth of a claim such as ∀xPx, which is determined to be true or false based on whether P holds of every object under consideration. But despite the fact that □ and ◊ are part of a logical notation we don’t have to interpret them in such a logically cut and dried way. Specifically we can interpret them in a way which is not settled based on the totality of facts, but rather the facts as known from a specific viewpoint. Instead of saying that □φ means that φ is provable without any axioms we can instead understand it as indicating that we possess a proof of φ. And thus since ◊φ = ~□~φ it (◊) indicates that we do not have a proof of ~φ, but is silent about whether we have a proof of φ or not. Or, in more natural terms, ◊φ states that φ is an open possibility, that we have not been able to rule it out yet. I would call such an interpretation epistemic possibility.

In such an interpretation of necessity and possibility many fewer things are necessary and more are simply possible then were under the purely logical interpretation. Of course in the logical interpretation there were a number of sentences (actually an infinite number) for which there was neither proof nor disproof. If G is one of those sentences the only thing we could claim about it would be that ◊G and ◊~G. But for every other φ we could assert □φ or □~φ. However, if we are working with the epistemic possibility interpretation then there only a finite number such φ which we can assert □φ or □~φ of. This follows from the simple fact that as beings of finite capacity we can only have seen a finite number of proofs (even collectively, if we wish to interpret what has been demonstrated as a collective exercise), and since it requires a proof to assert □φ obviously we can only make a finite number of such assertions. And so necessity is clearly much more sparse than it was under the logical possibility or possible worlds interpretations.

This may make epistemic possibility seem aesthetically unattractive. Why should we bother with this interpretation, or try to understand ordinary language claims about necessity and possibility through it? Well one reason might be that it is the right interpretation to understand these claims through, but I will put that aside for the moment. Consider instead how this interpretation can deal with nested possibility quantifiers. In our previous two interpretations nested quantifiers were, for the most part, redundant. If the decision as to whether a statement is necessary or just possible is determined by the complete body of facts then surely that claim about possibility is itself necessarily true, given that we have just shown that it followed definitely from the complete body of facts. However, this is not the case with epistemic possibility. A statement involving epistemic possibility says something about our possession of a proof (and not, it should be noted, whether a proof exists). And thus a claim about the epistemic possibility of such a statement is about whether we have a proof about whether can or can’t have a certain proof. Thus adding such quantifiers genuinely expresses a new idea. Let’s consider each of the four possibilities: □□φ says that we have a proof that we have a proof of φ, which is rather trivial although hard to establish (we must prove that our proof is without error); □◊φ says that we have a proof that we can’t disprove φ (prove ~φ), we might say this about G (and ~G), mentioned earlier, because it states that there is something that we cannot in principle prove about them, even though we aren’t able to make a positive claim; ◊□φ states that we do not have a proof stating that we can’t prove φ, or, in other words, that proving φ hasn’t been ruled out; and, finally, ◊◊φ asserts that we do not have a proof that we can disprove φ, which is to say very little positively except that certain proofs are not off the table yet.

Obviously we wouldn’t expect to come across such nested quantifiers except in the context of proof theory, or possibly other meta-theoretical inquiries. For example, a theory about black holes may rule out the possibility of determining anything about what is happening inside them. Thus we may, on the basis of that, claim that □◊ψ, where ψ is some statement about what is happening inside the black hole. Or, in normal language, that we have demonstrated the impossibility of ruling out certain claims about what is happening inside the black hole (due to our inability to look inside). Nested quantifiers might also occur when we wish to qualify our ability to reason. Let us say that we have deduced ~□φ, that φ cannot be logically demonstrated. But we might still concede that ◊□φ, that we admit that the process we reached that conclusion by was fallible, although in ways we cannot immediately detect, and thus at the level where we are considering our proofs themselves we cannot rule out the possibility that our proof was wrong. Of course such qualifications don’t add anything to our inquiries, they simply concede that we are imperfect.

September 26, 2007

Possibly Necessary

Filed under: Logic — Peter @ 12:00 am

Does it make sense to claim that something is possibly necessary, or necessarily necessary? I think the idea has intuitive appeal, but that it is a logically useless construction unless we are indicating something different with each possibility quantifier, in which case we are equivocating. Of the two options the idea that we are equivocating is probably a better possibility to pursues, if we feel that we absolutely must have the ability to make such statements. But while we might be able to rescue our ability to make claims using such statements I don’t think that the claims made in this way are very interesting, certainly they don’t have the same content as the intuitive understanding of such nested quantifiers.

Let’s begin by taking a look at what necessity means. The simplest way to understand necessity is as a purely logical construct, as a way of indicating which statements must logically be true, which can be true, and which cannot be true. Thus to assert □φ is to state then that φ is a tautology, □~φ is to state that φ is a contradiction, and ◊φ is to state that φ is neither a contradiction or a tautology, and that is truth cannot be determined by appeal to the axioms alone (which perhaps coincides with the idea that φ is an a posteriori truth). If ⇒ means “proves” then this means that ∅ ⇒ φ is the same as □φ, that φ follows from the rules of logic alone, no additional axioms needed. And thus ◊φ is true if and only if there is some consistent set of axioms S such that S ⇒ φ. If we wished to understand this in terms of possible worlds each axiom set would represent all the facts that are true in a particular possible world. Thus if some axiom set proves the sentence it is true in some possible world, and if every axiom set proves the sentence it is true in every possible world.

But when we stack such quantifiers we don’t add anything. Whether ∅ proves φ is not dependant on any axioms, only the rules of logic, which we assume are constant (lest we be unable to reason at all). Which implies that all claims about necessity and possibility are themselves necessary, because they don’t depend on any axioms. But there is no need to take this on faith, we can work through it logically. Assume then that ∅ ⇒ φ. Which means that if we start from ∅ we can produce a proof of φ. But doing that in turn proves that ∅ ⇒ φ. Thus (∅ ⇒ φ) → (∅ ⇒ (∅ ⇒ φ)). To put this back in terms of necessity and possibility this means that □φ → □□φ. Similarly if we can produce a proof of ~φ from ∅ then it follows equally obviously that □~φ → □□~φ. By being unable to prove ~φ from ∅ we can deduce that ◊φ (because if it isn’t a contradiction then {φ} is a consistent set of axioms that proves φ). Which means that ∅ ⇒ ◊φ, or, in other words, that ◊φ → □◊φ. And since this exhausts all the possibilities we must conclude that once we have introduced such a quantifier we can only add further □ quantifiers. ◊□φ and ◊◊φ are thus pointless because you can always assert something stronger. Which is why I say that in strictly logical terms nested quantifiers are useless, you can’t indicate anything more with nested quantifiers than you could with just one application of them.

But this interpretation rules out a posteriori logical necessity, if something is necessary you must be able to deduce it in an a priori fashion, because if you can’t then it must not be necessary after all. Obviously there are some (Kripke) who would argue that such a posteriori necessity does in fact exist. Thus such philosophers must mean something else by necessary than logically necessary. So instead of considering what is logically possible we consider instead an arbitrary set of possible worlds, bounded only by the fact that tautologies are true in each such world and contradictions true in none of them. But other than this the members of the set of worlds are arbitrary (in the sense that we can define the set however we like as long as we abide by those restrictions). Although a fact may be logically possible there may not be a possible world in this set where it is true, or it may be true in every one. In such a framework □φ simply means that φ is true in each world belonging to this set, and ◊φ means that φ is true in at least one world in the set. While I don’t object to such a construction by itself I would object to calling □ defined in terms of it necessity and ◊ possibility. What these quantifiers really denote depends on how the set is defined; how we determine which worlds belong to it. For example, the set may contain all worlds that could result from any set of initial conditions evolved over time according the physical laws we know. And I would call this a set of physical possibilities, and so □ would mean physically necessary and ◊ would mean physically possible. But perhaps this is simply a semantic quibble that arises form the fact that I naturally think of necessity, free of any context, as meaning logically necessary. So to sidestep this debate, let us simply agree to a principle of disambiguation, where we will refrain from referring to unqualified necessity from this point forward, and in terms of quantifiers □p will mean physically necessary, □l will mean logically necessary, and so on.

Let us proceed given this understanding. Consider an arbitrary □S, where S is a set of possible worlds, and □Sφ holds if and only if φ is true in each member of S. Now consider □SSφ. Obviously this holds only if □Sφ is true in each member of S. Here we are faced with a choice. One possibility is that S is defined in the same way in each member of S. This seems quite reasonable to me, we understand S as a kind of super-logical constant and thus it doesn’t vary (for example, the definition of what is physically possible is the same in every physically possible world, the definition of what is metaphysically possible is the same in every metaphysically possible world, and so on). If this is the case then we are back to where we were with logical possibility, each □Sφ implies □SSφ and each ◊Sφ implies □SSφ, thus making such nested quantifiers as redundant as in the case of purely logical necessity.

The more complicated choice to make is to say that S is not held constant within each member of S. In this case just because □Sφ holds doesn’t mean that □Sφ is true in each member of S. Such a situation is not impossible, per se, but now by nesting our quantifier we commit an error of equivocation. Previously we established that the meaning of □S was dependant on the definition of S. Thus when considering whether □Sφ holds within a particular member of S is actually to consider a different fact then □Sφ given our initial definition of S. And so a sentence such as □SSφ is deceptive since the first and second quantifiers are not in fact the same quantifier despite the fact that they are designated with the same notation. In fact the meaning of the second □S isn’t even fixed, a fact that is not reflected at all in our notation.

Now this does not rule out our ability to talk about nested quantification when we allow the definition of the set of possible worlds itself to vary, it just means that we have to be cautious in our reasoning involving such constructions. Which brings me to the assertion that ◊□φ implies □φ, a key claim in a version of a certain famous argument. The problem with that assertion is that either ◊□φ is equivalent to □□φ and thus □φ by definition, and hence the implication demonstrates nothing (if we are working under a definition in which □φ implies □□φ), or the implication cannot be established as true or false, thinking that it is necessarily true is to commit an error of equivocation. Simply consider a set Q containing two worlds: a and b (Q = {a,b}). And let us further stipulate that in a φ is true and Q = {a} and in b φ is false and Q = {b}. ◊QQφ is thus true because □Qφ is true in a. However □Qφ is not true because φ is not true in b. Now we could add further restrictions to the way Q may vary within possible worlds contained in that set and thus force the implication to be true, but the reasoning here points out the fact that we need not be committed to such an implication even if we want to allow meaningful nested possibility quantifiers. And thus an argument that relies on such an implication cannot succeed without showing something much more complicated about the structure of possible possibility and, in addition, arguing that we should even allow such complicated structures (since I for one am inclined to the view that sets of possible worlds should not be allowed to vary within the worlds that belong to them).

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