5 Jan 2015

Priest, (1) ‘Dialectic and Dialetheic’, section 1, “Why It Is Necessary to Argue This”

 

by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own, unless otherwise indicated.]




Graham Priest


“Dialectic and Dialetheic”


1 Why It Is Necessary to Argue This



Brief Summary:

Many scholars argue that Marx’s and Hegel’s dialectics involve a non-logical notion of contradiction or that contradiction is conceptual and does not obtain in reality. Priest, however, will argue that the logical sense of contradiction is fundamental to their philosophies of dialectic.




Summary

 

 

[Dialetheism is the philosophical position that true contradictions exist.] It would seem at first uncontroversial to say that Marx’s and Hegel’s dialectics were dialetheic, given their apparent contradictory structures. Priest offers some quotations from Hegel’s Science of Logic to exemplify this.


… common experience … says that … there is a host of contradictory things, contradictory arrangements, whose contradiction exists not merely in external reflection, but in themselves. (440.)

External sensuous motion is contradiction’s. immediate existence. Something moves, not because at one moment it is here and at another there, but because at one and the same moment it is here and not here, because in this “here,” it at once is and is not. [Hegel, cited in Priest 389. Italics in the original. Priest says this comes a few lines after the prior one cited at page 440. From the bibliography: Hegel, G. W. F. 1969 (1812). The Science of Logic. London: Allen and Unwin]


So it seems clear that Hegel is a dialetheist. However, many scholars do not take this interpretation. “many, if not most, interpretations of Hegel assert that where Hegel talks of contradiction, and even asserts one,· he must be understood as meaning something else” (389). Priest cites some examples. Acton seems to say that Hegel did not mean contradiction in its normal formal logical sense. Marxist philosophers such as Cornforth, also, instead of understanding contradiction logically, seem to think of it more in terms of forces acting against one another [which does not present a logical self-contradiction, but rather a conflict between separate entities]. Norman sees the contradiction as “the interdependence of opposed concepts” [rather than one term and its negation both being affirmed.'] Self conflict is merely a conflict internal to one political body [and thus still a conflict not between on things an itself but rather between two parts of one thing.] (390) Also very interesting is the notion of contradiction in the history of Soviet thought. Up until the early 1950s, contradiction meant a number of things, “including opposing· tendencies, diametrically opposed concepts, and logical contradictions,” and even logical contradiction was seen as something that obtains in reality. However, after the early 1950s, contradiction may be something conceptual but not something occurring in reality. Nonetheless, Hegel and Marx do use the notion of contradiction in its logical sense, and for them contradictions do obtain in reality. Priest will argue that it is this logic sense that is most primary in their conceptions of dialectic, and these other senses of contradiction that we discussed above are derivative of that logical sense.

Now, while there are certainly examples of Hegel and Marx using the notion of contradiction in other than its logical sense, to insist that they never meant what they said literally when they claimed that contradictions occur in reality, or even when they asserted - contradictions, inflicts such violence on their dialectics that the distorted product is but a pale shadow of its proper self. For the central theoretical notion of contradiction in Marx and Hegel is precisely-the logical one. Other uses are derivative, and usually derive their significance from the central notion.
(Priest 391)



Citations from:

Priest, Graham. ‘Dialectic and Dialetheiç’. Science & Society, 1989/1990, 53 (4) 388–415.


 



 

 

 

 

 

 

3 Jan 2015

Priest, (Intro) ‘Dialectic and Dialetheic’, Introduction, “Dialectics Requires Dialetheism”

 

by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own, unless otherwise indicated.]




Graham Priest


“Dialectic and Dialetheic”


Introduction: Dialectics Requires Dialetheism



Brief Summary:

Priest will argue that Hegel’s and Marx’s dialectics were based on dialetheia, that is, on true contradiction.



Summary


Priest will argue that there is an intimate link between dialectics and dialetheism. For dialectics, Priest will focus on Hegel’s and Marx’s uses of it. Dialetheism is the philosophical position that there are true contradictions. Dialetheic philosophy is especially interesting in logic, since it challenges the orthadoxy of classical logic, which for some reason, still seems to hold sway even today. [I personally believe this debate to be the most exciting and the most important controversies in philosophy today. It also has the potential, I think, to make logic interesting for a wider range of philosophers.]

A dialetheia is a true contradiction, where “contradiction” has its ordinary, logical, sense. Thus, a dialetheia is a true statement of the form A&~A. Dialetheism is, consequently, the view that there are true contradictions.
(388)

Priest will show that Marx’s and Hegel’s dialectics are based on dialetheism. Dialetheic logic is especially good for dealing with paradoxes of self-reference. And Hegel dealt with such philosophical matters of self-reference as thought thinking themselves and categories applying to themselves. Thus it should not be too much of a surprise that he is a dialetheist. (388)





Citations from:

Priest, Graham. ‘Dialectic and Dialetheiç’. Science & Society, 1989/1990, 53 (4) 388–415.




 

 

Priest, “Dialectic and Dialetheic”, entry directory

 

by Corry Shores
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Graham Priest


“Dialectic and Dialetheic”


Introduction: Dialectics Requires Dialetheism


1 Why It Is Necessary to Argue This




Priest, Graham. ‘Dialectic and Dialetheiç’. Science & Society, 1989/1990, 53 (4) 388–415.



2 Jan 2015

Priest (4.7) In Contradiction, ‘Truth or Falsity: Truth Value Gaps’, summary

 

by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own.]



Graham Priest


In Contradiction:
A Study of the Transconsistent


Part II. Dialetheic Logical Theory

Ch.4. Truth or Falsity


4.7 Truth or Falsity: Truth Value Gaps



Brief Summary:

Priest gives his reasons for thinking that there are no truth value gaps, that is to say, that there are no sentences which are neither true nor false. Some hope that by designating the liar sentence as valueless that this will do away with liar-like inconsistencies. However, Priest in section 1.3 showed that even if we assumed there to be value-gaps, we still can have the problem of the liar paradox. Now in this section Priest goes further to argue that there can be no truth gaps anyway. He refutes the arguments for value-gaps one-by-one, with a special focus here on sentences which fail to refer, such as ‘the King of France is bald.’ Some value-gappers say that this sentence is neither true nor false, since there is no state of affairs, no Fact, which could confirm or deny it. Priest shows that while this might be true, that very lack of a Fact is itself a Fact to which its negation refers. So while ‘the King of France is bald’ may be valueless, we know that ‘it is not the case that the King of France is bald’ is true, since we know there is no Fact to confirm its affirmative form.




Summary


Previously Priest was discussing truth. Now he turns to falsity. He defines falsity in terms of truth and negation. [First recall our conventions for the Tarski (T) scheme. Tarski uses quotations around a sentence to mean the name for that sentence. For example:

“snow is white” is true if and only if snow is white

Now, we can abbreviate sentence formulations to such letters as α, and we can use underlining instead of quotation marks and F and T as the formulations “is true” and “is false”.]

We will say that a sentence, α, is false, Fα, just if its negation is true. We might write this thus:

FαT¬α
(Priest, 64)

As we can see from the formulation, we are using negation to define falsity (a proposition is false if its negation is true). So we might now ask, what is negation? Priest admits he does not have a definition to offer. At best we have a circular definition: “Negation is that sentential function which turns a true sentence into a false one, and vice versa.” [This is circular of course because negation is defined as what changes a true sentence to a false one, and a false to a true; however, we already defined falsity as the value given to the negation of a true sentence. We understand negation as what generates falsity, but we understand falsity as what is generated by negation.] But this is a problem in classical logic as well. “Orthodox truth-tables define negation in terms of truth and falsity. But falsity can be defined only in terms of, or by using, negation.” (64)


So, we cannot define negation. However, we can still say certain intelligible things about it. For example, we can say things about the conditions “under which a negated sentence is true.” (64) [In the following we are perhaps assuming that because there are only two truth values, if something is not true, then it is false. Or put another way, if something is not true, then its negation is true.]

a sufficient condition for the truth of a negated sentence, ¬α, is the failure of the truth of α. In other words, if a sentence is not true, it is false:

¬TαFα  
(Priest, 64)

Priest then continues by explaining why there can only be two truth values, true and false. [Recall that Value-gappists argue that some sentences are neither true nor false. He will use an analogy.  His basic idea seems to be that when you assert something, you cannot be neither right nor wrong. In two person games, you can have a draw, when neither player achieves their goal, and thus neither one is winner; but, since neither player won some advantage over the other, neither player is loser as well. But asserting claims does not have this two person dynamic. In one person games, there is not another player whose success makes you fail and whose failure makes you win. So there is not a second player, and thus it is not the case that if she does not win and you do not win, neither one wins and neither fails. Rather, in one person games, you either achieve your aim or you do not, you either win or lose. This is similarly the case when asserting claims. There is just one ‘player’, the claim or claimer, and it either achieves its ‘aim’ of stating the truth, or it does not. You cannot have a ‘tie’ or ‘draw’ if there is only one ‘player’.]

This fact about falsity follows from the analysis of truth we have just had. To speak truly is to succeed in a certain activity. And in the context of asserting, anything less than success is failure. There is no question of falling into some limbo between the two. To use the game analogy again, a draw is possible in a two-player game, for neither player may achieve his end. In a one-player game either the player achieves his end or he does not: there is no third possibility. Asserting is a one-player game. The point, again, is Dummett’s. As he puts it,19 [the following is block quotation of Dummett] |

A statement, so long as it is not ambiguous or vague, divides all states of affairs into just two classes. For a given state of affairs, either the statement is used in such a way that a man who asserted it but envisaged that state of affairs as a possibility would be held to have spoken misleadingly, or the assertion of the statement would not be taken as expressing the speaker’s exclusion of that possibility. If a state of affairs of the first kind obtains, the statement is false; if all actual states of affairs are of the second kind, it is true.
(64-65)
[Footnote 19, quoting, “Dummett (1959a), p. 8 of reprint. Italics original.” From the bibliography:
Dummett, M. (1959a) ‘Truth’, Proceedings of the Aristotelian Society 59, 141–62. Reprinted in Dummett (1978).
Dummet, M. (1978) Truth and Other Enigmas, Duckworth.
(305-306)]


Some people do, on the contrary, argue that there are truth value gaps, “that is, a limbo between truth and falsity” (65). Priest will now explain why their arguments are faulty. Priest first notes that he, in section 1.3, already showed how the arguments for the valuelessness of the liar sentences did not eliminate the problem. The next example he gives is “Aristotle’s argument in De Interpretatione, chapter 9, concerning future contingents” (65). He does not here address it, because its lack of cogency is well established, and he cites Susan Haack’s treatment in her Deviant Logic: Some Philosophical Issues. [We will just look at her formulation of it for now:

(1) If every future tense sentence is either true or false, then, of each pair consisting of a future tense sentence and its denial, one must be true, the other false.
(2) If, of each pair consisting of a future tense sentence and its denial, one must be true, the other false, then, everything that happens, happens 'of necessity'.
(3) But not everything that happens, happens of necessity; some events are contingent.
∴ (4) Not every future tense sentence is true or false.
Clearly, this argument is a valid one. But, equally clearly, Aristotle's arguements for the premisses, particularly (2), need examination.
(Haack p74)

Although Priest does not here examine the argument, he does however discuss some of it in his Logic: A Very Short Introduction. There he uses modal logic to show the problem we find in step 2. See pages 39-46.] And, the other arguments for truth gaps

appear to be a motley crew concerning non-denoting terms and other kinds of ‘‘presupposition failure’’; category mistakes and other ‘‘nonsense’’; sentences undecidable by the appropriate mathematical or empirical techniques; and so on. (65)

Yet despite this variety, they share a similar rationale, which Priest will now outline. [The basic idea here seems to be the following. Consider ‘the King of France is bald.’ Its meaning is clear. But it cannot be true or false, since it fails to refer to any real state of affairs which could confirm or deny it. Thus some sentences are undecidable for this reason.]

The correspondence theory of truth may not be correct, but it captures an important insight concerning truth: for something to be true, there must be something in the world which makes it so. This need not be a state of affairs as traditionally conceived of by correspondence theorists. It might, in the case of a mathematical truth for example, be our possession (in principle) of a proof. In the case of a statement of legal right, it might be certain activities of a legislature. But there must be something, some Fact, such that if (counterfactually) it did not hold, the sentence would not be true. The rationale can now be stated simply thus: for certain sentences, α, there is no Fact which makes ¬α true, neither is there a Fact which makes ¬α true. For example, in the case of reference failure, there is no state of affairs which is either the King of France’s being bald, or his not being bald. For the case of undecidable empirical sentences, there is no possible experiment which would verify either that a particle has a certain momentum, or that it does not have it. And so on.
(65)


There is a general reason why this argument fails. [So again, the problem these value-gappers seem to have is that there is no state of affairs which can prove a claim like the King of France is bald. However, what about the claim, it is not the case that the King of France is Bald? What would need to happen for this claim to be true? We are assuming that making something true requires some Fact which affirms it as such. If there is no reference, like there is no King of France, then there is no Fact to make it true or not true, and thus it is valueless.

The lack of a Fact means we cannot affirm it. So we might say there is this Fact: we cannot affirm ‘the King of France’ is bald, since no Fact exists to affirm it. That then is the fact which would affirm the sentence:

it is not the case that the King of France is bald.

This insight here seems to be that we can say non-referring sentence are valueless, but their negations are not, since their lack of evidence confirms that they are not true and thus that their negations are true. Please interpret the following for yourself for a better reading.]

there is a general reason why this argument fails. In a nutshell, if there is no Fact that makes α true, there is a Fact that makes ¬α true, viz. the Fact that there is no Fact that makes α true. Less cryptically, the point is this. Suppose that α is a sentence, and suppose that there is nothing in the world in virtue of which α is true—no fact, no proof, no | experimental test. Then this is the Fact in virtue of which ¬α is true. We may not know that this Fact obtains, but this is irrelevant. And we might be able to distinguish between different kinds of Fact which make ¬α true. For example, in the case of denotation failure, we might distinguish between the case where ‘John’s brother is a butcher’ is false because John has no brother, and that where it is false because he has a brother who is a French-polisher. But this is not a significant difference as far as truth and falsity simpliciter go.
(65-66)

[Next Priest addresses an intuitionist reply. I cannot explain this section, given my current unfamiliarity with the topic. Priest explains intuitionism in his book Introduction to Non-Classical Logic, chapters 6 and 20. Perhaps the main idea of this paragraph the following. Intuitionist logic does not have a strong principle of excluded middle. Priest, after examining the proof conditions for sentences in intuitionist logic, writes in this other book:

Note that these conditions fail to verify a number of standard logical principles – most notoriously, some instances of the law of excluded middle: A ∨ ⇁A.
(Priest, Introduction to Non-Classical Logic, 104)

However, as we saw in section 1.3, Priest uses the principle of excluded middle to show why the value-gap argument does not work. Please read the following to interpret it properly for yourself.]

There is one important reply here: the intuitionist one. It may be argued that the point that we cannot, in general, recognise when α fails is important. For Facts of this kind cannot play the required semantic role. This is, I think, incorrect. However, to discuss this issue here would take us too far away from the central theme of the book, and so I will not do so. In view of my rejection of the intuitionist claim and my consequent endorsement of the law of excluded middle and related principles, the position I am advocating might be called ‘‘classical dialetheism’’. It would be equally possible to have an ‘‘intuitionist dialetheism’’, which took a constructive stance on negation (so that a proof of the impossibility of a proof of α was required for the truth of ¬α) and the other logical constants. (We noted in section 1.3 that the proofs of many logical paradoxes do not require the law of excluded middle or other intuitionistically invalid principles.) The paradoxical features of intuitionist implication, such as ¬α⊃(α⊃(β), could not be incorporated. But these have always been dubious features of intuitionism anyway.
(66)


[Thus given the fact that classic dialetheism makes use of the principle of excluded middle:]

if α is any atomic sentence of a kind whose members have been proposed as truth valueless, ¬α is true. Thus, ‘Julius Caesar is not a prime number’, ‘The man next door does not have a television set’ (when there is no man next door), and so on are simply true.
(66)

While these sentences may seem strange, Priest describes some situations where they would be appropriate.


[I am uncertain of the details in the final paragraph, partly because he makes reference to section 4.3, which as of this time I have not summarized. At any rate, he discusses a sentence we saw already in 1.3, namely, This sentence is true. In section 1.3, we noted that it would be a good candidate for a value-gap. Here Priest is saying that the sentence is false and its negation is true. This seems to be because there is no fact which could make it true, and as we said already before in this chapter, that makes it false.]

As a final application of the position, let us return to the example given in section 4.3 of the sentence 0; in effect, ‘This sentence is true’. We saw there that the truth conditions of this sentence imply neither the truth of this sentence nor its falsity. There is therefore no question of an a priori proof (or refutation) of it. By its nature, this is the only kind of Fact which could make it true. No experiment is going to decide the issue. Hence, by the previous discussion, this sentence is simply false and its negation is true.
(66)




Citations from:

Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Clarendon/Oxford University, 2006 [first published 1987].


Otherwise if indicated, from:

Priest, Graham. An Introduction to Non-Classical Logic: From If to Is. Cambridge: Cambridge University, 2001/2008.



Haack, Susan. Deviant Logic: Some Philosophical Issues. Cambridge: Cambridge University, 1974.

 

Priest (1.3) In Contradiction, ‘Truth Value Gaps’, summary

 

by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own.]



Graham Priest


In Contradiction:
A Study of the Transconsistent


Part I. The Logical Paradoxes



Ch.1 Semantic Paradoxes


1.3 Truth Value Gaps



Brief Summary:

Priest is arguing that inconsistencies like the ones produced by liar-like paradoxes are inevitable in a natural language. He addresses arguments against this position. The strongest among them is the argument that such paradoxical sentences are neither true nor false, since they are valueless. Priest demonstrates how even if we take this assumption, we still do not avoid the liar-like paradoxes.



Summary


[Priest’s basic aim in this chapter is to show that dialetheias, true contradictions, are generated necessarily by both formal and natural languages.] Previously Priest showed how the Tarski conditions lead to a liar-like paradox, that is, to a dialetheia. Those three conditions, basically, are that 1) the language in question is able to give names to all its sentences, that 2) names for sentences can be equated with the sentence they name (such that self-reference is also possible), and 3) we can draw the inference that if a formula holds if and only if it does not hold, then you have both that formula and its negation. Now in this section Priest notes that in order to deny that English satisfies the Tarski conditions and thus does not fall prey to the inconsistencies it leads too, then one  must deny that at least one of these conditions holds for English. (12) It would seem very difficult to deny conditions 1 and instead claim that not every sentence in English can have a name in English; for, all we need to do is put quotation marks around any phrase in English and we get its name. So we must look to see if we can deny conditions 2 and 3, and Priest will begin first with 3. (12d)


So recall condition 3.

(3) The rule of inference {α ↔ ¬α} ⊢ α ∧ ¬α is valid in the logic underlying the theory.
(Priest 11)

[Priest will claim that the best and maybe the only reason to reject this is if you think there are truth value gaps, perhaps meaning that some sentences are neither True nor False. This would perhaps have to be the case, since if α is True, then ¬α is False, and vice versa. Thus if you have both, then you have a sentence (their conjunction) which is both True and False. If you accept the validity of dialetheias, this is fine. But if you do not, then you cannot say that α is True nor can you say that it is False. Thus you must think it has a third value which is neither True or False or that it has no value. What Priest says about intuitionism I will have to return to later when I understand it better. Priest’s book Introduction to Non-Classical Logic looks promising for grasping better this topic.]

There is one (and perhaps only one), plausible reason for rejecting the reductio principle of 3, and this is the existence of truth value gaps, sentences that are neither true not false. Not that an intuitionist will think that these cause the principle to fail: the principle is valid intuitionistically.
(13)

[In the next part, Priest will use the term “gap-in/gap-out conditional”, which seems to mean that say you have p → q. And suppose also that p for example does not have a truth value. The question is, does that mean the whole conditional does not have a truth value? If the whole would not have a value as a consequence of only one part not having a value, then it is a “gap-in/gap-out” conditional (called such perhaps because we put one gap into the formulation and we get a gap output for the whole). But if one part can be valueless but the whole have a value, then it is not a “gap-in/gap-out” conditional. I am guessing. I am also not certain about the rest, so this is another guess to his meaning. But let us return to his formulation for condition 3:

{α ↔ ¬α} ⊢ α ∧ ¬α

The final point he might be making is that if we allow for truth gaps, then the left side of the formula can be true but the right side not true. I am not sure how to do this. But maybe he is saying that with regard to this idea of truth gaps, we might say that α is true but ¬α has no value. Nonetheless, a conjunction requires both terms to be true. So α ∧ ¬α would not then be true. Now, recall the only way that a conditional can be untrue. That would happen if the antecedent is true while the consequent is false. Assume again that α is true but ¬α has no value. That means

α → ¬α

is: true implies no value (so not false), and thus the conditional would still be true. And,

¬α → α

would be: no value implies true, and thus it would be true. Then put them together into the biconditional (or just conjunction of the two conditionals) and you still would have true. Thus if you thought that there could be truth gaps (valueless terms), then you can reject the third condition, which would then allow you to say that the dialetheias produced by the three conditions together are not necessary for a language that can have truth gaps. Priest is defending the idea that dialetheias are necessary. He later argues that truth gaps cannot exist. But for now, we will suppose they do, and still he will show that dialetheias will result from the three conditions]

But suppose we are thinking in more classical terms and that we have a sentence, α, such that both α and ¬α fail to be true. Then α ∧ ¬α will fail to be true (assuming a normal conjunction, as I will do throughout). But, given a conditional that is not simply a gap-in/gap-out conditional (where the valuelessness of a part spreads to that of the whole), α → ¬α and its converse may hold, and their conjunction may be true. In this case the inference fails. In fact, under very weak conditions the reductio scheme is equivalent to the law of excluded middle, whose failure can very naturally be taken to express the existence of truth value gaps. Hence if we may take paradoxical sentences to be neither true nor false, this particular argument to dialetheism may be blocked. I shall argue in section 4.7 that there are no truth value gaps. However, for the present let us suppose, at least for the sake of argument, that there are. I will argue that dialetheism is not to be avoided in this way.
(13)


Priest then notes that there are two main sorts of theses holding that sentences may be truth-valueless.

1) Some sentences are neither true nor false, even though sentences are the sort of thing that can be either true or false.

2) It is not sentences themselves that can be true or false, but rather only that which they express can have truth value. And some uses of sentences fail to express something that can have truth value. This theses also has two subvarieties, depending on whether the holder thinks it is statements or propositions that are being expressed. Priest will address both parties simultaneously in the following way:

I intend my discussion to apply to all versions and subversions of the thesis. To this end, I will now write ‘true’, ‘false’, and their cognates with initial capitals. Those who think sentences are true/false can read ‘True/False sentence’ in the obvious way. Those who think that it is statements or propositions that are true/false can read it as ‘sentence (the use of) which makes a true/false statement/proposition’, depending on their preferred theory. The thesis that there are truth valueless sentences can now be expressed as: there are some (indicative) sentences that are neither True nor False. Let us call such sentences ‘Valueless’.
(13)


Priest has some main points. The first is that:

even granted that Valueless sentences vitiate the reductio scheme, this does not, per se, solve the paradoxes.
(13)


Priest now looks at some arguments for why the paradoxical sentences are valueless, but “none of them is very satisfactory.” (14) [Understanding the following seems to require a gasp of the referenced texts and ideas. I do not have this familiarity, but let us make sense of what Priest is doing as best we can for the time being. Priest cites Ryle as one philosopher who thinks that one of the paradoxical sentences are valueless. The counter example Priest gives is this. My father says that all one-legged men in town lie (do not tell the truth), and the one and only one-legged man in town says that my father always tells the truth. Here we have a liar paradox. If either claim is true then it is also false. Priest then says that these sentences stand all the tests for making a statement: “I understood what he said; I can draw inferences from it; I can act on the information contained in it, and so on” (14). So Ryle seems to be saying that the expressions can be statements, meaning that their content is obvious enough that it can be affirmed or denied, and yet there somehow is no truth value to them. Before going further, let us look at a little of the Ryle text that Priest cites. What Ryle seems to be saying is that in the sentence, ‘This sentence is false’, the part ‘this sentence’ fails to refer to something, because of an infinite recursion of substitutions. So Ryle has us consider the statement:

That statement is false.

He says that such sentences come with a ‘namely-rider’. So which other sentence is this sentence referring to? How about, ‘Today is Tuesday.” So we might substitute:

That statement [namely that today is Tuesday] is false.

So far, no problem. But what happens with sentences like:

This statement is false.

? Let us try to insert the namely-rider:

This statement [namely that this statement is false] is false.

But here again we have another reference, ‘this statement’. So we need another namely-rider to know what it is referring to. So we get:

This statement [namely that this statement {namely that this statement is false}] is false.

But still again in curly brackets we have another instance of ‘this statement’ that requires a namely-rider. This will have no end, and thus we never get to the predication ‘is false.’ Since it does not succeed in making a reference that would establish the subject of the sentence, we cannot give it a truth value (it seems more like a sentential function like x is false, where it would only have a truth value when we substitute a value for x.) Here are some passages from the Ryle text:

The same inattention to grammar is the source of such paradoxes as ‘the Liar’, ‘the Class of Classes . . .’ and ‘Impredicability’. When we ordinarily say ‘That statement is false’, what we say promises a namely-rider, e.g. ‘ . . . namely that to-day is Tuesday’. When we say ‘The current statement is false’ we are pretending either that no namely-rider is to be asked for or that the namely-rider is ‘ . . . namely that the present | statement is false’. If no namely-rider is to be asked for, then ‘The current statement’ does not refer to any statement. It is like saying ‘He is asthmatic’ while disallowing the question ‘Who?’ If, alternatively, it is pretended that there is indeed the namely-rider, ‘ . . . namely, that the current statement is false’, the promise is met by an echo of that promise. If unpacked, our pretended assertion would run ‘The current statement {namely, that the current statement [namely that the current statement (namely that the current statement . . .’. The brackets are never closed; no verb is ever reached; no statement of which we can even ask whether it is true or false is ever adduced.
(Ryle 67-68)

Many of the Paradoxes have to do with such things as statements about statements and epithets of epithets. So quotation-marks have to be employed. But the mishandling which generates the apparent antinomies consists not in mishandling quotation-marks but in treating referring expressions as fillings of their own namely-riders.
(Ryle 69)

Priest’s point seems to be that in the case of the one-legged man, we do not have this problem of infinite self-nestings of namely riders. We would have something like, the father says “All one-legged men lie,” and the one-legged man says “Everything (namely that all one legged men lie) that the father says is true.” Here when we think of truth values, we do not necessarily have self-reference problems. If we say the father’s claim is true, that means the one-legged man’s claim is false, which means the father’s claim is false. Same if we assume that the one-legged man’s claim is true: if that is true, then what the father says is false, meaning that what the one-legged man said is also false. Perhaps Priest is saying that because the statement passes all tests for being a statement, we cannot say that it fails to refer. Priest then mentions another philosopher, Kripke, who takes a similar strategy. I am not familiar with this, but Priest seems to be saying that for Kripke, you can have a sentence which at some ‘fixed point’ has  no truth value. But if the sentence has  no truth value, then it cannot be true, so it is untrue. However, even though it would have this negative truth value, we do not assign it any truth value at all. So in sum, those who try to say that the sentences in the liar paradox do not have value still encounter inconsistency. Thus we do not have these as strong justifications to deny that natural language necessarily leads to inconsistency. See the first full paragraph on p.14 for Priest’s discussion, as you should interpret it better for yourself. I provide some of it here in the following:]

Suppose that my father asserts the mendacity of all one-legged men in town; suppose also that there is only one one-legged man in town who, unbeknown to us, has asserted the veracity of my father. If Ryle is right, then either my father or this one-legged man failed to make a statement. Without loss of generality, let us suppose it to be my father. Yet, by all the standard tests for making a statement, he did. I understood what he said; I can draw inferences from it; I can act on the information contained in it, and so on. Alternatively, take Kripke’s position. Let α be any sentence that obtains no truth value at a fixed point. Then, obviously, ‘α is not true’ should be true at the fixed point (at least if truth at the fixed point models the behaviour of truth in English!), though in the construction it receives no truth value. Hence it seems that none of the motivations will do what is required.
(14)


[Priest will now give more reason to think that even if we have truth value gaps, we would still obtain paradoxical sentences. I do not grasp this paragraph sufficiently, but let us work through the ideas. He will give two sentences:

(1) This sentence is true

(2) This sentence is false

Of these, perhaps only the second one is paradoxical or  inconsistent, since if it is true it is false and if it is false it is true. Priest’s point however has to do with the semantic rules that govern the meanings of the sentence parts and that determine its truth value. I am not sure exactly how to understand this, but let us try the (T) scheme first for a non-problematic sentence:

“y is white” is true if and only if y is white

So if y is snow, then “y is white” is true, but if y is coal, then it is false. Perhaps the important thing here is that we can determine the second half of the formulation, since we can know whether or not the substitution for y is white or not. But what happens if our formulation is

“y is true” is true, if and only if y is true.

and we substitute “This sentence” in for y?

“This sentence is true” is true if and only if this sentence is true.

Recall in the case of

“y is white” is true if and only if y is white

we could know whether or not the substitution in the underlined part held or not, depending on whether or not the substitution was something white. However in

“This sentence is true” is true if and only if this sentence is true.

We are not talking about something outside this formulation, like snow is to the variable y, but rather something inside the formulation and whose meaning and truth value are conditioned by that sentence. So we do not have enough information to know if the underlined part is a true substitution. I think that might be Priest’s point, but I am unsure. Priest explains it another way in his book Logic: A Very Short Introduction

suppose someone says: This very sentence that I am now uttering is true. Is that true or false? Well, if it is true, it is true, since that is what it says. And if it is false, then it is false, since it says that it is true. Hence, both the assumption that it is true and the assumption that it is false appear to be consistent. Moreover, there would seem to be no other fact that settles the matter of what truth value it has. It’s not just that it has some value which we don’t, or even can’t, know. Rather, there would seem to be nothing that determines it as either true or false at all. It would seem to be neither true nor false.
(Priest, Logic: A Very Short Introduction, p.32)

Now consider the second sentence again:

(2) This sentence is false

Which in the (T) scheme would be:

“This sentence is false” is true if and only if this sentence is false.

Recall that for sentence 1, if it is true, then it is true, and if it is false, then it is false. It gives out a consistent value, but it is hard to know which one. But in the liar paradox, if it is true then it is false and if it is false then it is true. So regardless of what you assume, you get two truth values rather than one like sentence 1 had. So the second sentence is not a truth value gap but rather it is a glut, because there is too much value output. Priest’s argument here seems to be that those who would want to say there are truth value gaps would not apply it in the cases which produce the liar paradox. This is because the problem is not that we are lacking a way to determine their truth output – we know the output is both true and false – but rather the problem is that there is too much output. Hence again arguing for truth value gaps does not do away with the liar paradox, since it is not a case where truth value is lacking or is underdeterminable. Please interpret the paragraph for yourself to be sure of what it means.]

Any doubts that we might have that, even if there are Valueless sentences, paradoxical sentences are not among them are magnified when we consider the pair

This sentence is True          (1)

This sentence is False         (2)

| There is something odd about both these sentences, but, prima facie at least, it is not the same in both cases. In the case of (1) the semantic rules governing the use of the demonstrative ‘this sentence’ and those governing the predicate ‘is True’ appear not to be sufficient to determine the Truth value of the sentence. In other words, the semantic rules involved underdetermine its Truth value. Such a sentence is an obvious candidate for a Truth value gap. By contrast, in the case of (2) the semantic conditions of the words involved seem to overdetermine its Truth value. (2) would therefore seem a much more plausible candidate for a Truth value ‘‘glut’’ than a truth value gap, which is exactly, of course, what it is.
(14-15)


[Now Priest will now make another point in his argument against those who claim the liars paradoxes can be resolved by saying some sentences are valueless. This section is beyond my ability to summarize. But let us work through it gradually to at least follow the points in his argument. First we suppose there can be valueless sentences. So

‘This sentence is False’ is True iff it is False.

We can say that this sentence is valueless. So it is neither true nor false. That means we do not encounter the contradiction that results when we assume it is true then find it is false and vice versa. Now Priest will show why this argument will not work. So consider a sentence like

This sentence is not true

which we will call α and consider as valueless. Now, what  happens if we say:

‘α is not True’ is not True

? That means α is True. But that contradicts our assumption that α has no value. So we cannot say that. Or what if we say:

‘α is not True’ is Valueless

? We are already saying that α is Valueless. That means it is neither True nor False that α is not True. But that cannot be right, because we know that since α is Valueless, it cannot be True or False. By definition, it is not True, so we cannot be indifferent about whether or not it is not True. We rather have to admit that:

‘α is not True’ is True

This is fine, because α is not True, as well, it is also not False. But does this then imply that:

‘α is True’ is False

? No, because there is a third value, Valuelessness. (I am confused here. It seems that we can rightly assert that

‘α is True’ is False

since α cannot have the value of Truth. Perhaps Priest is saying that

‘α is not True’ is True

does not directly imply

‘α is True’ is False

because there is the third option of valueless. But I am not sure.) Priest says that it is beyond question that

if α is not True, ‘α is not True’ is True    (3)

(We will see in a bit why he establishes this.)

Then Priest has us consider the “extended” or “strengthened” liar paradox:

(4) is not True.       (4)

This can take one of three values, true, false, and valueless. If it is true, then it is not True. If it is not true (because it is either false or valueless), then it is True (since it says of itself that it is not True). But someone might say that if we suppose 4 is valueless, that does not imply it is true. However, recall again 3, which was undeniable:

if α is not True, ‘α is not True’ is True    (3)

4 is an instance of α, that is, of a valueless statement, hence:

if (4) is not True, then ‘(4) is not True’ is True

Since ‘(4) is not True’ is identical to 4 itself, we get:

i.e. if (4) is not True, then (4) is true.

In sum, Priest seems to be saying that even if you claim that the liar sentence is valueless, you still are committed to saying that it is not True, and in the end this will create the inconsistency of saying that the sentence which is not true is true. Please read this paragraph yourself to find a better interpretation.]

The second main point against Value gap solutions to the semantic paradoxes concerns extended paradoxes. Let us suppose that there are Valueless sentences, and that the claim that paradoxical sentences are Valueless can be substantiated. This allows us, in effect, to maintain that, although a paradoxical sentence such as ‘This sentence is False’ is True iff it is False, since it is neither, the derivation of a contradiction is blocked. There is, however, a standard argument to show that this ploy will not work. Some sentences are neither True nor False. Obviously we are capable of expressing this idea in English: we have just done so. (Moreover anyone who maintains that paradoxical sentences are Valueless must accept this on pain of obvious self refutation.) In particular, for any sentence a that is neither True nor False, ‘a is not True’ must be True. (Again, anyone who maintains a Value gap solution to the paradoxes must accept this, or face a devastating ad hominem argument.) This does not necessarily mean that ‘a is True’ is False, since it is possible to maintain that ‘a is True’ is Valueless, and that negation transforms a Valueless sentence into a True one. It is beyond question, though, that

if a is not True, ‘a is not True’ is True.       (3)

Now consider the ‘‘extended’’ (or ‘‘strengthened’’) liar paradox:

(4) is not True.                                (4)

This sentence is either True, False or Valueless. If it is True, then (by the T-scheme, which is not here at issue) it is not True. Similarly, if it is not True (i.e. False or Valueless), then it is True. Hence, whatever it is, we have a contradiction. One might object to the inference from (4)’s being Valueless to its being True. If, for example, we suppose that (4) makes no statement, then it should not follow that it makes a true one (see Goddard and Goldstein 1980). Yet we have agreed (and the Valuegappist is committed) to (3), an instance of which is

if (4) is not True, then ‘(4) is not True’ is True

i.e. if (4) is not True, then (4) is true. Hence there is no way out here.
(Priest 15)


[Now notice that when we deny something being true, we say it must then either be False or Valueless. This means we do not allow for an additional option, meaning that we still use the law of excluded middle. It is not clear to me why this might constitute an objection, or how Priest defends against that objection. Perhaps the basic idea is that a value gappist has no other options, but I am unsure. Please consult the following:]

It may be objected that the above argument still uses the law of excluded middle in the form of the assumption: (4) is True or it is not True (False or Valueless). However, this is just an instance of the law of excluded middle, and | one, moreover, that is unimpeachable for classical logic augmented with truth value gaps. (For the intuitionist the situation might be different, but we have already dealt with him.) Indeed, given that the Valuegappist is committed to the view that (4) is Valueless, and hence that it is not True, he can hardly deny that it is either True or not True.
(15-16)

There is another objection the Valuegappist might make, which is that “the notions necessary for the formation of the paradox (and in particular the notion of Valuelessness) are not expressible in the language in question” (16). [This might have something to do with the meta-language object language distinction. Priest replies by writing:]

But if this is right, it is an admission that the language for which the semantics has been given is not English, since these notions obviously are expressible in English. Thus the problem, which was to show how the English concepts are consistent, has not been solved.
(16)


[Again recall that Priest’s strengthened liar formulation and argument used the principle of excluded middle, since not-True implied either False or Valueless.] Still someone might deny the law of excluded middle [and thus affirm the invalidity of the strengthened liar paradox argument above]. Priest says this will not eliminate the problem, since there are “proofs of contradictions which do not use it” (16). Priest offers as an example Barry’s paradox:

Take Berry’s paradox, for example: English has a finite vocabulary. Hence there is a finite number of noun phrases with less than 100 letters. Consequently there can be only a finite number of natural numbers which are denoted by a noun phrase of this kind. Since there is an infinite number of natural numbers, there must be numbers which are not so denoted. Hence there must be a least. Consider the least number not denoted by a noun phrase with fewer than 100 letters. By definition, this cannot be denoted by a noun phrase with fewer than 100 letters, but we have just so denoted it. Contradiction. This argument appeals nowhere to the law of excluded middle. Both horns of the dilemma are given a direct proof. Reductio, or its equivalent, the law of the excluded middle, is not appealed to at all.
(16)





Citations from:

Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Clarendon/Oxford University, 2006 [first published 1987].



Priest, Graham. Logic: A Very Short Introduction. Oxford / New York: Oxford, 2000.



Ryle, G. (1950) ‘Heterologicality’, Analysis 11, 61–9.



 

1 Jan 2015

Priest (1.2) In Contradiction, ‘The Semantic Paradoxes’, summary

 

by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own.]



Graham Priest


In Contradiction:
A Study of the Transconsistent


Part I. The Logical Paradoxes



Ch.1. Semantic Paradoxes


1.2 The Semantic Paradoxes



Brief Summary:

Tarski tried to give a semantic definition of truth for formal languages, and these formal languages would approximate natural languages. There are certain conditions in his system for doing this that lead to a dialetheia (a true contradiction). This is a sort of liar paradox. Priest here shows how Tarski’s conditions necessarily lead to the inconsistency. In the next section, he will argue that they apply to English, thus dialetheias are inherent to both natural and formal languages.



Summary


Previously Priest looked at the logical paradoxes of self-reference, and he distinguished semantic ones from set theoretical ones. Now he will focus on the semantic ones. We noted how these paradoxes generate dialetheias, or true contradictions. Many people are against saying there exists true contradictions. But to maintain that stance, they must show what is invalid in the arguments producing the paradoxes. Priest’s “aim here is to defend the view that the semantic paradoxes are bona fide sound arguments.” (10) Priest will first “state a set of conditions sufficient for contradiction and then defend against all comers the view that natural language satisfies these conditions (or if not these, then others which have the same effect).” (10) One benefit from the following is that we will better see that solutions to the paradox fail, and we will also learn a bit as to why they do.


Priest explains that “Tarski (1936) located the root of the semantic paradoxes” in closure conditions. [Recall that a language is semantically closed when within it are contained all its components and generative rules. Such semantically closed languages prove inconsistent, as seen in the liar paradox.] Priest will now give the three Tarski conditions for a formal theory (and for simplicity we concern ourselves just with formulas with a single free variable).

(1) For every formula, α, there is a term of the language, α, its name.
(Priest, 11)

[Recall again the (T) scheme:

(T) X is true if, and only if, p

with the example:

“snow is white” if and only if snow is white

As we can see, the ‘meta-language’ on the right side of the equation contains the formula: snow is white, or instead, p. Its name, given in quotes “snow is white” or symbolized as X, is the name for that formula, articulated in the object language. Priest uses underline for quotation and α instead of p.]

(2) There is a formula of two free variables, Sat(x y), such that every instance of the scheme

Sat(t α) ↔ α(v/t)               (*)

is a theorem, where t is any term, α is any formula of one free variable, v, and α(v/t) is α with all free occurrences of ‘v’ replaced by ‘t’ (with the usual precautions concerning the binding of variables free in t).
(Priest, 11)

[Recall the formula and our discussion from  the prior section:

x satisfies α ↔ α(y/x)

We said that means, x satisfies the formula named α (that is, makes its value true when substituted into that sentential function) if and only if we can rightly substitute x in for the free variable y in the formula having that name. Let us look again at Priest’s formulation: “There is a formula of two free variables, Sat(x y)”. In the above, we said there was one free variable in α, which is y. But given the construction, which seemingly means, x satisfies y, this might be equivalent to: x satisfies α . And thus the ‘y’ here might stand for the formula name α . So again:

Sat(t α) ↔ α(v/t)  

Here, α is a formula with one free variable, v. Then t is the term that substitutes for v. So again our formulation says, t satisfies the formula named α (that is to say, the formulation with that name is true), if and only if t can be rightly substituted into the actual formulation.]

(3) The rule of inference {α ↔ ¬α} ⊢ α ∧ ¬α is valid in the logic underlying the theory.
(Priest 11)

[It seems simply that the third condition is that there is a particular rule of inference involved. Consider if we have a formula if and only if we have its negation. From this it is provable (we may infer) that we have both that formula and its negation. I am not sure how better to understand this. Using logical equivalences (as I find them on this wiki page) we might go about it like this:

α ↔ ¬α ≡ (α → ¬α) ∧ (¬α → α)

{from the first logical equivalence of biconditionals:
p↔q≡(p→q)∧(q→p)}

α ↔ ¬α ≡ (¬α ∨ ¬α) ∧ (¬¬α ∨ α)

{from the first logical equivalence of conditionals:
p→q≡¬p∨q}

α ↔ ¬α ≡ (¬α ∨ ¬α) ∧ (α ∨ α)

{double negation law: ¬(¬p)≡p}

α ↔ ¬α ≡ ¬α ∧ α

{from the idempotent laws: p∨p≡p}

]

Now Priest will show that “any theory which satisfies the Tarski closure conditions is inconsistent.” (11)

[So recall again our formulation in step 2:

Sat(t α) ↔ α(v/t)               (*)

It seems we will make a complicated sort of substitution. So we first consider a formulation like

v does not satisfy itself

or perhaps, 

v does not satisfy formula v

We would write it:

¬Sat(v v)

This would be the α formulation. Then t would be the name for this

“v does not satisfy formula v”

And we would write this:

¬Sat(v v)

Then we make our substitions into

Sat(t α) ↔ α(v/t)

and we obtain

Sat(¬Sat(v v) ¬Sat(v v)) ↔ ¬Sat(¬Sat(v v) ¬Sat(v v))

I am not sure how to make this more conceivable with an example. I would think that it should be something like:

“This sentence is false” is true if and only if this sentence is false.

Let us begin with:

v is white” is true if and only if v is white.

Now, we use satisfaction.

t satisfies “v is white” if and only if t can be substituted for v in the formula with this name (v is white).

Instead of ‘v is white’, we have ‘v does not satisfy v’. So

t satisfies “v does not satisfy v” if and only if t can be substituted for v in the formula: v does not satisfy v.

Then we substitute “v does not satisfy v” for t.

v does not satisfy v” satisfies “v does not satisfy v” if and only if “v does not satisfy v” can be substituted for v in the formula:  v does not satisfy v (hence making: “v does not satisfy v” does not satisfy “v does not satisfy v”)

Thus we have

v does not satisfy v” satisfies “v does not satisfy v” if and only if  “v does not satisfy v” does not satisfy “v does not satisfy v”.

Now consider again:

This sentence is false.

Since it is self referential, it is equivalent to saying in our formulations,

This sentence does not satisfy itself.

When we put the liar sentence into the (T) scheme using our different term for satisfaction, we then get:

“This sentence is false” is true if and only if “this sentence is false” is false.

]

Priest then discusses some other issues that complicate the matter [see page 11], but concludes: “the point is shown: these closure conditions give rise to contradiction.” (11)


Now consider the English language, and ask if the three conditions hold for it. It would seem they do. However, since English is not a formal language, the jargon “formula”, “term”, “theorem” and so on will not apply to it, since they only apply to formal languages. Yet,

Still, it is easy enough to rephrase the conditions while retaining their spirit. A natural language satisfies the Tarski conditions iff:
(1) For every phrase  α, there is a noun phrase α, its name.

(2) There is a phrase Sat, requiring two noun phrases to be inserted to make a sentence, such that every sentence of the form

Sat(t α) iff α(t)

is true, where a is any phrase requiring a noun phrase, t, to be inserted to make a sentence, and parentheses mark insertion.

(3) The following rule of inference is truth preserving:
α iff it is not the case that aα:
Hence, α and it is not the case that α.
(Priest, 12)

Thus “We can now proceed, exactly as before, to establish that any natural language that satisfies the Tarski conditions contains true sentences of the form ‘α and it is not the case that α.’ ” (11) And so “A natural language which satisfies the Tarski conditions therefore contains true contradictions.” (11) Next Priest will turn to the issue of whether or not natural languages can satisfy these Tarski conditions.

 


 



Citations from:
Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Clarendon/Oxford University, 2006 [first published 1987].

 

 



 

Hardegree (§§1-8) “Basic Set Theory”, summary up to §7, “Set Abstraction” and §8, ‘Set-Abstract Conversion’


by Corry Shores
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Gary Hardegree


“Basic Set Theory”
(A course text for his class “Philosophy 595 - Formal Semantics”)


§§1-8


Brief Summary:

Set Theory describes the structures and logical properties of groupings. A set may be made of ‘ur-elements’ and/or of other sets. But two sets cannot have the same members, because in fact they would be identical and would never have been separate sets to begin with. We may define a set using a ‘set-abstraction’ method, meaning that we describe the members not by actually listing each of them between curly brackets but rather by defining the conditions for the set’s membership, by means of a formulation like {v : ℱ}. What this means is that the set (between the curly brackets) includes those members v that fulfill formula ℱ. For example, the set of happy things would be formulated as: {x : x is happy}. More specifically the formulation would read:

{v : ℱ} =df   Sv(v S ↔ ℱ) 

which means, the set of v things fulfilling formula ℱ is defined as that specific (℩) set S which contains all the items v that fulfill ℱ. [Or: the set made of those things v which fulfill formula ℱ is defined as being that one specific set S such that for all v, v is a member of S if and only if it fulfills that formulation ℱ.] However, set theory does not normally use the description operator ℩ seen above. Instead it is ‘hidden’ in a ‘set-abstract conversion’:

v(v {v : ℱ} ℱ)     [set=abstract conversion]

Here, v is any variable, and ℱ is any formula. The following is a simple example.

x(x {x : x is happy} x is happy)

This says that something is an element of {x : x is happy} if and only if it is happy.
(page 6, quotation)

[This set-abstraction is reminiscent of the Tarski (T) scheme, which reads:

(T) X is true if, and only if, p.

For example,

“Snow is white” is true if, and only if, snow is white.

Here truth is defined as satisfaction, meaning that some sentence is true only if what it says is true. Similarly, the set-abstraction seems to be defining set membership in terms of satisfaction, namely, that something is a member of a set if it satisfies the conditions for membership in that set.]



Summary



1. Introduction


Formal semantics makes use of set theory. We will look at ideas in set theory “that have bearing on formal semantics” (2).



2. Membership


“sets have members, also called elements” (2). Epsilon ‘∈’
symbolizes membership. [It takes the Greek letter equivalent to ‘e’, for ‘elements.] If we want to say that element a is a member of set S, we write:

a S

If we want to say it is not a member, then we write:

aS


Conventionally we use lower case Roman-Italic letters to denote ‘points’ (elements or members), and we use Roman-Italic capital letters for sets whose elements are points. Some sets contain other sets as members. For them we use script letters or some other “gaudy font” (2). So in the following, we have an element included in a set, and that set is included in a set which contains other sets.

aB & B ∈ ℂ

[Hardegree then seems to suggest that we need not follow that convention necessarily.]

Note, however, that the following is equally legitimate.

ab & b ∈ c

(2)



3. Extensionality


“Sets have members, just like clubs.” Yet, consider how “two different clubs can have the same membership” and thus “a club is not identified with its membership, nor even by its membership” (2). However, this cannot be so in set theory: “two different sets cannot have the same membership” (2). [So, if two sets are said to have the same members, then they are identical, and thus there is only that one set with two names.] This restriction is called the Principle of Extensionality.

The Principle of Extensionality:
for any set A, and for any set B:
x(xA x B) → A = B            [extensionality]

(Hardegree, p.3, bracketing his)

[The basic insight of the above formulation is that one set of members can have different names, but that set can only ever be one set. For, two different sets may not have the same membership. Informally it says that if set A and set B have the same members, then they are the same set. Less informally it say: for any set A and for any set B, for all members x, if all such x’s are included in A if and only if they are included in B, then A equals B.]


4. The Empty Set


All sets have members, with one exception, the empty set.

Empty set:
there is a set S such that ∼∃[x S]       [empty set]
(Hardegree, p.3, bracketing his)

As the principle of extensionality says that there may be only one set for any unique group of members, this means there can only be one empty set. “It is fittingly called the empty set, and is denoted ∅.” (4)


5. Simple Sets; Singletons, Doubletons, etc.


Normally when we denote a set with only a few elements, we merely list those elements within curly brackets and separate the items with commas. Here are some of Hardegree’s examples. (3)

{Mozart}
{Mozart, Jupiter}
{Mozart, Jupiter, 41}

Hardegree then gives the following informal definitions for this structure.

{a}       =df    the set whose only element is a
{a,b}    =df    the set whose only elements are a and
b
{a,b,c} =df    the set whose only elements are a, b, and c

(page 3)

Then Hardegree provides the following principles to formally summarize these above definitions. [The basic insight of these formulations is that something is included within a set if it is a member of that set. Informally they read something like, for all x, x is included in the set whose members are a, b, and c, if and only if x is either a, b, or c.]

x ( x ∈ {a} ↔ x=a )
x ( x ∈ {a,b} ↔ x=ax=b )
x ( x ∈ {a,b,c} ↔ x=ax=bx=c )

(page 3)


Next Hardegree provides the following terminology:

{a} is called the singleton (unit set) of a.
{a,b} is called the doubleton (unordered-pair) of a and
b.
{a,b,c} is called the tripleton (unordered triple) of a, b, and c.
etc.

(page 3)


If a, b, and c are all well-defined, then sets {a}, {a,b}, {a,b,c} will as well be well-defined. [Now, for some reason, probably having to do with what is meant by well-defined] “set theory postulates the existence of infinitely-many sets, including the following, just for starters.

∅, {∅}, {{∅}}, {{{∅}}}, etc.
(page 3)

[which perhaps means, the null set, the set containing the null set as a member, the set containing the set containing the null-set as a member, and so on]

Hardegree says it is important that we “appreciate how many sets are alluded to by the above list”, so he will demonstrate two things, that, 1) “the above list has infinitely-many entries,” and 2) “the above list contains no duplicates!” (page 4). [The proof is excluded in our summary but can be read on page 4.]



6. Pure and Impure Sets


A set which contains only other sets is called pure. However, a set can be

constructed from an underlying universe of ‘ur-elements’, which are presumed not to be sets, as in the following (from earlier).

{Mozart, Jupiter, 41}

(page 5)

Such sets made of non-sets are called impure.



7. Set Abstraction


Many sets have more than three members, and so it is impractical to use the listing convention we employed above. Instead,

a more concise notation is employed – set-abstraction, whose basic form is

{v : ℱ}

where v is a variable, and ℱ is a formula.
(page 5)

[It seems it is saying that ℱ is a formula containing the free variable v, and that formulation may for example predicate that variable.]

The following are simple examples.

{x : x is happy}
{x : x is happy and x is virtuous}
{x : the mother of x is taller than x}

(page 5)

[It seams that the first set in the list is the set of (all) happy things. Perhaps it could be read, ‘the set containing members x such that x are happy’, or maybe something like ‘the set containing members x where x are happy’.]

The intuitive idea is quite simple – {v : ℱ} consists of exactly those things that satisfy the condition described by the formula ℱ. For example, {x : x is happy} consists of exactly those things that satisfy the condition of being-happy.
(page 5)

[To understand the following formulation, let us consider some things. {v : ℱ} is a set of things which satisfy the formulation of ℱ, for example, ‘is happy’. We will use the definite-description-operator, ‘℩’, which seems to be a way to refer to one specific thing. In this case, it seems we give the set {v : ℱ} the name S. And it seems we define {v : ℱ} as being the one particular S in question. And this S has only those members that fulfill the formula ℱ. So in the following,

S

seems to mean, the (one and only, specific) set S ….

And,

v(v S ↔ ℱ)

would mean, for all v, v is a member of set S if and only if it fulfills the formulation ℱ.

So together, 

{v : ℱ} =df   Sv(v S ↔ ℱ) 

seems to mean, the set made of those things v which fulfill formula ℱ is defined as being that one specific set S such that for all v, v is a member of S if and only if it fulfills that formulation ℱ. However, I am not sure if this is correct. Does the ℱ standing alone in the biconditional mean “v fulfills ℱ”? (Some text below seems to indicate this.)  Also, I cannot explain the next bracketed text, reading “[S not free in ℱ]”. Does this mean ‘where S is not free in ℱ”? How could that be, if S is not a variable? How can the set S be ‘in’ the formula ℱ which conditions S’s members? Please read Hardegree’s text yourself to find a better interpretation.]

The following is the official explicit definition.

{v : ℱ} =df   Sv(v S ↔ ℱ) 
[S not free in ℱ]              [set-abstract]

Here, the symbol ‘℩’ (upside-down iota) is the definite-description-operator, informally defined as
follows.
vℱ =df   the v such that ℱ

(Hardegree, p.5, bracketing his)



8. Set-Abstract Conversion


So above we saw the official definition for set-abstraction, which again is (in part):

{v : ℱ} =df   Sv(v S ↔ ℱ) 

And as we noted, it uses the description-operator, ℩.

However, the description-operator is almost never employed in set theory. Rather, it usually gets hidden under an associated principle of set-abstract conversion.
(6)

[The following formulation seems to be saying, for all variables v, v is a member of set {v : ℱ} (the set whose members fulfill formula ℱ) if and only if v fulfills formula ℱ. Then the example might be read as, for all x, x is included in the set of all happy things if and only if x is happy.]

v(v {v : ℱ} ℱ)     [set=abstract conversion]

Here, v is any variable, and ℱ is any formula. The following is a simple example.

x(x {x : x is happy} x is happy)

This says that something is an element of {x : x is happy} if and only if it is happy.
(Hardegree, p., bracketing his)



[Note: I changed to Hardegree’s
double-struck F symbol :

image

to script capital F,  ℱ , because it was not showing in the published version of this blog post.]




Hardegree, Gary. “Basic Set Theory”. A course text for his class “Philosophy 595 - Formal Semantics”.
http://people.umass.edu/gmhwww/595/text.htm
http://people.umass.edu/gmhwww/595/pdf/set%20theory/Set-Theory-Chap0.pdf