Priest, Ch5 of Logic: A Very Short Introduction, “Self Reference: What is this Chapter About?”, summary

by Corry Shores

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Summary of

Graham Priest

Logic: A Very Short Introduction

Ch.5
Self Reference: What is this Chapter About?

Very Brief Summary:
Self-reference can present problems in logic, which leads us to conclude that there are actually four and not just the first two of the following possibilities. A sentence can be either 1) just true, 2) just false, 3) both true and false, or 4) neither true nor false. The liar sentence, “This sentence is false,” is a candidate for the third option, and its “cousin,” “This sentence is true,” for the fourth. There are certain inferences that intuitively seem valid, but under the first two “classic” assumptions they are deemed invalid. To their credit, the new assumptions make them valid. However, other inferences that the classic assumptions rightly deems valid are counter-intuitively deemed invalid under the new assumptions. There are further problems with the new assumptions. The valueless “cousin” sentence is assumed to be not true (and as well not false), but it in fact says of itself that it is true. And a stronger version of the liar sentence, “This sentence is not true,”  is not just true and false but in a more problematically contradictory way it is as well both true and not-true.

Brief Summary:
Paradoxical and otherwise problematic instances of self-reference lead us to suspect that we have more options than the following two: 1) a sentence can be just true, or 2) a sentence can be just false. Consider the “liar” sentence, ‘This sentence is false.’ If it is true, then it is false; but if it is false, then it is true. Either way, it’s truth-value will contradict what it says its truth-value is. So we have option 3) a sentence can be both true and false. Or consider the “liar cousin” sentence, ‘This sentence is true.’ Normally the terms in such a declarative sentence refer to things or situations by which we may determine the truth or falsity of the statement, that is to say, whether or not the indicated situation holds in reality or not. So if we say, “this chair is red,” we look to the indicated chair and its color, and we determine if the sentence is true or not. However, the terms in “this sentence is true” does not point us to such a determining situation, since we are only able to make two equally viable assumptions about its truth value, namely, that it is either true or that it is false; but, we have no way to make the determination one way or another, since it will always be consistent with what it says of itself under both assumptions. It would seem that we have no grounds that would allow us to determine whether it is true or false, and thus we have option 4) a sentence may be neither true nor false. The classical assumptions 1 and 2 lead us to conclude certain inferences are valid when our intuitions say otherwise. For example, “The Queen is rich,” “The Queen isn’t rich,” therefore, “Pigs can fly” (q, ¬q/p). Our intuitions tell us this seems invalid. But by just using assumptions 1 and 2, it is valid, since structurally speaking there is no situation where the premises are true and the conclusion is false. For, the premises can never all be true anyway. However, under the new assumptions, particularly that sentences can be both true and false, q, ¬q/p can be valid, if q is both true and false and p just false. For, q is at least true and ¬q is also at least true. However, our intuitions tell us that q∨p, ¬q/p is valid, but the new assumptions deem it invalid. Yet, perhaps it only seems intuitively valid if we forget that there are exceptional situations where sentences can be both true and false. There are other problems with the assumptions. When we assume that the liar cousin, “This sentence is true,” is neither true nor false, that means it cannot be true, but it says of itself that it is true. And while we might go along with saying that “This sentence is false” is both true and false, we might not feel the same way about “This sentence is not-true”. Here, we might conclude that it is both true and not-true (and not just true and false), which is a stronger contradiction that we may not want to accept.

Summary

 

This issue of reference is not a simple one, especially in cases of self-reference. Sometimes a name refers to something larger that it is a part of. “For example, consider the sentence ‘This sentence contains five words’. The name which is the subject of this sentence, ‘this sentence’, refers to the whole sentence, of which that name is a part” (31). We also have self-reference in the following other cases. There is the name “These regulations” in the sentence, “These regulations may be revised by a majority decision of the Department of Philosophy”. [Here the larger body of regulations of which this stipulation is a part is referred to by the name “These regulations”.] And there is the name “this thought” in the thinking of the person who says “If I am thinking this thought, then I must be conscious” (31).

All these instances above are not problematic cases of self-reference. But there are ones that are. Consider:

This very sentence that I am now uttering is false.
(31d)

We will call the sentence above λ. Now we ask, is it true or false?

Well, if it is true, then what it says is the case, so λ is false. But if it is false, then, since this is exactly what it claims, it is true. In either case, λ would seem to be both true and false.
(32a)

But matters are not much better with this statement:

This very sentence that I am now uttering is true.
(32)

Why? It would seem consistent. If it is true, then it is true, since it says it is true. And if it is false, then it is false, since it claims instead to be true. But, how would its truth or falsity be determined? [The only way it seems it can be determined is by assumptions, and even this does not determine it as one or the other, because both are equally valid. Since its truth or falsity cannot be determined anyway, it perhaps has neither value.]

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.
(32)

These are ancient paradoxes. The first one, “This sentence is false” is a form of the liar paradox, which was discovered by the ancient Greek philosopher Eubulides. Liar type paradoxes also have appeared in recent debates, “some of which play a crucial role in central parts of mathematical reasoning” (32). [The next paradox involves the mathematical and logical notion of a set. Sets are abstract in a way that allows for self-reference.] One such important paradox is found in set theory, and it involves the set of all non-self-including sets. To arrive at this paradox, we need to walk through some other concepts.

A set is a collection of objects. Thus, for example, one may have the set of all people, the set of all numbers, the set of all abstract ideas.
(32)

[But as we mentioned, there is an element of abstraction to sets that allows for them to be taken as members of other sets.]

Sets can be members of other sets. Thus, for example, the set of all the people in a room is a set, and hence is a member of the set of all sets.
(32)

[Furthermore, a set can even include its very own self. To understand this, we of course cannot imagine a set as being like a physical container like a jar, because then it could not physically fit within itself. Also, there is a strange doubling that seems to be happening which could not be understood with physical metaphors. We are dealing a self-inclusive set abstractly, which means there is one set, and thus it has just one name, but it is regarded doubly, namely, as being the set that is including itself and also the set that is included in itself.]

Some sets can even be members of themselves: the set of all the objects mentioned on this page is an object mentioned on this page (I have just mentioned it), and so a member of itself; the set of all sets is a set, and so a member of itself.
(32)

Many sets, however, cannot be self-inclusive.

And some sets are certainly not members of themselves: the set | of all people is not a person, and so not a member of the set of all people.
(32-34 [The text skips page 33, which is entirely an image])

[So we have the following progression of concepts, with a  new addition: 1) some simple set of things, 2) a set included in another set, as for example the sets included in the set of all sets, 3) a set that is included in itself, and thus the set to which this set belongs is not really other to it, and now, 4) a set that does not include itself, and furthermore 5) the set of all sets that do not include themselves.] So now we “consider the set of all those sets that are not members of themselves” (34). We will call this set of all non-self-inclusive sets R. We now ask, “Is R a member of itself, or is it not?” (34). [The problem will be that like the liar sentence, its status of self-inclusion is indeterminable. To write out the following more fully: If the set of all non-self-inclusive sets is included in itself, then it is not really a non-self-inclusive set, since it is self-inclusive. If the set of all non-self-inclusive sets is not included in itself, then it would belong within itself as as a member, because it is non-self-inclusive.]

If it is a member of itself, then it is one of the things that is not a member of itself, and so it is not a member of itself. If, on the other hand, it is not a member of itself, it is one of those sets that are not members of themselves, and so it is a member of itself. It would seem that R both is and is not a member of itself.
(34)

This paradox is called Russell’s paradox, named after its discoverer, Bertrand Russell. Just as we saw with liar paradox, it is also problematic to have the set of all self-inclusive sets.

Like the liar paradox, it has a cousin. What about the set of all sets that are members of themselves. Is this a member of itself, or is it not? Well, if it is, it is; and if it is not, it is not. Again, there would seem to be nothing to determine the matter either way.
(34)

[Recall what we said in Chapter 2. There we were looking at truth conditions and truth functions. On page 9 for example, Priest has us assume when making a truth table for negation “that every sentence is either true or false, but not both” (9).] These problematic examples of self-reference challenge our assumption that we made in Chapter 2 that “every sentence is either true or false, but not both. ‘This sentence is false’, and ‘R is not a member of itself’ seem to be both true and false; and their cousins seem to be neither true nor false” (34). [In section 1.3 of Priest’s In Contradiction, he explains these two situation using the terms gaps and gluts. To understand this distinction, we need to note a semantics issue. I think to follow through these ideas, we might consider a different class of statements altogether, namely, ones that refer to things other than themselves. So, “this chair is red” is true if the chair is red and false if it is not. Why is this different than “this statement is false”? We take note that the terms in “this chair is red” refer to a situation that may determine the truth value of the statement. How that truth value is determined is another matter. But the fact is, presumably, if certain basic conditions are met, that to which the terms refer can definitely determine the statement’s truth value. Now, what is it that terms in “This sentence is true” refer to? They refer back to the sentence itself. But the problem is that the sentence itself, unlike the chair’s color, cannot determine the truth value of the statement. As Priest writes in In Contradiction, “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 Contradiction 15). In this case, there is a truth-value “gap” since it can be determined neither as true nor as false. What about “This sentence is False”? Here we have the same structure of self-reference with the term “this sentence” and now we have the predication “is False”? For some reason that I do not quite grasp clearly, here the situation is different. I do not understand so well, because one could say that this second sentence is not doing anything different than the first. For the first case, the “truth-teller”, if we assume it is true then it is true and if we assume it is false it is false. In the second case, the “liar”, if we assume it is true it is false and if we assume it is false it is true. On those grounds, why do we not say that the liar also is neither true nor false, since we as well cannot on the basis of the terms determine one value or the other? This I do not understand. Perhaps the idea is the following. The truth-teller’s truth-value cannot be rightly found. It can only be endowed by means of assumption, which means that it intrinsically has no value on its own. The liar sentence always outputs the opposite of your assumptions, which contradicts what it says it should be. For the liar it does matter what your input is, because you get a self-consistent consistent output. But since the liar’s output is always inconsistent with its meaning, it does not matter what you input. For, its output can be inputted again to once more get the opposite output value. If it is true, then it is false, but if it is false, then it is true. So given this problematic circularity, it does not matter what assumption you begin with. But as you can see I am not certain what justifies us in distinguishing them fundamentally. At any rate, taking it for granted that the liar sentence is both true and false, we here have a “glut” since there is too much determination of its truth value, rather than a lack of it like in the truth-teller.]

Priest says we can accommodate this problematic situation by taking these other truth-status possibilities into account.

Assume that in any situation, every sentence is true but not false, false but not true, both true and false, or neither true nor false.
(34)

We recall the truth conditions for negation, conjunction, and disjunction from chapter 2.

In any situation:

¬a has the value T just if a has the value F.
¬a has the value F just if a has the value T.

a & b has the value T just if both of a and b have the value T.
a & b has the value F just if at least one of a and b has the value F.

|

a ∨ b has the value T just if at least one of a and b has the value T.
a ∨ b has the value F just if both of a and b have the value F.
(34-35)

[Instead of following the prior restriction that allowed only for true or false values,] we will “work out the truth values of sentences under the new regime” (35). [It seems here Priest is selecting as exercises three possible truth situations for negation, conjunction, and disjunction. 1) First we suppose a classical logic situation where a is false but not true. Here, negation simply flips the value. 2) Second we suppose a glut situation where a is both true and false, while b is just true, and the two are joined conjunctively. Here, the whole conjunct is both true and false. It seems the reasoning is this. Since a is at least true, that makes a & b true. But since it is also false, that makes a & b false as well. 3) Third we suppose a gap situation where a is just true, but b is neither true nor false, and the two are joined disjunctively. Here a ∨ b is merely true. The reasoning is as follows. What would make it false is only if both a and b are false. But since at least a is true, it does not matter that b is neither, and so the disjunction is always true.] [In the following, for the “clauses” (c1/c2), I insert them in curly brackets for convenience.]

• Suppose that a is F but not T. Then, since a is F, ¬a is T (by the first clause for negation).
{c1: ¬a has the value T just if a has the value F.} And since a is not T, ¬a is not F (by the second clause for negation). Hence, ¬a is T but not F.
{c2: ¬a has the value F just if a has the value T.}

• Suppose that a is T and F, and that b is just T. Then both a and b are T, so a & b is T (by the first clause for conjunction).
{c1: a & b has the value T just if both of a and b have the value T.}
But, because a is F, at least one of a and b is F, so a & b is F (by the second clause for conjunction). So a & b is both T and F.
{c2: a & b has the value F just if at least one of a and b has the value F.}

• Suppose that a is just T, and that b is neither T nor F. Then since a is T, at least one of a and b is T, and hence a∨b is T (by the first clause for disjunction).
{c1: a∨b has the value T just if at least one of a and b has the value T.}
But since a is not F, then it is not the case that a and b are both F. So a∨b is not F (by the second clause for disjunction). Hence, a∨b is just T.
{c2: a∨b has the value F just if both of a and b have the value F.}).
(35)

Now we wonder what this means for validity. Recall that “A valid argument is […] one where there is no situation where the premisses are true, and the conclusion is not true” (35). This is still the case, as is the fact that “a situation is […] something that gives a truth value to each relevant sentence” (35). The only difference is that situations now may also give either two truth values or none. We will now ask if the inference q/q∨p is valid. [The basic idea here seems to be that we cannot have a situation where the conclusion is not true while the premise is true. This is because we assume that q is true. That is enough to make the disjunct true, where q appears again. Thus it is valid. If q is false, then we cannot determine the validity anyway, so those cases do not matter. I wonder, what if q is both true and false? Perhaps that does not change the situation, since insofar as it is false, it has no bearing on the test for validity of the inference. I am not sure about this, but that might be what Priest is suggesting below in parentheses.]

So consider the inference q/q∨p. In any situation where q has the value T. (It may have the value F also, but no matter.) Thus, if the premiss has the value T, so does the conclusion. The inference is valid.
(35)

[Recall another inference from chapter 2: q, ¬q/p. We noted that we cannot have any situation where all the premises are true and the conclusion false. This is because we have both q and its negation, which means always at least one premise will be false. Because we do not even need to relate the premises to the conclusion, we called it vacuously valid. Here was the truth table:

] Under the old assumptions, q, ¬q/p is valid, but under the new ones, it is invalid. [The reasoning for this seems to be the following. q can be both true and false, and p just false. This means that ¬q is both true and false. Now since both ¬q and q are both true and false, they are both at least true, while the conclusion is false, thus making the inference invalid. Of course a concern could be that this reasoning does not work, since ¬q and q are also both false, and thus this case of glut values does not allow us to test the validity of the inference. Priest says that their additional falsity does not matter. I am not exactly sure why. It again could be the fact that even though they are no less false as true, that falsity is not relevant to the test of validity, and only their truth is relevant. I find these paraconsistency ideas absolutely fascinating philosophically. We do not take the joint truth and falsity as an unbreakable pair of values. They both stand independently on their own. Both values are absolutely affirmative in the sense that the one does not subtract from the other. I find this affirmative concept of “both” in application to truth and falsity to be quite interesting and powerful. It is addition without mixture or contamination, but the things being combined you would normally think would interfere with each other’s value.] Priest will explain why under the new assumptions q, ¬q/p is an invalid inference. [The fact that the new assumptions correspond more with our intuitions about the inference suggest that classical logic is inadequate and that we instead should consider a non-classical logic.]

just take a situation where q has the values T and F, but p has just the value F. Since q is both T and F, ¬q is also both | T and F. Hence, both premisses are T (and F as well, but that is not relevant), and the conclusion, p, is not T. This gives us another diagnosis of why we find the inference intuitively invalid. It is invalid.
(35-36)

Priest then says that “As we saw in Chapter 2, this inference follows from two other inferences,” namely, q/q∨p and

q∨p, ¬q
       p

[I recall the discussion of these other inferences, but at the time I did not realize that q, ¬q/p followed from them. I am still unsure how this is, but perhaps the idea is the following. The inference q, ¬q/p seems to throw in p at the end, and to all appearances it comes out of nowhere. So it would make sense if we introduce it in the premises, hence the need for q/q∨p, which seems to justify introducing other terms. But now that there are two terms, and we infer from them merely one of the two, we need a way to eliminate one of them. Hence the q∨p, ¬q/p. Most likely the above reasoning is not what Priest means by q, ¬q/p follows from these other two inferences. Perhaps he is just saying that if you begin with these other two inferences, you can combine them to get in essence q, ¬q/p.] Priest will now find a way to invalidate q∨p, ¬q/p by finding an instance (using our new assumptions) where the premises are true and the conclusion false. So we assume that p is just false, and p is of course the conclusion. But we assume that q is both true and false. This means “that both premisses get the value T (as well as F). But the conclusion does not get the value T. Hence the inference is invalid” (36).

[Previously these new assumptions allowed us to determine q, ¬q/p as invalid, which matched our intuitions about the inference. This was one advantage over the old (classical) assumptions. But now Priest acknowledges this case where the new assumptions make q∨p, ¬q/p invalid, which goes against our intuitions. Priest will still defend the new assumptions. His basic point seems to be that really it does in fact match out intuitions, but only when we are keeping in mind instances where q can be both true and false, as in the liar paradox. Then the inference intuitively seems valid.]

In Chapter 2, I said that this inference does seem intuitively valid. So, given the new account, our intuitions about this must be wrong. One can offer an explanation of this fact, however. The inference appears to be valid because, if ¬q is true, this seems to rule out the truth of q, leaving us with p. But on the present account, the truth of ¬q does not rule out that of q. It would do so only if something could not be both true and false. When we think the inference to be valid, we are perhaps forgetting such possibilities, which can arise in unusual cases, like those which are provided by self-reference.
(36)

Priest invites us to think about which explanation (the current one or the one from chapter 2) we find more compelling. Priest then notes other problems with the new assumptions. [I do not grasp the main ideas here clearly enough to restate them properly. The main idea is that even with our new ‘gap’ and ‘glut’ assumptions, we still have unresolved problems with the liar and its cousin. Regarding gaps, Priest discusses many problems with them in his In Contradiction. See section 1.3 and section 4.7. In our current treatment here, Priest shows that we still have a contradiction with the gap assumption applied to the cousin. Even though we begin by assuming that it has neither a true  nor a false value, we know from this that it is at least not true (for if it were true, then we are not using the gap assumption). However, it says of itself that it is true.]

Consider the liar paradox and its cousin. Take the latter first. The sentence ‘This sentence is true’ was supposed to be an example of something that is neither true nor false. Let us suppose that this is so. | Then, in particular, it is not true. But it, itself, says that it is true. So it must be false, contrary to our supposition that it is neither true nor false. We seem to have ended up in a contradiction.
(36-37)

Then he turns to the liar sentence, but now under a different formulation, “This sentence is not true,” which also presents a contradiction. [I think I do not adequately grasp the point here. We will conclude that the sentence results in a contradiction. I had thought that by saying it is both true and false we were already acknowledging there is a contradiction. Also, we are  making a distinction between not-true and false, which I do not know how to make. I will quote it below, because I cannot convey the meaning well in my own words and thinking. He does not present it this way, but I let me offer the following formulation. We begin with “This sentence is not true”. We say it is both true and false. Insofar as it is true, what it says of itself holds, and thus it is also not true. Insofar as it is false, what it says of itself does not hold. Thus it is not the case that it is not true, therefore it is true, but it says of itself that it is not true. So we have more than just the sentence being both true and false, as per our assumptions. It is as well both true and not true, in accordance with its stated self-determinations. So the idea here might be the following. Someone could think that it is one thing to say that a sentence is both true and false. But that is not as strong and as evident a contradiction as to say that it is both true and not true. So perhaps we might be willing to go along with saying that a sentence has both the values 1 and 0, or T and F. But we might not feel so sure if we take it another step to say that it is both 1 and not 1, or T and not T. I am not sure why someone would accept the first articulation but reject the second. And as I said, I also do not know how to distinguish not-T from F. If we only have two values, I would think that they would be equivalent. Perhaps the idea is that with the new assumptions they are not equivalent. The liar cousin under the gap assumption is not T but also not F. Thus not-T and F are not equivalent there. So his point might be that we need these extra values, like not-T vs. F, and thus we have extra complications.]

Or take the liar sentence, ‘This sentence is false’. This was supposed to be an example of a sentence that is both true and false. Let’s tweak it a bit. Consider, instead, the sentence ‘This sentence is not true’. What is the truth value of this? If it is true, then what it says is the case; so it is not true. But if it’s not true, then, since that is what it says, it is true. Either way, it would seem to be both true and not true. Again, we have a contradiction on our hands. It’s not just that a sentence may take the values T and F; rather, a sentence can both be T and not be T.
(37)

Priest concludes: “It is situations of this kind that have made the subject of self-reference a contentious one, ever since Eubulides. It is, indeed, a very tangled issue” (37).

 

[The following is quotation.]

 

Main Idea of the Chapter

● Sentences may be true, false, both, or neither.
(quoted from Priest, 37, boldface his)

 

 

From:

 

Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.

Also mentioned:

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

 

 

Priest, Ch4 of Logic: A Very Short Introduction, “Descriptions and Existence: Did the Greeks Worship Zeus?”, summary

by Corry Shores

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Summary of

Graham Priest

Logic: A Very Short Introduction

Ch.4
Descriptions and Existence: Did the Greeks Worship Zeus?

Very Brief Summary:
We may specify a thing by describing it in terms of its properties, thus creating a “definite description”. This thing which is itself described by predicates may also receive predication. There is a problematic principle called the Characterization Principle that is sometimes used to make arguments, thereby invalidating them in certain cases. The “CP” says that whatever is in the thing’s description is also something that can be predicated to that described thing. For example, the ontological argument says that God can only be described as having existence, thus by the CP, God does in fact exist. However, there is a rule that the predicates to non-existing described things are false, thus the ontological argument begs the question; for, it only works if you first assume that God is a real thing, but that is what was to be proven. However, the rule does not apply to fictional entities like Greek gods who on the one hand do not exist, while on the other hand, we are still able to rightly assign each of them their proper attributes.

Brief Summary:
A definite description specifies a thing satisfying certain conditions, for example, “the man who first landed on the Moon”. Descriptions can be formulated symbolically by the use of variables that are predicated. The overall formulation takes the form ιxc_x. Here, the ιx means, “the object x, such that…”, and the c_x gives the conditions specifying the object. In our example we could write ιx(xM & xF) to mean, “the object x such that x is a man and x first landed on the Moon”. Furthermore, we may treat the whole description as something that can take predicates, and we can use Greek letters to stand for the whole description, thus possibly making the above formulation simply μ. This abbreviation will help us examine the validity of the Characterization Principle (CP), which is used in the Ontological Argument for God. We describe God as having a variety of properties that specify God, with the final one being “exists”: ιx(xP₁ & … & xP_n). The CP says that a thing characterized by certain properties in fact has those properties, and thus the whole described thing is predicated by the properties given in the description. Symbolically this involves substituting all cases of x in the description with that description itself. In this formulation we would get: ιx((xP₁ & … & xP_n)P₁ & … & (xP₁ & … & xP_n)P_n), which in part says that the object that is omniscient etc., and exists, is in fact omniscient, etc., and does really exist. Using the Greek letters we can render the above substitution as: γP₁ & … & γP_n. But there is an important rule this argument breaks, namely that any predication to a non-existing entity is false. If there is a God, then the predication that God exists is true; but if there is no God in reality, then this predication is false. This means that for the argument to work, it must assume the truth of its conclusion at the outset, and is thus invalid. Yet there are cases where this rule does not apply, for example in instances of fictional entities like Greek gods whose properties can rightly be predicated to their description even though the thing described does not exist.

Summary

 

Priest will discuss something called “definite descriptions” or just “descriptions,” which can be the subjects of sentences (24). But by “description” we do not here mean it in its normal broad sense, but instead it is a technical term.

Descriptions are phrases like ‘the man who first landed on the Moon’ and ‘the only man-made object on the Earth that is visible from space’. In general, descriptions have the form: the thing satisfying such and such a condition.
(24)

We will formulate descriptions following Bertrand Russell. [For this formulation, we keep in mind the idea of satisfying a condition. The man on the moon is an object. We call him x. We say, “there is an object x.” The condition that makes this man the one we have in mind is that he was the first to land on the moon. Before we get to this description that specifies this man, we need to introduce that description. So we say, “the object, x, such that…”. We write this as ιx(…). The “…” is not meant to be there. It is where the description should go. So in this case: 

ιx(x is a man and x landed first on the Moon).

We will further abbreviate this. M will stand for “is a man” and F will stand for “landed first on the Moon” (p.24). We now have:

ιx(xM & xF)

Priest gives this formalization:

In general, a description is something of the form ιxc_x, where c_x is some condition containing occurrences of x.
(24)

Descriptions can be subjects to which predicates are assigned. We will use U to mean “was born in the USA.” This means we would write, “the man who first landed on the Moon was born in the USA” as:

ιx(xM & xF)U

Now we would like to make this look like a normal predication with just two symbols. We will abbreviate ιx(xM & xF) as μ. We now have

μU

Instead of our “was born in the USA” predicate U, we will return to the Moon examples. Thus we would write “The first man to land on the Moon is a man and he landed first on the moon” as

μM & μF

Recall what we said about quantifiers in the last section. In those cases, the truth of the formulations had something to do with the quantities of the things described. So for “All people are happy” to be true, that happiness must be shared by every person. However, descriptions are names and not quantifiers. So,

The man who first landed on the Moon was born in the USA”, μU, is true just if the particular person referred to by the phrase μ has the property expressed by U.
(26)

Now consider proper names, like “Annika” or “the Big Bang” (26). These merely designate the thing. They do not contribute additional information about what they refer to. But definite descriptions do carry this extra info. “Thus, for example, ‘the man who first landed on the Moon’ carries the information that the object referred to has the property of being a man and being first on the Moon” (26). This is actually not so trivial. Descriptions prove useful in certain mathematical and philosophical arguments. Priest will use the example of the Ontological Argument for the existence of God, which can take the following simple form:

God is the being with all the perfections.
But existence is a perfection.
So God possesses existence.
(26)

In other words, God exists. Perfections include omniscience, omnipotence, moral perfection… “[i]n general, the perfections are all those properties that it is a jolly good thing to have” (26). Now we wonder, why does the second premise say that existence is a perfection? The answer is a bit complicated and involves Plato’s philosophy. Priest says we can work around this issue by 1) making “a list of properties like omniscience, omnipotence, etc., include existence in the list, and simply let ‘perfection’ mean any property on the list” [as we will see, there is a problem with the structure of the argument, and it would not matter really how we justify this premise], and 2) we will “take ‘God’ to be synonymous with a certain description, namely, ‘the being which has all the perfections (i.e., those properties on the list)’” (p.27). Now in that light, consider the first premise: “God is the being with all the perfections”. This is the same as our second stipulation. So we can eliminate it for now. The second premise also: “Existence is a perfection” is the same as the first stipulation, and so we can eliminate it too. The argument now has one line:

The object which is omniscient, omnipotent, morally perfect, … and exists, exists.
(27)

We will make this more apparent by abbreviating. We will write God’s list of properties as P₁, P₂, …, P_n. The last one on the list, P_n, is the property of existence. Now let us use our definite description notation. We will predicate all these properties to the description for God:

ιx(xP₁ & … & xP_n)

We will abbreviate this whole description as γ. [In the following, it seems we regard the full sentence that we are abbreviating to be: “The object which is omniscient, omnipotent, morally perfect, … and exists, is omniscient, omnipotent, morally perfect, … exists.”] The one-line conclusion then becomes

γP₁ & … & γP_n (from which γP_n follows).
(p.27)

Priest explains that we are dealing here with an instance of something called the Characterization Principle. It can be stated as “a thing has those properties by which it is characterized” or “the thing satisfying such and such a condition, satisfies that very condition” (27). We will abbreviate the Characterization Principle as CP. Above we had another instance of it, namely, “The first man to land on the Moon is a man and he first landed on the Moon”, or μM & μF.

In general, we obtain a case of the CP if we take some description, ιxc_x, and substitute it for every occurrence of x in the condition c_x.
(27)

[So take for example again “The first man to land on the Moon is a man and he landed first on the Moon”. We begin simply with the description

ιx(xM & xF),

that is, “the object x who is a man and who first landed on the Moon.” Here, the condition c_x is (xM & xF). We see there are two occurrences of x. So now we substitute (xM & xF) into the x’s in the parentheses of

ιx(xM & xF).

We then get

ιx((xM & xF)M & (xM & xF)F)).

In other words, we get, “the first man to land on the Moon is a man and the first man to land on the moon is the first to have landed on the moon”.]

[So the formulation of CP again is “the thing satisfying such and such a condition, satisfies that very condition.” This seems tautological. Thus,] CP looks true by definition. But actually it is false, since it implies many things that are no doubt untrue (27).

One problem with its implications is that on the basis of the CP we can “deduce the existence of all kinds of things that do not really exist” (28). We can for example on its basis conclude that there is a greatest integer, when of course there is not [there can always be one greater by adding 1.] So we first consider the non-negative integers, going from 0, 1, 2, 3, and on and on. Now we select a condition, which will be our c_x in ιxc_x. We then make c_x be “x is the greatest integer and x exists.” [Let us formulate this so it is similar to the above, where we had the substitution:

ιx((xM & xF)M & (xM & xF)F)),

which stood for: “the first man to land on the Moon is a man and the first man to land on the moon is the first to have landed on the moon”. So let us in this case make ιxc_x be:

ιx(xG & xE),

meaning “the object x that is the greatest integer and it exists”. Now we substitute (xG & xE) for each case of x in the conditions. This gives us

ιx((xG & xE)G & (xG & xE)E).

So here perhaps we have “the greatest integer, which does exist, is the greatest integer, and it exists.” It is not entirely clear to me why the CP was needed for the absurdity. Before applying it, we already said that it exists. Perhaps the idea is the following. In the first case, “exists” is not predicated to x as much as it is just a description said to apply to it, putting aside whether x is a real thing or not. So we could for example formulate legitimate descriptions of things that could not possibly exist, like a square circle: x is a circle and x has sides like a square. We could even add that it exists. This is perhaps merely just ascribing properties, putting aside whether the described thing can actually take such predicates as “exists” or “is a possible object”. But with the CP, perhaps what we are doing is adding predication to the description. We are saying that the thing which is circular, square, and existing is a real thing. Here is perhaps where the problems arise, since we are moving away from combining properties to saying something about the thing which is affirmable or deniable. But while still just describing it, we are not yet interested in the affirmability or deniability of these combinations of properties. This is not how Priest describes it, however. Perhaps for him, in both cases there is predication that is affirmable or deniable, or that can be true or false. He will say that the second level of predication can only be true if the predications in the description refer to a real thing.]

Let c_x be the condition ‘x is the greatest integer & x exists’. Let δ be ιxc_x. The the CP gives us ‘δ is the greatest integer, and δ exists’.
(28)

There are even more absurdities that the CP leads to. [Let us also put the next example into our longer form just to make the mechanics visible. So again, we begin with the formulation ιxc_x. We will make the condition “x married the Pope,” and let us here write that xP. So we have ιx(xP). Now we substitute xP in for each case of x in the condition. This gives us now  ιx(xP)P, which might be read something like, “the one who is married to the Pope is married to the Pope”.  Again, it would seem that there is enough for the absurdity without the repetition of the qualification that this individual is married to the Pope. Once would seem to be enough. So I again wonder if the issue is that description is not on the level of truth and falsity, but the second instance of the predication is, since it is perhaps no longer part of the description but rather is a predication to a described object. Or, in light of what Priest later says, perhaps description is not a matter of truth or falsity, but rather one of referring or not-referring. And predications to non-referring descriptions will in most cases be false.]

The absurdities do not end there. Consider some unmarried person, say the Pope. We can prove that he is married. Let c_x be the condition ‘x married the Pope’. Let δ be the description ιxc_x. The CP gives us ‘δ married the Pope’. So someone married the Pope, i.e., the Pope is married.
(28)

Priest then addresses what is going on that makes these instances problematic. His answer seems to be that in those absurd instances, the description does not refer to anything. [Perhaps this then invalidates the predication. Since there is no person who married the Pope, it does not matter which predications we give to it, since there is nothing to which those predicates may apply.]

What is to be said about all this? A fairly standard modern answer goes as follows. Consider the description ιxc_x. If there is a unique object that satisfies the condition c_x in some situation, then the description refers to it. Otherwise, it refers to nothing: it is an ‘empty name’. Thus, there is a unique x, such that x is a man and x landed first on the moon, Armstrong. So ‘the x such that x is a man and x landed first on the moon’ refers to Armstrong. Similarly, there is a unique least integer, namely 0; hence, the description ‘the object which is the least integer’ denotes 0. But since there is no greatest integer, ‘the object which is the greatest integer’ fails to refer to anything. Similarly, the description ‘the city in Australia which has more than a million people’ also fails to refer. Not, this time, because there are no such cities, but because there are several of them.
(28)

This means that the CP is not problematic in those cases when the ιxc_x, that is, the unique object satisfying c_x, actually exists. In those cases, the CP holds. (28d)

[This next point gets very interesting I think. If the described object does not exist, this makes all predications to it false. Why this is so might be interesting to discuss. For, could the predications not be neither true nor false, or be things to which affirmation or denial cannot rightly apply? To say it is false might carry with it metaphysical assumptions, namely, that nothing true can be said of things that do not exist. I wonder what to do with the notion that ‘the Pope’s wife does not exist”? Would the predication, ‘is non-existent’ be a false predication too? Perhaps the idea is that for non-existing things, all ‘predications’ would be part of the description. So maybe we can describe a non-existing thing all we want using predicates, but we cannot predicate the description itself. I am not sure. Here is what Priest writes:]

But what if there is no unique object satisfying c_x? If n is a name and P is a predicate, the sentence nP is true just if there is an object that n refers | to, and it has the property expressed by P. Hence, if n denotes no object, nP must be false. Thus, if there is no unique thing having the property P, (if, for example, P is ‘is a winged horse’) (ιx xP)P is false. As is to be expected, under these conditions, the CP may fail.
(28-29)

Priest now asks how all this applies to the Ontological Argument? [The basic idea here will be that on the basis of the description, we do not know whether or not the described thing exists. The argument only works if we assume that there really is the thing described. But God’s existence was to be proved, and thus it cannot be assumed.] Recall that we described God with a series of predications, among which is that God exists: ιx(xP₁ & … & xP_n). We then made γ stand for the description, and we applied the CP to get γP₁ & … & γP_n. This in effect predicated existence to God. But, before we move to these predications of the description, we first need to establish whether or not the described thing exists. For otherwise the predications will be false.

So γ refers to this thing, and γP₁ & … & γP_n is true. If there is not, then γ refers to nothing; so each conjunct of γP₁ & & . . . & γP_n is false; as, therefore, is the whole conjunction. In other words, the instance of the CP used in the argument is true enough if God exists; but it is false if God does not exist. So if one is arguing for the existence of God, one cannot simply invoke this instance of the CP: that would just be assuming what one is supposed to be proving. Philosophers say that such an argument begs the question; that is, begs to be granted exactly what is in question. And an argument that begs the question clearly does not work.
29)

Priest ends by noting a problem with this rule that no true predications can be given to non-existing entities. [It seems similar to what we said above in brackets. In this case, we] consider a  mythological figure, Zeus. [The insight seems to be that there are non-existing things that rightly have certain predicates.] Zeus’ description could be “the most powerful of the ancient Greek gods,” and his predicates could be “lived on Mount Olympus,” “was worshipped by the Greeks,” and so on (29). [These predicates are true, but the description refers to a non-existing entity. It seems here that instead of saying that on the basis of its non-existence the predicates are false, we instead say that the predicates are true despite its non-existence.] So if it is right that no Greek gods existed,

then the description ‘the most powerful of the ancient Greek gods’ does not refer to anything. But in that case, there are true subject/predicate sentences in which the subject term fails to refer to anything, such as ‘The most powerful | of the ancient Greek gods was worshipped by the Greeks’. To put it tendentiously, there are truths about non-existent objects, after all.
(29-30)

[I wonder then how this would apply to the God example. Could it be said that God is a fictional entity to which it rightly can be said that God exists? Is there another notion of truth at work here, perhaps something that could be called, fictional truth?]

 

[The following section is entirely quotation.]

 

Main Idea of the Chapter

● ιxc_x is true in a situation just if, in that situation, there is a unique object, α, satisfying c_x and αP.
(quoted from Priest, 30, boldface his)

 

 

From:

 

Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.

 

 

Priest, Ch3 of Logic: A Very Short Introduction, “Names and Quantifiers: Is Nothing Something?”, summary

by Corry Shores

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Summary of

Graham Priest

Logic: A Very Short Introduction

Ch.3
Names and Quantifiers: Is Nothing Something?

Brief Summary: 
When we speak of things, we might refer to some specific thing by name, like if we say, “Marcus came to the party”. In this case, what we are saying refers just to this one named person or thing. Or we might speak broadly and universally of all of a group of things, like if we said, “everyone came to the party”. In this case, what we say of the people or things applies to all of them. Or, we might refer to some thing, but without designating it specifically with a name, like when we say, “Someone came to the party”. Here we are saying something about a person or thing, but we are not specifying which one. When we want to speak of some thing or another, as in, “someone is happy,” we could use the existential quantifier and formulate this as, ∃x xH, meaning, there is some x such that x is happy. Or if we wanted to say, “Everyone is happy,” we could write ∀x xH, meaning, for all x, x is happy. Note that from just one quantified sentence an inference can be drawn. For example, if all people are happy, then there is some person who is happy. By using quantification, we can settle debates in mathematics and philosophy.

Summary

Previously we examined inferences with phrases like “or” and “it is not the case that” (17). These words are joined to “whole sentences to make other whole sentences” (17). But there are inferences that work in ways different from these. Consider this one:

Marcus gave me a book.

Therefore,

Someone game me a book.

[Previously we determined validity not on the basis of what was within each sentence but rather on the basis of rules of the combinations, additive modifications, and inferences of whole sentences. Here, however we are not combining or modifying sentences.] “Neither the premiss nor the conclusion has a part which is itself a whole sentence. If this inference is valid, it is so because of what is going on within whole sentences” (17).

The simplest whole sentences have a subject and predicate. Priest has us consider these examples:

1) Marcus saw the elephant.
2) Annika fell asleep.
3) Someone hit me.
4) Nobody came to the party.

The subject tells us what the sentence is about, and the predicate tells us what is said about the subject (17-19). We now wonder, what makes these sentences true? They would be true if the subject really does have the property ascribed to it by the predicate. “Take the second example. It is true if the object referred to by the subject ‘Annika’ has the property expressed by the predicate, that is, fell asleep” (19).

But, consider again sentence 3: “Someone hit me”. What is its subject? Perhaps it is the person who hit you. But what if the speaker is lying, and thus no one hit you. Sentence four, “Nobody came to the party” is even less certain about who the subject would be, since “’nobody’ does not refer to a person – or to anything else” (19). While ‘Marcus’ and ‘Annika’ are proper names that refer to some specific person, ‘nobody’, ‘somebody’, and ‘everyone’ are quantifiers [since they refer to some quantity of subjects] (19).

We now look at the standard modern explanation for how quantifiers work. To do this, we will use notation to simplify the matter. A subject will be represented with a lower case letter, and the predicate with an upper case one. We combine the lower and upper to signify the predication of the subject. That predication is true if the subject has that predicated property.

A situation comes furnished with a stock of objects. In our case, the relevant objects are all people. All the names which occur in our reasoning about this situation refer to one of the objects in this collection. Thus, if we write m for 'Marcus', m refers | to one of these objects. And if we write H for 'is happy', then the sentence mH is true in the situation just if the object referred to by m has the property expressed by H. (For perverse reasons of their own, logicians usually reverse the order, and write Hm, instead of mH. This is just a matter of convention.)
(19-20)

Now, when we use the quantifier “someone”, it is like a sort of variable. We mean that there is some object in the collection which has that predicated property, and we use ∃x, the particular quantifier, to represent this object.

Now consider the sentence ‘Someone is happy’. This is true in the situation just if there is some object or other, in the collection of objects, that is happy – that is, some object in the collection, call it x, is such that x is happy. Let us write ‘Some object, x, is such that’ as ∃x. Then we may write the sentence as: ‘∃x x is happy’; or remembering that we are writing ‘is happy’ as H, as: ∃x xH. Logicians sometimes call ∃x a particular quantifier.
(20)

The universal quantifier, then, would be when we speak of every object, as in everyone, and we write it ∀x.

What about ‘Everyone is happy’? This is true in a situation if every object in the relevant collection is happy. That is, every object, x, in the collection is such that x is happy. If we write ‘Every object, x, is such that’ as ∀x, then we can write this as ∀x xH. Logicians usually call ∀x a universal quantifier.
(20)

For ‘Nobody is happy’, that is, for cases where there is no object, x, in the relevant collection, such that x is happy, we merely write:

¬∃x xH

rather than make a new symbol, “For to say that no one is happy is to say it is not the case that somebody is happy” (20).

Names and quantifiers work very differently. The fact that we write ‘Marcus is happy’ and ‘Someone is happy’ in these two different ways:

mH
∃x xH

tells us that “not all grammatical subjects are equal” (20d). Now, recall the original inference:

Marcus gave me a book.

Therefore,

Someone game me a book.

Which we may write:

  mG    
∃x xG

We see now why it is valid. If at least one person, Marcus, gave me a book, that means someone game me a book. Now let us look at an inferences from sentences with “nobody.” Previously Priest quoted these famous lines from Lewis Carroll’s Through the Looking Glass:

‘Just look along the road, and tell me if you can see . . . [the Messenger]’
’I see nobody on the road.’ said Alice.
‘I only wish I had such eyes,’ the King remarked in a fretful tone. ‘To be able to see Nobody! And at that distance tool! Why, it's as much as I can do to see real people, by this light!’
(qtd. in Priest 19)

Here, Alice sees nobody, and from that fact the king infers that she sees somebody, namely, “Nobody.” So let us write the predicate “is seen by Alice” as A to formulate this inference:

   ∃x xA  
  ¬∃x xA

This of course is invalid. There is a “relevant domain” [presumably, the things that are visible to Alice in that situation]. And we are saying there is no object in that relevant domain. Obviously then, it is not true that there is some object in that domain (21).

Priest notes that quantifiers are important also in very serious debates in mathematics and philosophy. Priest proceeds to show how they are useful in a certain debate regarding the existence of God. We begin by assuming that there is a reason or explanation for everything. “people don’t get ill for no reason; cars don’t break down without a fault” (21).  So everything has a cause. But then we ask, what is the cause of everything? [The use of ‘everything’ might seem ambiguous here, since perhaps we are thinking about a long chain of causal relations and we want to know what the first one is, or perhaps it means something like, ‘the reason why all things are here in the first place and are acting causally upon one another’. But this ambiguity, as we will see, is what is at issue here.] The cause of everything cannot be a physical thing like a person for example. Could it be something like the Big Bang of cosmology? No, because even this would have a cause. It must be something metaphysical, and “God is the obvious candidate” (21).

The above is the idea behind an argument for the existence of God called the ‘cosmological argument’. As we will see, it is based on a logical fallacy that we can uncover using quantification. “Everything has a cause” ambiguously means two things: 1) that for each event, there is yet another event which caused the first one we mentioned, “that is, for every x, there is a y, such that x is caused by y,” or 2) that there is one single thing or event that causes each and every other one, “that is, there is some y such that for every x, x is caused by y” (21-22). So we have sort of relation, which is one thing being caused by another thing. We will call this relation C. And we will then write ‘x is caused by y’ as xCy. We can then write formulate the two above meanings using quantification, which will make plain their very different meanings.

1. ∀x ∃y xCy
2. ∃y ∀x xCy

[I will venture a rewording. The first one is saying that for all x’s, there is a y such that x is caused by y. In other words, consider one x or another. Every one you consider will have a cause, y. However, we do not know if there is just one y for all of them, or if there is a different y for each x, or perhaps if some x’s share a common y. The second one says that there is a y such that for all x’s, each one is caused by y. In other words, there are many x’s, and they all are caused by the same y. So it seems what is important is the order of the quantification. For, the question could be, why is it just by switching the order of the quantifications does the meaning change? I am guessing here, but perhaps the first one sets a context for the following quantifications. When we begin with a universal, like in the first case, the focus is on all members of this domain. Then when we follow that with the particular quantification, it is in reference to all those members, and so there can perhaps be many of these particular things. However, when we begin with the particular quantification, we are now talking about this one thing. Then when we follow with the universal, all these many things are now understood in terms of the first singular one.]

Priest then observes that the two formulations we made are not logically equivalent, since from the second one we can infer the first; however, from the first we cannot infer the second. If, as the second one says, there is one thing that causes everything else, then we also know, like the first says, that everything has some cause. However, from the fact that everything has a cause does not mean that we can infer that there is one cause for everything. For, there could be a different cause for each effect.

The Cosmological Argument, then, operates incorrectly on the basis of the ambiguity in the formulation.

From this example we can see why we must be clear about our quantifiers. Also, we see that for the most part “something” and “nothing” “do not stand for objects, but function in a completely different way” (22). But Priest then notes that in certain cases they can stand for some certain thing. We consider two claims: 1) the cosmos goes back infinitely into the past, and thus has no beginning, or 2) the cosmos came into existence at some particular time. Now we pose a formulation, “the cosmos came out of nothing.” Which of the two does it apply to? It does not apply to the first one, since here the cosmos was always there, and thus, it did not come out of nothing. Rather, it applies to the second one. For, if the cosmos comes about at a particular time, then presumably there was nothing before it, and thus it comes out of nothing. Now, let us formulate “the cosmos came out of nothing” using quantifiers. We will write ‘x came into existence out of y’' as xEy, and we will call the cosmos c. What do we get?

1)   ¬∃x cEx

So the formulation applies both to “the cosmos came out of nothing” and “the cosmos came to be at some particular time.” This is what we wanted. We would also hope that it does not apply to the first case. Does it? In the first case, where there is no beginning to the cosmos, there also is nothing coming before it. So, ‘to come out of nothing’ cannot simply mean that there was not some thing that came before everything else. Rather,

When we say that in the second cosmology the cosmos came into existence out of nothing, we mean that it came into being from nothingness. So nothing can be a thing. The White King was not so foolish after all.
(23)

[The following is entirely quotation.]

Main Ideas of the Chapter
● The sentence nP is true in a situation if the object referred to by n has the property expressed by P in that situation.
● ∃x xP is true in a situation just if some object in the situation, x, is such that xP.
● ∀x xP is true in a situation just if every object in the situation, x, is such that xP.
(23)

 

From:

Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.