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by Corry Shores
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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own.]
Summary of
John Nolt
Logics
Part 3: Classical Predicate Logic
Chapter 6: Classical Predicate Logic: Syntax
6.3 Identity
Brief summary:
We can use the ‘=’ operator – meaning “is identical to” or “is the same thing as” – to articulate a number of things normally inexpressible in the language of predicate logic. It is an operator, because its meaning remains constant in all cases. But it dually is a two-place predicate, relation, or function, because it relates one term to another in a semantic manner where the extension of one term is said to be no different than the extension of the other. The non-identity of terms is written normally as ‘~a=b’, and so this does not mean ‘not-a is identical to b’ but rather that ‘a is not identical to b’. This is because for atomic formulas, when they take a negation operator, its scope extends over the entire formula. We also can write the non-identity of terms as ‘a≠b’ (but ‘~a=b’ is less problematic and thus preferred). Using the equality operator, we can render the following meanings.
Else
‘God is more perfect than anything else’
∀x(~x = g → Pgx)
Other than
‘Al will go with anyone other than Beth’
∀x(~x = b → Gax)
Except
(for one term)
‘Al will go with anyone except Beth’ [...]
∀x(~x = b → Gax) &~Gab
or
∀x(~x = b ↔ Gax)
Except
(for two terms, where the predication of the terms is not determined)
‘Everyone except Al and Beth is happy’
∀x((~x = a & ~x = b) → Hx)
Except
(for two terms, where the predication is denied to the exceptional terms)
∀x((~x = a & ~x = b) → Hx) & (~Ha & ~Hb)
or
∀x((~x = a & ~x = b) ↔ Hx)
Superlatives
‘Al is faster than all other runners’
Ra & ∀x((Rx & ~x = a) → Fax)
‘There is no largest number.’
~∃x(Nx & ∀y((Ny & ~y = x) → Lxy))
Only
(for one term)
‘Only Al is happy’
Ha & ∀x(Hx → x = a)
Only
(for two terms)
‘Only Al and Beth are happy’
∀x(Hx ↔ (x = a ∨ x = b))
At least
(for one term)
‘There is at least one mind.’
∃xMx
At least
(for two terms)
‘There are at least two minds.’
∃x∃y((Mx&My) & ~x = y)
At least
(for three terms)
‘There are at least three minds.’
∃x∃y∃z(((Mx&My) & Mz) & (~x = y & ~y = z) & ~x = z)
At most
(for one term)
‘ There is at most one mind.’
∀x∀y((Mx & My) → x = y)
or
~∃x∃y((Mx&My) & ~x = y)
or
∃x∀y(My → y = x)
At most
(for two terms)
‘There are at most two minds.’
∃x∃y∀z(Mz → (z = x ∨ z = y)
At most
(for three terms)
‘There are at most three minds.’
∃x∃y∃z∀w(Mw → ((w = x ∨ w = y) ∨ w = z))
At most
(for all things)
‘There is at most just one thing.’
∃x∀y y = x
Numerical Quantifiers
(for one term)
‘There is exactly one mind.’
(∃xMx & ∃x∀y(My → y = x)
or
∃x∀y(My ↔ y = x)
Numerical Quantifiers
(for two terms)
‘There are exactly two minds.’
∃x∃y(~x = y & ∀z(Mz ↔ (z = x ∨ z = y)))
Definite Description
(affirmative)
‘The present king of France is bald’
or
‘There is at least one thing which is presently king of France, it alone is presently king of France, and it is bald.’
∃x((Kx & ∀y(Ky → y = x)) & Bx)
Definite Description
(negating existence of)
‘It is not the case that there is at least one present king of France, who alone is presently king of France, and who is bald.’
~∃x((Kx & ∀y(Ky → y = x)) & Bx)
Definite Description
(negating the predication of)
‘There is at least one present king of France, who alone is presently king of France, and who is not bald.’
∃x((Kx & ∀y(Ky → y = x)) & ~Bx)
A definite description, like “the present king of France,” seems like a name, but in fact it has the more complex logical structure of “the F is G”, or
∃x((Fx & ∀y(Fy → y = x)) & Gx)
When we formulate propositions using definite descriptions for non-existing entities, we can fall into error by not translating the ambiguous English formulations into the unambiguous formalizations. This allows us to avoid paradoxes that really are not inherent to such instances. For example, we have “The present king of France is bald.” We have the first intuition that it is false, because there is no such thing which takes that predicate, so the predication fails. We then have the second intuition that since it is false, its negation must be true. And so we think, “The present king of France is not bald” is true. But we as well have the third intuition, which is basically the same as the first, that the sentence “The present king of France is not bald” is false, because it refers to a non-existing entity and the predication therefore fails. So the same sentence, “The present king of France is not bald” is both true and false, under two equally valid intuitions. But Bertrand Russell’s analysis of definite descriptions shows that in fact we are not affirming and denying the same sentence. Rather we are saying that “There presently is no bald king of France” is true while “There is presently a bald king of France” is false. [See the last two formalization in the above listing.]
Summary
Nolt will discuss the use of the logical operator of ‘=’ in predicate logic. What makes it unique as an operator is that unlike the others he has discussed so far, this operator is a predicate, specifically a two-place predicate. But it cannot be conceived just simply as a predicate like ‘F’ or ‘L’. Such predicates as these can have variable meanings, but ‘=’ always has a fixed meaning, namely, “is identical to” or “is the same thing as” which makes it an operator as well (Nolt 174). Another reason for considering it as an operator is that unlike with predicates, we often write the equals sign between the terms it relates. One important thing to note is that this custom of placing the equals sign between the terms is that it can create misunderstandings when identities are negated. So suppose ‘a’ means ‘Alice’ and ‘b’ means ‘Bob’. If we want to write ‘Alice is not Bob’, we would write:
~a=b
But in seeing this we might be tempted to incorrectly read it as ‘not-Alice is identical to Bob.” The reason this is incorrect has to do with the scope of the negation operator. Nolt says that for atomic formula, “the negation operator’s scope extends over the entire formula” (Nolt 175). [So this could explain why the scope would be different were we to write for propositions ~A&B.] We might also wonder, what if we wanted to write, ‘not-Alice is identical to Bob’? Nolt seems to be saying that this is not really a situation we would find ourselves in, because there is really no way to conceptualize what not-Alice is.
To avoid the misreading we noted above, sometimes people place brackets around the formulation like: ~(a=b). Nolt objects to this for two reasons. It adds too much extra writing, and it “further differentiates ‘=’ from other two-place predicates” (175). [I am not sure why that would be a problem, however.] The other solution is to use the not-equals sign ‘≠’ as in ‘a≠b’, and Nolt will sometimes use it in metalanguage text. However, this solution has a problem. Suppose you wanted to double negate and identity formulation. [I think the idea is that maybe we are asking, how do we write, “it is not the case that a does not equal b” or something like that. I suppose that normally we could write ~~a=b, which would not be problematic. However,] when we use the ‘≠’ symbol, it would be ‘~a≠b’. But this does not look like a double negation, and it can also lead to misreadings [which I suppose would be like not-Alice does not equal Bob.]
Now that we have the identity predicate in our language of predicate logic, we will need to modify our definition of an atomic formula.
Any predicate followed by zero or more names or any formula of the form α = β, where α and β are names, is an atomic formula.
(Nolt 175)
As we will see, the identity predicate will enable us to express many sorts of concepts (Nolt 175).
Else, Other Than, Except
We sometimes express the notion of nonidentity with the terms ‘else’, ‘other than’, and ‘except’. Nolt gives an example. Here is the key for the letter symbols.
‘g’ is ‘God’
‘P’ is ‘more perfect than’
Now suppose we wanted to make the absurd claim first that God is more perfect than everything, including himself. In other words, included in our meanings would be the absurd claim that Pgg. We would write that as ∀xPgx, meaning, God is more perfect than all things (or, for all x, God is more perfect than x). But that is not what we want to say. Instead we want to say that God is greater than all things, except himself. Nolt has it written as:
‘God is more perfect than anything else’
∀x(~x = g → Pgx)
(Nolt 175)
[I am not entirely sure how this would be read and understood. The basic idea is that we are saying for all things that are not identical with God, God is greater than them. So the idea here is that we might be saying, ‘for all x, if x is not identical with g, then g is greater than x.]
Nolt now will explain how to render the expression ‘other than’. He has us consider the example,
‘Al will go with anyone other than Beth’
∀x(~x = b → Gax)
(Nolt 175)
[This has the same structure as before.] If we want to give the meaning of ‘except’, we will need extra notation, because it has a stronger meaning than ‘other than.’
‘Al will go with anyone except Beth’ [...]
∀x(~x = b → Gax) &~Gab
– or more compactly
∀x(~x = b ↔ Gax)
[This compact formulation is equivalent to the first, since ‘~Gab’ is equivalent to ‘∀x(Gax → ~x = b)’]
(Nolt 175, bracketed text his)
[Let me first comment on the compact version. Nolt seems to be saying that the larger and the more compact formulations are equivalent, because an equivalent form of ~Gab is ‘∀x(Gax → ~x = b)’, which, when taken together with ∀x(~x = b → Gax)’, can derive ∀x(~x = b ↔ Gax).]
Nolt then shows how we can apply ‘else,’ ‘other than,’ and ‘except’ to more than one individual. He gives this example:
‘Everyone except Al and Beth is happy’
∀x((~x = a & ~x = b) → Hx)
(Nolt 176)
[Here the idea seems to be something like, for all people x, if x is not Al or x is not Beth, then x is Happy.] And if we mean in addition to everyone other than Al and Beth being happy that also Al and Beth are not happy, then:
∀x((~x = a & ~x = b) → Hx) & (~Ha & ~Hb)
or
∀x((~x = a & ~x = b) ↔ Hx)
(Nolt 176)
Superlatives
Nolt will show how to render superlatives by using both comparatives and the identity predicate. Here is an example:
‘a’ is ‘Al’
‘R’ is ‘is a runner’
‘F’ is ‘is faster than’
‘Al is faster than all other runners’
Ra & ∀x((Rx & ~x = a) → Fax)
(Nolt 176)
[Here the idea seems to be that we are saying, Al is a runner, and for all things x, if they are runners and they are not identical to Al, then Al is faster than them.] Here is another example.
N = is a number
L = is larger than
‘There is no largest number.’
~∃x(Nx & ∀y((Ny & ~y = x) → Lxy))
(Nolt 176)
[This one is a little complicated. Let us read it out first, and then process it. There is no x such that x is a number and for all y, if y is a number and y is not identical with x, then x is the larger than y. Looking overall at the structure, we seem to be saying that there is not a number for which there is no other number that is larger than it. In terms of the x and y, we seem to be saying with the ‘for all y’ that we mean, given any possible y number, there is not some number that is larger than all those y numbers. I am also not exactly sure why we need the ~y = x part, but of course the largest number we are positing, and also denying that it exists, would not be identical to any of the other numbers that it would have been larger than.]
Only
We use only to mean that some particular thing alone has some property. This furthermore means that “anything that has that property is identical with it”. For this reason, we would formulize the following sentence in this manner.
‘Only Al is happy’
Ha & ∀x(Hx → x = a)
(Nolt 176)
[Here we are saying that Al is happy and that were anyone else happy they would really just be Al himself.] It is not as simple as we might think if we want to say, “Only Al and Beth are happy”. We cannot write it as:
∀x(Hx ↔ (x = a & x = b))
(Nolt 176)
[This seems to be saying that the only happy people are Al and Beth.] The problem with this is that implies something is happy only if it is both Al and Beth. We need instead the following formulation when we mean that Al and Beth are different people.
∀x(Hx ↔ (x = a ∨ x = b))
(Nolt 176)
At least
We already have seen how to write ‘at least one’. For example,
‘There is at least one mind.’
∃xMx
(176)
What about, “There are at least two minds”? We cannot simply write, ∃x∃y(Mx&My), because x and y under this formulation could be the same object. Thus we would have instead:
‘There are at least two minds.’
∃x∃y((Mx&My) & ~x = y)
‘There are at least three minds.’
∃x∃y∃z(((Mx&My) & Mz) & (~x = y & ~y = z) & ~x = z)
(Nolt 176)
At most
If we say there are ‘at most’ so many things, that means there are not more than that quantity. So if we say ‘there is at most one mind’, “this means that no two distinct things are minds; that is, if we choose any object x and any | object y and they turn out both to be minds, then they are identical” (Nolt 176-177).
‘There is at most one mind.’
∀x∀y((Mx & My) → x = y)
(Nolt 177)
We might also notice that to say there is at most one mind is also to deny that there are at least two minds.
~∃x∃y((Mx&My) & ~x = y)
(Nolt 177)
Nolt now will give a schema for formulating “at most” structures where there are more than two things, as the patterns above become too ungainly in these cases. He first shows the pattern for at most two things.
‘There are at most two minds.’
∃x∃y∀z(Mz → (z = x ∨ z = y)
(Nolt 177)
[This formulation seems to give us the following possibilities: {1} There is no object z that is a mind (deny the antecedent; total minds = 0), {2a} There are some minds, with one being object x and the other being object y (affirm the antecedent, affirm both disjuncts, x≠y; total minds = 2), {2b} There is some mind, with it being both object x and also being object y (affirm the antecedent, affirm both disjuncts, x=y; total minds = 1), {3a} There is some mind, with it being object x but not object y (affirm the antecedent, deny the right disjunct, x≠y; total minds = 1), {3b} There is some mind, with it being object y but not object x (affirm the antecedent, deny the left disjunct, x≠y; total minds =1). As we can see, there is no possibility for there being more than two minds under this structure.] Here is how we can use this structure for other numbers of at most things.
‘There is at most one mind.’
∃x∀y(My → y = x)
‘There are at most three minds.’
∃x∃y∃z∀w(Mw → ((w = x ∨ w = y) ∨ w = z))
(Nolt 177)
Certain mystics argue that there is just one thing in the cosmos. We could write that:
‘There is at most just one thing.’
∃x∀y y = x
(Nolt 177)
Numerical Quantifiers
[Our above formulations for ‘at most’ tell us the upper limit, but there could be any quantities below that limit in those cases.] We now will make formulations for when we specify the exact number of things. Suppose we want to say, ‘there is exactly one mind’. This means both that there is at least one mind and at most one mind. Thus we write it:
‘There is exactly one mind.’
(∃xMx & ∃x∀y(My → y = x)
(Nolt 177)
[This seems to say that there is some thing that is a mind and were anything else to also be a mind, it actually would just be the same one just mentioned.] Nolt then gives a more compact formulation of this.
‘There is exactly one mind.’
∃x∀y(My ↔ y = x)
This combines into a single quantified biconditional the statement ‘there is at least one mind’ – which is normally just ‘∃xMx’ but may be equivalently written as ‘∃x∀y(y = x → My) – and ‘there is at most one mind’ – ∃x∀y(My → y = x)’.
(Nolt 177)
Nolt then explains how we can use this structure to write:
‘There are exactly two minds.’
∃x∃y(~x = y & ∀z(Mz ↔ (z = x ∨ z = y)))
(Nolt 177)
[Here we seem to be saying the following. Firstly there is some object x and some object y, but they are not the same object. Next, we are saying that were there any other object, it must be identical to at least one of these other two. Thus there are exactly two minds.] This structure allows us to “formalize numerical quantities, which are phrases of the form ‘there are exactly n...’, where ‘n’ stands for a number” (Nolt 177).
Russell’s Theory of Definite Descriptions
We consider the famous sentence, “The present king of France is bald.” Since France is no longer a monarchy, what are we to think of this sentence? Is it true or false? And what makes it either true or false. To deal with these situations, Bertrand Russell created the notion of definite description. A definite description, Russell says, takes the form ‘the so-and-so’. Here are some examples:
Examples of definite descriptions:
“The present king of France”
“The positive square root of two”
“The supreme being”
“The woman I met last week”
(Nolt 178)
Now take the example of ‘The present king of France is bald’. It seems to have the subject predicate form we are used to, with the present king of France, ‘k’, being predicated by ‘is bald’, ‘B’, and thus we might write, ‘Bk’. Nolt notes that there are problems with this formulation. We have a name [which could only be a name of an object], but it is unclear what object it names, if there be any such thing. The metaphysics of the situation is very unclear, because we may not be sure that we understand exactly what kind of entity the present king of France is (Nolt 178).
This furthermore leads to a paradox. Consider if we say that ‘the present king of France is bald’ is false [v(Bk) = F], perhaps because there is no such king. But that means the negation of the statement must be true [v(~Bk) = T]. So, ‘the present king of France is not bald’ is true. But if we use the same reasoning that gave us our first valuation, namely that when the sentence predicates a non-existing object it is false, that means ‘the present king of France is not bald’ would also be false [v(~Bk) = F], because it is predicating a non-existing object. But now we have ~Bk as having both the values true and false, which is a paradox. [Note. Graham Priest has a similar discussion of this sentence in the context of evaluating the truth-value gaps sort of non-classical logic, in section 4.7 of In Contradiction. He also discusses definite descriptions in general in chapter 4 of his Logic: A Very Short Introduction.]
Russell diagnosis the problem as lying in the fact that we should not render ‘The present king of France is bald’ as ‘Bk’ (Nolt 178). This is because “Definite descriptions, he argues, are not simple names; sentences containing them have a logical structure quite different from their surface grammar” (Nolt 178). [In the case of Bk, we have a sentence of the structure ‘the F is G’, in our case, the k is B.] Nolt explains that sentences of the structure ‘the F is G’ have three important components to what they mean:
1. There is at least one F,
2. it alone is F, and
3. it is G.
In symbols this is
∃x((Fx & ∀y(Fy → y = x)) & Gx)
(Nolt 178)
[In the formalization, we are first saying that there is some thing x which is F. So already we see that instead of just saying ‘k’ for the king of France, we are saying that there is an object which is the king, or perhaps something like Fa. The next part of the formalization is saying that there is only one such thing that is F, because were there any other thing, it would be identical to that thing which is F. And finally it is saying that this x is G. So for our example we might put this as, there is a thing which is the king of France, and were there any other king of France it would really have to be no other than this one same king, and this king of France is bald.] As we can see, “Russell’s analysis dissolves the purported name ‘k’ into a complex of logical relationships involving the one-place predicate ‘is presently king of France’” (Nolt 178). Nolt now renders the formalization using our example.
K = is presently king of France
B = is bald
[the following now is quotation]
(A) ∃x((Kx & ∀y(Ky → y = x)) & Bx)
Translating back into English we get
1. There is at least one thing which is presently king of France,
2. it alone is presently king of France, and
3. it is bald.
(Nolt 178)
This formulation is false. [I think this is because there is no such x that fulfills those qualifications.] This makes its negation true. So
~∃x((Kx & ∀y(Ky → y = x)) & Bx)
(Nolt 178)
is true. And thus its translated form
(i) It is not the case that there is at least one present king of France, who alone is presently king of France, and who is bald,
is true. This accounts for the intuition that ‘The present king of France is not bald’ is true.
(Nolt 178)
So we have the intuition that “The present king of France is not bald” is true [here the intuition came from us negating what we had the intuition of as being true.] We also had the intuition that this same sentence is false [because it refers to something that does not exist.] But what Russell will do is not to affirm and deny the same sentence. Rather, he will distinguish a different formulation to which the second intuition corresponds. [The two ways are to either negate the initial existential quantifier and to say that it is not the case that there presently exists a bald king of France. Here we are saying, no bald king of France exists. This is true. The second formulation, which is for the second insight, says that there is a present King of France who is not bald. This is false. So it is not the same sentence which is both true and false. The sentence that is true is that there is presently no bald king of France. The sentence that is false is that there is presently a king of France who is bald. But both of those sentences do not also have the opposing truth value. The confusion arises because both intuitions are built into the ambiguous sentence “The present king of France is bald.”]
But what of the intuition that this same sentence is false? That is also right, according to Russell! For the sentence is ambiguous; it can also mean
(ii) There is at least one present king of France, who alone is presently king of France, and who is not bald,
that is, in symbols:
(C) ∃x((Kx & ∀y(Ky → y = x)) & ~Bx)
Statement (ii) and its formalization (C) are both false for the same reason that (A) is: There is no present king of France. The difference between (B) and (C) lies in the scope of the negation operator. The ambiguity is therefore a matter of scope, like the ambiguities that arise from the mixing of universal and existential quantifiers in English. Formalization reveals the ambiguity and untangles conflicting intuitions; the English sentence that appears to be both true and false is from a logical point of view really two sentences. What troubled us, then, was merely a grammatical illusion.
In summary, Russell shows how certain sentences containing expressions that seem to refer to nonexistent entities can be analyzed into formulas containing only well-understood quantifiers and predicates. This both dispels logical paradox and forestalls a metaphysical snipe hunt.
(Nolt 179)
From:
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
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