The sign “=” expresses identity. The sentence “a=a” tells us that the object a is identical with (the same as, not different from) itself; “a=b” – that a and b are actually the same thing; “∀x x=x” – that everything is identical with itself; “∃x x=a” – that there exists something that is identical with a, i.e. that a exists.
As things are identical or not with things, to the left or right of the identity (equality) sign can be a constant or a variable – constants and variables are the two types of symbols in the language of predicate logic that denote things (predicates, respectively predicate letters, do not denote things).
a and b are different things if and only if they are not identical, ¬a=b. For short, instead of “¬a=b” we may write “a≠b”; “…≠…” is an abbreviation for “¬…=…”.
The addition of the identity sign to the language of predicate logic is accompanied by the addition of a new rule to both its syntax and its semantics. The syntactic rule is that if i and j are constants or variables (one can be a constant and the other a variable), then “i=j” is a well-formed formula. The semantic rule (which, like the other semantic rules, is formulated with respect to some structure consisting of a universe of discourse, interpretation, and possibly assignment) is that “i=j” is true in the structure if the interpretation (respectively, the assignment) relates “i” and “j” to the same thing in the universe of discourse; and it is not true if it is not so. For example, the sentence “a=b” will be true in some universe of discourse for some interpretation of “a” and “b” if “a” and “b” denote, according to the interpretation, the same thing, and it will be false if they denote different things. The definitions of logical inference, logical equivalence, logical validity, etc. remain the same. For example, logically valid are still the formulas that are true in every universe of discourse for every interpretation and assignment. For example, the formula “∃x x=a” is logically valid, because whichever universe of discourse we take and no matter how we interpret “a” in it, there will always be something that is identical to what we have given as an interpretation to “a” – namely, this very thing.
The identity (equality) sign enriches the language of predicate logic. It can be used to affirm that there are n, at least n, or at most n things (of some kind or absolutely), where n is an arbitrary natural number (1, 2, 3, …). For example, “∃х∃у х≠у” tells us that there are at least 2 things. Literally: “There exist x and y, which are not the same thing”. “∀х∀у∀z (х=у ∨ x=z ∨ y=z)” tells us that there are at most 2 things. Literally: “Whatever x, y and z we take, at least two of them will be identical – thus there can be no three or more things.
If we interpret “F” as “…is a genius” and limit the universe of discourse to humans, the following formulas will correspond to the following natural language sentences:
∃хFx – There is at least one genius. |
∀х∀y[(Fx ∧ Fy) → x=y] – There is at most one genius. |
Literally: “Whatever persons x and y we take, if each of them is a genius, they will be the same person” – thus it is claimed that there is no more than one genius, without claiming that geniuses exist at all.
∃x[Fx ∧ ∀y(Fу → x=y)] – There is exactly one genius. |
Literally: “There is a person x who is a genius, and whatever person we take, if he (she) is a genius, he (she) is identical with x” – the first part of the sentence guarantees that geniuses exist and the second excludes that they are more than one.
∃х∃y(Fx ∧ Fy ∧ x≠y) – There are at least two geniuses. |
Literally: “There is a person x and a person y who are geniuses and are not the same person”.
∀х∀y∀z[(Fx ∧ Fy ∧ Fz) → (x=y ∨ х=z ∨ y=z)] – There are at most two geniuses. |
Literally: “Whatever people x, y, and z we take, if they are geniuses, then at least two of them are the same person” – this excludes the existence of three or more geniuses without assuming that there is one or two.
∃x∃y{Fx ∧ Fy ∧ x≠y ∧ ∀z[Fz → (z=x ∨ z=y)]} – There are exactly two geniuses. |
Literally: “There is some person x and some person y who are geniuses and are not the same person (so far it is guaranteed that at least two geniuses exist), and whatever person we take, if he (or she) is a genius, he (or she) is identical to x or y (thus ensuring that there is no third genius)”.
In a similar way, we could continue with 3, 4, ..., 1000, ... geniuses.
Sentences such as “There are exactly two geniuses” or “There are at most two geniuses” cannot be symbolized without the use of the identity sign. There are many other such sentences. Such is, for example, the sentence “There is a man who despises all other people”. If we limit the universe of discourse to humans and symbolize “…despises…” with “F”, the formula
∃x∀yFxy |
will not correspond to “There is a man who despises all other people” but to “There is a man who despises all people” (literally: “There is a person x such that whatever person we take, x will despise him (her)”). The latter implies that the person in question despises himself (herself) as he (she) is also a person. By inserting the word “other”, we do not want to say this; we want to say that there is a person x such that whatever person we take, if he (she) is different from x, then x despises him (her). The latter is symbolized with the help of the identity sign as follows:
∃x∀y(y≠x → Fxy) |
In order to be able to apply our proof procedure (see the previous section) to inferences involving the sign of identity, we need to provide it with two additional principles, which we will call Law of identity and Indiscernibility of identicals^{1}.
The law of identity expresses the obvious truth that everything is identical with itself. If “a” stands in the place of an arbitrary constant, it is stated as follows:
Law of identity (LI): | a = a |
The low of identity allows us to use as axioms sentences such as “a=a”, “b=b”, “c=c”, etc. in one proof at any time. Whenever we wish, we can write such a sentence on a new line without having to deduce it from previous lines.
Indiscernibility of identicals is the principle that if a and b are the same thing, then everything that is fulfilled for a will be fulfilled for b. Thus, if “a” and “b” are arbitrary constants and α(a) is an arbitrary formula that contains “a”, this principle corresponds to the following valid inference scheme:
Indiscernibility of identicals (Indis): | a = b |
α(a) | |
α(b) |
a may occur more than once in α(a). α(b) is the formula obtained from α(a) by replacing one, several or all occurrences of a with b. Note that not all occurrences of a need to be replaced by b. The following inferences are applications of Indiscernibility of identicals.
a = b |
Fa |
Fb |
From the fact that a is the same thing as b and a is F, it follows that b is also F.
b = c |
Fbc |
Fcc |
From the fact that b and c are the same thing and b is in the relation F to c, it follows that c is also in the relation F to c, i.e. to itself.
a = b |
∃z(z≠a ∧ Fza) |
∃z(z≠b ∧ Fza) |
Here α(a) is “∃z(z≠a ∧ Fza)” and α(b) is “∃z(z≠b ∧ Fza)”. α(b) is obtained by replacing only the first occurrence of “a” in α(a) with “b”.
Law of identity and Indiscernibility of identicals express all that is contained in the concept of identity. This means that the two inference schemes^{2} can be used to prove the validity of any logically valid inference and of any logically valid formula in which equations occur. Such are, for example, the following formulas expressing the properties of symmetry and transitivity of the identity relation (see the previous section):
∀х∀у(x=y → y=x) |
∀x∀y∀z[(x=y ∧ y=z) → x=z}] |
Let us prove their validity. First about the symmetry of the identity relation:
1. ¬∀х∀у(x=y → y=x) assumption |
2. ∃х∃у¬(x=y → y=x) 1, RbQ |
3. ∃у¬(a=y → y=a) 2, EI |
4. ¬(a=b → b=a) 3, EI |
5. a=b ∧ b≠a 4, Imp |
6. а=b 5, S |
7. a=a LI |
8. b=a 6, 7, Indis (the first occurrence of “a” in “a=a” is replaced by “b” because of “a=b”) |
9. b≠a 5, S – contradicts 8 |
10. ∀х∀у(x=y → y=x) 1–9, reductio ad absurdum |
A proof that the identity relation is transitive:
1. ¬∀x∀y∀z[(x=y ∧ y=z) → x=z] assumption |
2. ∃x∃y∃z¬[(x=y ∧ y=z) → x=z] 1, RbQ |
3. ¬[(a=b ∧ b=c) → a=c] 2, EI (three times)^{3} |
4. a=b ∧ b=c ∧ a≠c 3, Imp |
5. b=c 4, S |
6. a=b 4, S |
7. а=c 5, 6, Indis |
8. a≠c 4, S – contradicts 7 |
9. ∀x∀y∀z[(x=y ∧ y=z) → x=z] 1–8, reductio ad absurdum |
As an example of a proof of the logical validity of a natural language argument involving the concept of identity, consider the following inference:
If Schopenhauer loved anything at all, it was his dog. |
His dog was faithful to him. |
Schopenhauer loved only those who were faithful to him. |
We symbolize the singular terms “Schopenhauer” and “Schopenhauer’s dog” with “a” and “b”, respectively, and the predicates “…loved…” and “…was faithful to…” with “F” and “G”, respectively. The first premise can be paraphrased with the simpler sentence “If Schopenhauer loved something, it was his dog”, which is symbolized by “∀x(Fax → b=x)”. The conclusion can be paraphrased with the sentence “If Schopenhauer loved something, it was true to him”, which is symbolized by “∀x(Fax→Gxa)”. Thus, the whole inference is symbolized by
∀x(Fax → b=х) |
Gba |
∀х(Fax → Gxa) |
Here is proof of its validity:
1. ∀x(Fax → b=х) |
2. Gba / ∀х(Fax → Gxa) |
3. ¬∀х(Fax → Gxa) assumption |
4. ∃х¬(Fax → Gxa) 3, RbQ |
5. ¬(Fac → Gca) 4, EI |
6. Fac ∧ ¬Gca 5, Imp |
7. Fac → b=c 1, UI |
8. b=c 6, 7, S and MP |
9. Gca 8, 2, Indis |
10. ¬Gca 6, S – contradicts 9 |
11. ∀х(Fax → Gxa) 3–10, reductio ad absurdum |
Here are some examples of definite descriptions: “the prime minister of Bulgaria”, “the deepest ocean on Earth”, “the author of Waverley”^{4}. These are singular terms that, like proper names, serve to denote things but, unlike them, have meaning and logical structure. Each definite description contains a simple or compound predicate (in the above examples these are “…is a prime minister of Bulgaria”, “…is an ocean than which there is no deeper on Earth”, “…is an author of Waverley”). Predicates are general terms, not singular terms. What makes definite descriptions singular terms is the definite article in front of the predicate. Its use indicates that it is assumed that the predicate is true of a single object in the universe of discourse. If we symbolize the predicate by “F”, any definite description can be paraphrased with the expression “the only thing that is F”.
Since defined descriptions are singular terms, we have symbolized them with constants. For example, we would symbolize the sentence “The author of Waverley is a Scotsman” with “Fa”, where “a” corresponds to the definite description “the author of Waverley” and “F” to the predicate “…is a Scotsman”. This is not incorrect but there are cases where the logical structure of a definite description, which is lost when it is symbolized by a constant, is important for the logical validity of an inference. Consider, for example, the following obviously valid inference:
The author of Waverly is author of only quality works. |
Waverly is a quality work. |
If we symbolize the definite description “the author of Waverley” with a constant, the logical validity of the inference cannot be proven, as its symbolic representation will look like this:
∀x(Fax → Gх) |
Gb |
The predicates “…is author of…” and “…is a quality work” are symbolized by “F” and “G”, respectively, and the singular terms “the author of Waverley” and “Waverly” are symbolized by “a” and “b”, respectively. In the symbolic representation, the connection between “the author of Waverley” (symbolized by “a”) and “Waverly” (symbolized by “b”), which is important for the validity of the inference, is lost. In such cases, it is necessary to take into account the predicates or singular terms involved in the description. Russell’s theory of definite descriptions, which will be presented shortly, shows how to do this.
In addition, definite descriptions are related to the following philosophical and logical problem. Like proper names of natural languages, the only function of constants (“a”, “b”, …) in the symbolic language of logic is to denote things. For example, in the symbolic representation of the above inference, the constant “a” denotes a certain human (the author of Waverley, i.e. Sir Walter Scott) and the constant “b” denotes a certain novel (Waverley). If we symbolize singular terms only with constants, this may lead to the view that a singular term has meaning (its usage makes sense) only insofar as it denotes a certain thing. However, this view has the absurd consequence that for a sentence of the form “a exists” to be meaningful, it must be true. The reason is as follows: a sentence containing a singular term “a” will be meaningful only if that term is meaningful. But since the meaning of “a” is the object it signifies, if the latter does not exist, “a” has no meaning. Therefore, if the sentence “a exists” is false, “a” will not designate anything and will be meaningless, which makes the sentence itself meaningless. So, such sentences make sense only if they are true. We have the absurd consequence that if someone claims that something he has named in some way exists (be it God, Pegasus, the present king of France, etc.), no one who wants to be logical can disagree with him (her). Otherwise, saying, for example, “God does not exist” would mean talking nonsense.
This problem manifests itself also in the symbolic language of logic. The sentence “a exists” is symbolized with the formula “∃x x=a” (“There exists a thing that is identical with a”). Since every possible interpretation of “a” consists in choosing something in the universe of discourse that “a” will denote (see “3.4 Syntax and semantics of predicate logic”), for every interpretation of “a” in every universe of discourse it will be true that there is something identical with a (namely a itself) and therefore the formula “∃x x=a” (“a exists”) will be true in any structure, i.e. it will be logically valid. Here is proof of its logical validity:
1. ¬∃x x=a assumption |
2. ∀x x≠a 1, RbQ |
3. a≠a 2, UI |
4. a=a LI – contradicts 3 |
4. ∃x x=a 1–4, reductio ad absurdum |
The logical validity of “∃x x=a” means that if we use only constants to symbolize singular terms of natural languages, we cannot claim of quite a lot of things that they do not exist, even though we know that it is so, because (as negations of logically true sentences) the claims turn out to be logically false. Such are, for example, the following:
Pegasus does not exist. |
The present king of France does not exist. |
The largest number does not exist. |
A solution to the problem would be to never use (when trying to speak the truth) singular terms that do not denote existing things (such as “Pegasus” or “the present king of France”) because anyone could use the above proof to show that these non-existent things exist. But how can we always know what exists and what does not, so as not to talk about the non-existent?
The classic solution to this problem is Bertrand Russell’s theory of definite descriptions (Russell, 1905). It is as follows. When someone claims something using a definite description – imagine that he (she) seriously claims that the present king of France is bald – the speaker makes (explicitly or implicitly) not one, but in fact three statements. First, he (she) believes that there is a present king of France, i.e. there is an implicit claim of existence. Second, the use of the definite article “the” shows that the speaker believes that there is no more than one present king of France, i.e. there is also an implicit claim of uniqueness. And third, the speaker believes that the existing and unique thing in question is bald – the explicit statement. Combining and symbolizing these three statements, we obtain the following:
(1) | ∃x[Fx ∧ ∀y(Fy → x=y) ∧ Gx] |
“F” symbolizes the definite description’s predicate “…is a present king of France” and “G” the predicate “…is bald”. (1) tells us literally the following: “There is a thing x that is a present king of France (“∃x[Fx…” – the claim of the existence of a present king of France); and each thing that is a present king of France is identical with x (“…∀y(Fy → x=y)…” – the claim of the uniqueness of the present king of France); and x is bald (“…Gx…” – the explicit statement)”. In other words, (1) states that there is a present king of France, that there is no more than one present king of France, and that he (she) is bald.
Following Russell’s analysis, the sentence “The present king of France exists” is analyzed as (1) but without the last member of the conjunction, related to the king’s baldness:
(2) | ∃x[Fx ∧ ∀y(Fy → x=y)] |
(2) tells us that there exists something that is a present king of France and that only it is a present king of France. Unlike “∃x x=a”, (2) is not a logically valid formula and can easily be negated if we want to argue that, since France is a republic, the present king of France does not exist:
(3) | ¬∃x[Fx ∧ ∀y(Fy → x=y)] |
(3) is true if there is no present king of France or if there are two or more present kings of France. Unlike “¬∃x x=a”, (3) is not meaningless if the present king of France does not exist, because the constant “a” does not occur in it – its meaningfulness obliged us to accept the existence of the thing it denotes. In (3) there are no constants, only predicates.
In this way, Russell’s analysis solves the problem with the singular terms that denote nothing when they are definite descriptions (such as the “the present king of France”). However, the analysis can also be applied to all other singular terms that denote nothing – for example, to proper names such as “Pegasus” or pronoun-containing expressions such as “this man in the distance” (it may have just seemed to me that there is a person in the distance). For this purpose, it is sufficient to paraphrase the singular term in question with a definite description. Russell himself believes that all proper names are hidden definite descriptions, but even if we do not accept this thesis, we can always find a definite description with which a given proper name or other singular term can be replaced in the context. For example, we may paraphrase “Pegasus” with “the winged horse caught by a Greek named Bellerophon”; “this man” – with “the man who is currently standing in the distance”, etc. Once the singular term is replaced by a definite description, we can apply the Russell’s analysis as above.
In addition to solving the problem with the singular terms denoting nothing, Russell’s theory of definite descriptions makes it possible to prove the validity of inferences that depend on predicates or singular terms contained in definite descriptions, such as that about the author of Waverley above. As we saw, the validity of the inference
The author of Waverly is author of only quality works. |
Waverly is a quality work. |
cannot be demonstrated without going into the structure of the definite description “the author of Waverley”. The theory of definite descriptions shows how to do this. If we symbolize “Waverley” with “a”, “…is author of…” with “F”, and “…is a quality work” – with “G” and apply Russell’s analysis, the symbolic representation of the inference will be as follows:
∃x[Fxa ∧ ∀y(Fya → y=x) ∧ ∀z(Fxz → Gz)] |
Ga |
The premise is literally this: “There is an x who is an author of Waverley; anyone who is an author of Waverley is identical to x; and anything x has written is a quality work”. The logical validity of the inference can now be shown:
1. ∃x[Fxa ∧ ∀y(Fya → y=x) ∧ ∀z(Fxz → Gz)] / Ga |
2. Fba ∧ ∀y(Fya → y=b) ∧ ∀z(Fbz → Gz) 1, EI |
3. ∀z(Fbz → Gz) 2, S |
4. Fba → Ga 3, UI |
5. Fba 2, S |
6. Ga 4, 5, MP |
(1) Symbolize the sentences. |
1) | There are at least three sages. |
2) | There are at most three sages. |
3) | There are exactly three sages. |
(2) Symbolize the sentences using the notations. |
1) | No one loves anyone but himself (herself). (F – …loves…; D – the set of humans) |
2) | Everyone loves someone else. (as above) |
3) | There is something that is more perfect than anything but itself. (F – …is more perfect than…) |
4) | Nothing is against God except God himself. (F – …is against…, a – God) |
5) | For each thing there is another thing, more perfect than it. (F – …is more perfect than…) |
6) | If something is more perfect than everything but itself, it is God. (F – …is more perfect than…, а – God) |
7) | No one who desires the good of everyone but himself is selfish. (F – …desires the good of…, G – …is selfish; D – the set of humans) |
8) | No one respects someone who respects no one but himself. (F – …respects…; D – the set of humans) |
9) | If two things are different, then one has a property that the other does not. (F – …is a property, G – …has…) |
10) | Peter respects only Mary and John. (F – …respects…, а – Peter, b – Mary, c – John) |
(3) Prove the validity of the inferences using the notations. |
1) | John swims faster than the team’s coach. | |
No one swims faster than himself (herself). | ||
John is not the team’s coach. | ||
F – …swims faster than…, а – John, b – the team’s coach |
2) | No one has access to the security cameras except the manager and the head of security. | |
The documents were stolen by someone who has access to the security cameras. | ||
The manager or the head of security stole the documents. | ||
F – …has access to the security cameras, G – …stole the documents, a – the manager, b – the head of security; D – the set of humans |
(4) Prove the following logical equivalences: |
1) | ∀x(a=x → Fx) ⇔ Fa |
2) | ∃x(Fx ∧ x=a) ⇔ Fa |
(5) Symbolize the sentences using the notations. |
1) | The wife of John loves flowers. (F – …is a wife of…, G – …loves flowers, a – John) |
2) | The father of Alice knows Bob. (F – …is a father of…, G – …knows…, a – Alice, b – Bob; D – the set of humans) |
3) | This town where I was born is probably boring for many. (F – …is a town, G – …was born in…, H – …is probably boring for many, а – me) |
4) | The castle where Dracula was born is in Carpathians. (F – …is a castle, G – …was born in…, H – …is in…, a – Dracula, c – Carpathians) |
5) | The chess player that no computer can beat is a Hungarian. (F – …is a chess player, G – …is a computer, H – …can beat…, I – …is a Hungarian) |
6) | If the author of Waverley is a Scotsman, then the author of Ivanhoe is a Scotsman. (F – …is an author of…, G – …is a Scotsman, w – Waverley, i – Ivanhoe) |
(6) Prove the validity of the inferences using the notations. |
1) | The author of Waverley wrote Ivanhoe. | |
There is someone who wrote both Ivanhoe and Waverley. | ||
(F – …wrote…, w – Waverley, i – Ivanhoe) |
2) | The soprano I listened to last night performed Queen of the night aria. | |
Everyone who performs this aria can reach F_{6}. | ||
Some sopranos can reach F_{6}. | ||
(F – …is a soprano, G – I listened to…last night, H – …performed Queen of the night aria, I – …can reach F_{6}; D – the set of humans) |