by Corry Shores
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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part II:
Quantification and Identity
13.
Free Logics
13.6
Identity in Free Logic
Brief summary:
(13.6.1) In our free logics, identity will be defined like in classical logic as: v(=) = {⟨D, d⟩ :d ∈ D}, and it will have the same properties and tableau rules as in classical logic too.
| Double Negation Development (¬¬D) |
| ¬¬A ↓ A |
| Conjunction Development (∧D) |
| A ∧ B ↓ A ↓ B |
| Negated Conjunction Development (¬∧D) |
| ¬(A ∧ B) ↙ ↘ ¬A ¬B |
| Disjunction Development (∨D) |
| A ∨ B ↙ ↘ A B |
| Negated Disjunction Development (¬∨D) |
| ¬(A ∨ B) ↓ ¬A ↓ ¬B |
| Conditional Development (⊃D) |
| A ⊃ B ↙ ↘ ¬A B |
| Negated Conditional Development (¬⊃D) |
| ¬(A ⊃ B) ↓ A ↓ ¬B |
| Biconditional Development (≡D) |
| A ≡ B ↙ ↘ A ¬A B ¬B |
| Negated Biconditional Development (≡D) |
| A ≡ B ↙ ↘ A ¬A ¬B B |
| Negated Existential Development (¬∃D) |
| ¬∃xA ↓ ∀x¬A |
| Negated Universal Development (¬∀D) |
| ¬∀xA ↓ ∃x¬A |
| Universal Instantiation Development (UI,D) |
| ∀xA ↙ ↘ ¬ℭa Ax(a)
where a is any constant on the branch (choosing a new constant only if there are none there already) |
| Particular Instantiation Development (PI,D) |
| ∃xA ↓ ℭc ↓ Ax(c)
where c is a constant new to the branch |
| Principle of Identity Development (=D) |
| . ↓ a = a
(You can always add a line of the form a = a) |
| Substitutivity of Identicals (SI,D) |
| a = b ↓ Ax(a) ↓ Ax(b)
where A is any atomic sentence distinct from a = b. |
(6-8; 266, 272, 291, with names and additional text at the bottom made by me)
(13.6.2) Negative free logics are constrained by the Negativity Constraint, which says that if something belongs to a predicate, it needs to be an existent thing. That means whenever a does not exist, a = a is false, because we cannot predicate it of identity when it is non-existent. (13.6.3) For negative free logics, the extension of the identity predicate is limited to existent things: v(=) = {⟨d, d⟩ : d ∈ E}. We must change the first tableau identity rule to the “Self-Identity of Existents,” which requires a line of the form ℭa before we can obtain a = a.
| Self-Identity of Existents (SIE) |
| ℭa ↓ a = a
(You can always add a line of the form a = a if you already have ℭa) |
The second identity rule on the substitutivity of identicals stays the same. But to close a branch, it is not enough to merely have a ≠ a. It would only close on that basis if in addition to a ≠ a we also have ℭa. (Note that we probably also need the Negativity Constraint Rule too:)
| Negativity Constraint Rule(NCR,D) |
| Pa1 ... an ↓ ℭa1 ⫶ ℭan |
(p.293, section 13.4.3, with name added at the top)
(13.6.4) Priest next gives an example tableau for a valid formula in negative free logic and another for an invalid formula. (13.6.5) We form a counter-model from an open branch in the following way: “given a bunch of identities, a = b, b = c, . . . on a branch, one chooses a single object for all the constants in the bunch to denote. For every predicate, P, excluding identity (but including ℭ), ⟨∂a1, . . . , ∂an⟩ ∈ v(P) iff Pa1 . . . an is on the branch; E = v(ℭ); and v(=) comprises the set of all pairs ⟨d, d⟩, where D is any object in E” (298-299). (13.6.6) These tableaux for identity are sound and complete. (13.6.7) The application of the Negativity Constraint to identity would make certain formulations false, in correspondence with our intuitions. For example, it makes Sherlock Holmes = Pegasus false. However, it still produces counter-examples similar to ones normally found in negative free logics. So we would want Father Christmas = Santa Claus or Santa Claus = Santa Claus to be true, but the negativity constraint makes them false. (13.6.8) The substitutivity of identicals remains valid regardless of which treatment of identity we choose, and thus we still have problems dealing with situations where something develops so much that it remains identical but is not substitutable with regard to its predications at the beginning and end of its development. For example, one same person begins as a baby and ends as an adult. But it seems odd that we are allowed then to say that the adult is a baby. (13.6.9) “With just outer quantifiers, free logic is just classical logic plus a distinguished predicate for existence. And in positive free logic, even this predicate satisfies no special semantic conditions. The only difference is therefore simply one of informal interpretation” (299). (13.6.10) “With just inner quantifiers, consider a free logic interpretation – positive or negative, with or without identity – where D = E; this is a classical interpretation. Hence, any inference (not involving ℭ) that is valid in the logic is valid in classical logic” (299). (13.6.11) But, “there is a limited relationship in the other direction;” for example, “∀xPx ⊭ ∃xPx […] but this is classically valid” (299).
[Identity in Free Logics: The Basics]
[Identity Under the Negativity Constraint in Negative Free Logics]
[Rules for Negative Free Logics]
[Two Example Tableaux]
[Counter-Models]
[The Soundness and Completeness of the Tableaux]
[Problems with the Negativity Constraint Applied to Identity]
[Problems of Growth as Still Holding]
[Similarities Between Classical Logic and Free Logic with Only Outer Quantifiers]
[Similarities Between Classical Logic and Free Logic with Just Inner Quantifiers]
[Dissimilarities in the Other Direction]
Summary
[Identity in Free Logics: The Basics]
[In our free logics, identity will be defined like in classical logic as: v(=) = {⟨D, d⟩ :d ∈ D}, and it will have the same properties and tableau rules as in classical logic too.]
[Recall from section 12.5 the identity predicate in classical logic.
(12.5.1) The identity predicate is a binary predicate symbolizes as ‘=’ and formulated as ‘a1 = a2’. We can write ‘¬a1 = a2’ as ‘a1 ≠ a2’. (12.5.2) “In any interpretation, ⟨D, v⟩, v(=) is {⟨d, d⟩: d ∈ D}. That is, ⟨d, e⟩ is in the denotation of the identity predicate, just if d is e.”
(p.272. The above from the brief summary of section 12.5).
So the identity predicate, like all predicates, has a denotation, which is a set of n-tuples of the members of the domain. Since it is a binary predicate, its denotation will be a set of couples, (at least) one for each member of the domain, where that member is both the first and second part of the couple. (I am a little bit more confused by “That is, ⟨d, e⟩ is in the denotation of the identity predicate, just if d is e”. I am not sure if we are saying that the couples can have different symbols for each part, so long as they are said to be identical, or if we are saying that ⟨d, e⟩ is to be understood as a structure, but any actual case would have to have identically the same symbols for both parts of the couple.) Next recall that we are dealing with free logics, where there is the main domain D of all objects and the “inner domain” E of existent objects, which is a subset of D, with the remainder in D being non-existing objects (see section 13.2.2). In section 13.2.4, we learned that the semantics for free logics are mostly the same as in classical first-order logic except that the truth conditions for quantifiers concern only the domain of existents:
v(∀xA) = 1 iff for all d ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some d ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
(p.291, section 13.2.4)
But as we can see, the quantifiers here only range over the domain of existents. In section 13.5.2, we expanded the quantifiers so that we could have ones that range over the full domain, which can include non-existent objects: Quantifiers ranging over the outer domain D are called the outer quantifiers, and they are written as ∃ and ∀. The quantifiers that range over the inner domain E are called inner quantifiers, and they are written as ∃E and ∀E (see section 13.5.2). Priest will now discuss adding the identity predicate to free logics, and he says that “The situation is the same whether the language has outer quantifiers or merely inner quantifiers.” We still will define the identity predicate as:
v(=) = {⟨D, d⟩ :d ∈ D}
And the tableau rules will be the same too as for classical logic (below copying from section 12.5.3)
| Double Negation Development (¬¬D) |
| ¬¬A ↓ A |
| Conjunction Development (∧D) |
| A ∧ B ↓ A ↓ B |
| Negated Conjunction Development (¬∧D) |
| ¬(A ∧ B) ↙ ↘ ¬A ¬B |
| Disjunction Development (∨D) |
| A ∨ B ↙ ↘ A B |
| Negated Disjunction Development (¬∨D) |
| ¬(A ∨ B) ↓ ¬A ↓ ¬B |
| Conditional Development (⊃D) |
| A ⊃ B ↙ ↘ ¬A B |
| Negated Conditional Development (¬⊃D) |
| ¬(A ⊃ B) ↓ A ↓ ¬B |
| Biconditional Development (≡D) |
| A ≡ B ↙ ↘ A ¬A B ¬B |
| Negated Biconditional Development (≡D) |
| A ≡ B ↙ ↘ A ¬A ¬B B |
| Negated Existential Development (¬∃D) |
| ¬∃xA ↓ ∀x¬A |
| Negated Universal Development (¬∀D) |
| ¬∀xA ↓ ∃x¬A |
| Universal Instantiation Development (UI,D) |
| ∀xA ↓ Ax(a)
where a is any constant on the branch. (If there are not any, we select one at will.) |
| Particular Instantiation Development (PI,D) |
| ∃ xA↓ Ax(c)
where c is any constant that does not occur so far on the branch. |
| Principle of Identity Development (=D) |
| . ↓ a = a
(You can always add a line of the form a = a) |
| Substitutivity of Identicals (SI,D) |
| a = b ↓ Ax(a) ↓ Ax(b)
where A is any atomic sentence distinct from a = b. |
(6-8; 266, 272, with names and additional text at the bottom made by me)
And so “identity has exactly the same properties as it does in classical logic.” Note that it is not stated explicitly whether or not we use the free logic rules for instantiations or the classical logical rules. My guess from the wording of the quotation is that the identity rules are the same as for classical logic, but the tableau rules in general are the same as for free logic, and thus we should instead use the following two free logic instantiation rules from section 13.3.1:
| Universal Instantiation Development (UI,D) |
| ∀xA ↙ ↘ ¬ℭa Ax(a)
where a is any constant on the branch (choosing a new constant only if there are none there already) |
| Particular Instantiation Development (PI,D) |
| ∃xA ↓ ℭc ↓ Ax(c)
where c is a constant new to the branch |
(p.291, section 13.3.1, with names and additional text at the bottom made by me)
]
Let us now consider how the addition of the identity predicate affects free logic. The situation is the same whether the language has outer quantifiers or merely inner quantifiers. The simplest and most natural treatment of identity in free logic is exactly the same as in classical logic. In any interpretation, v(=) = {⟨D, d⟩ :d ∈ D}. The tableau rules for it are then exactly the same as in classical logic. In particular, identity has exactly the same properties as it does in classical logic.
(297)
[Identity Under the Negativity Constraint in Negative Free Logics]
[Negative free logics are constrained by the Negativity Constraint, which says that if something belongs to a predicate, it needs to be an existent thing. That means whenever a does not exist, a = a is false, because we cannot predicate it of identity when it is non-existent.]
[First recall from section 13.2.1 the existence predicate:
Free logics have the same vocabulary as classical first-order logics. But in free logics we have the one-place existence predicate ℭ. We can think of ℭa as meaning ‘a exists’.
(from the brief summary of section 13.2.1)
And from section 13.2.3:
In any interpretation, v(ℭ) = E.
(p.290, section 13.2.3)
With that in mind, let us now recall negative free logics from section 13.4.2:
Some might still want to use free logics to accommodate non-existing things, but they might think that non-existing things should not have positive properties. For, while existing things have such tangible, physical properties that allow them to be seen and be physically interactable, non-existing things do not. (So we might want to say that Sherlock Holmes is in our domain, but we might also want to say that as a non-existing object, he cannot actually live on Baker St. For, only physically real things can have spatial location.) To disallow non-existing objects from having positive properties, we could apply the negativity constraint: If ⟨d1, . . . , dn⟩ ∈ v(P) then d1 ∈ v(ℭ), and …and dn ∈ v(ℭ). (In other words, if something belongs to a predicate, it needs to be an existent thing.) Free logics with the negativity constraint are called negative free logics.
(from the brief summary of section 13.2.3)
So the important ideas for us there are that we will want to say that only existent things can be predicated. We do that with the negativity constraint, which says that any predicated things must also be in the domain of the ℭ predicate, which is comprised of all the existent things in our domain. We call logics with the negativity constraint ‘negative free logics.’ Now, recall again that we are dealing with the identity predicate. Suppose we are using a negative free logic. That means identity cannot hold for non-existent things. (For, only existent things can be predicated, and identity requires predication.) So a = a will be false whenever a does not exist. The negativity constraint ensures that.]
In a thoroughgoing negative free logic, however, this approach will not be satisfactory. For we will need to apply the Negativity Constraint of 13.4.2 to all predicates, including identity. Thus, a = b will be false if either a or b does not exist. In particular, a = a will be false if a does not exist.
(297)
[Rules for Negative Free Logics]
[For negative free logics, the extension of the identity predicate is limited to existent things: v(=) = {⟨d, d⟩ : d ∈ E}. We must change the first tableau identity rule to the “Self-Identity of Existents,” which requires a line of the form ℭa before we can obtain a = a. The second identity rule on the substitutivity of identicals stays the same. But to close a branch, it is not enough to merely have a ≠ a. It would only close on that basis if in addition to a ≠ a we also have ℭa.]
[Now we will modify the semantic and tableau rules so that identity will no longer hold for non-existent things. We change the definition of identity so that it only involves items among the set of existent things, E: v(=) = (⟨d, d⟩ : d ∈ E}. Since identity only holds necessarily for existent things, we will change the first identity rule to:
| Self-Identity of Existents (SIE) |
| ℭa ↓ a = a
(You can always add a line of the form a = a if you already have ℭa) |
The second identity rule remains the same. This also means however that it is no contradiction for a non-existing thing a to be not identical to itself. So in our tableau, simply having a ≠ a does not close a branch. It would only close on that basis if in addition to a ≠ a we also have ℭa. Priest lastly shows how we can establish the symmetry of identity in negative free logics (see the tableau in the quotation). Let me note one further thing. There does not seem to be any explicit mention of it, but I wonder if for negative free logics we use the alternate rules for instantiations that we mention above in section 13.6.1, and add the from section 13.4.3:
| Negativity Constraint Rule(NCR,D) |
| Pa1 ... an ↓ ℭa1 ⫶ ℭan |
(p.293, section 13.4.3, with name added at the top)
See section 13.6.4 below where this rule is employed in the example tableau.]
The semantic and tableau rules for identity must therefore be changed to make this possible. In particular, the extension of identity must be restricted to those things in E; so v(=) = {⟨d, d⟩ : d ∈ E}.3 For the | corresponding tableaux, the first identity rule must be changed to:
Self-Identity of Existents (SIE)
ℭa
↓
a = a
(You can always add a line of the form a = a if you already have ℭa)
(One can call this rule the Self-Identity of Existents, SIE.) The other, SI, remains the same. We cannot now close a branch simply if we find a line of the form a ≠ a. But we can, if we find a line of the form ℭa as well. So, in practice, we may close a branch under those conditions. Note that we can still establish the symmetry of identity, as follows:
| Symmetry of identity | ||
| 1. .
2. .
3. .
4. | a = b ↓ ℭa ↓ a = a ↓ b = a | P .
P .
2SIE .
1,2SI |
(enumeration and step accounting are my own and are probably mistaken)
The second line is the NCR.
(297-298)
3. Thus, the new relation x = y could be defined in terms of the old one as follows: x = y ∧ ℭx ∧ ℭy (or just x = y ∧ ℭx).
(297)
[Two Example Tableaux]
[Priest next gives an example tableau for a valid formula in negative free logic and another for an invalid formula.]
[(ditto)]
To illustrate the new rules, consider the following tableaux, which demonstrate that ¬ℭa ⊢ ¬a = b and ⊬ (ℭa ∨ a = a) ∧ (¬ℭa ∨ a = a):
| ¬ℭa ⊢ ¬a = b | ||
| 1. .
2. .
3. .
4. | ¬ℭa ↓ ¬¬a = b ↓ a = b ↓ ℭa × | P .
P ..
2¬¬ ..
3NCR (4×1) valid |
(enumeration and step accounting are my own and are probably mistaken)
| ⊬ (ℭa ∨ a = a) ∧ (¬ℭa ∨ a = a) | ||
| 1. .
2. .
3. .
4. .
5. | ¬((ℭa ∨ a = a)∧(¬ℭa ∨ a = a)) ↙ ↘ ¬(ℭa ∨ a = a) ¬(¬ℭa ∨ a = a) ↓ ↓ ¬ℭa ¬¬ℭa ↓ ↓ a ≠ a a ≠ a ↓ ℭa × | P .
1¬∧ .
2¬∨ .
2¬∨ .
3b¬¬ (4,5×=) open invalid |
(enumeration and step accounting are my own and are probably mistaken)
In the first tableau, line four is obtained by applying the NCR to line three. In the second tableau, the right branch closes, but the left branch, which would have closed with the classical rules for identity, remains open.
(298)
[Counter-Models]
[We form a counter-model from an open branch in the following way: “given a bunch of identities, a = b, b = c, . . . on a branch, one chooses a single object for all the constants in the bunch to denote. For every predicate, P, excluding identity (but including ℭ), ⟨∂a1, . . . , ∂an⟩ ∈ v(P) iff Pa1 . . . an is on the branch; E = v(ℭ); and v(=) comprises the set of all pairs ⟨d, d⟩, where D is any object in E” (298-299).]
[(ditto). Note: see also section 12.5.9.]
Given an open branch of a tableau of this kind, one reads off a counter-model by combining the procedures for negative free logic (13.4.4) with those for identity (12.5.8). In particular, given a bunch of identities, a = b, b = c, . . . on a branch, one chooses a single object for all the constants in the bunch to denote. For every predicate, P, excluding identity (but | including ℭ), ⟨∂a1, . . . , ∂an⟩ ∈ v(P) iff Pa1 . . . an is on the branch; E = v(ℭ); and v(=) comprises the set of all pairs ⟨d, d⟩, where D is any object in E. The left-hand branch of the second tableau of 13.6.4 gives the interpretation where D = {∂a}, E = v(ℭ) = φ = v(=), and v(a) = ∂a. It is not difficult to check that this interpretation makes the whole formula false, since it makes the left conjunct false.
(298-299)
[The Soundness and Completeness of the Tableaux]
[These tableaux for identity are sound and complete.]
[(ditto)]
The tableaux for identity, with and without the NCR, are sound and complete with respect to the appropriate semantics. This is proved in 13.7.
(299)
[Problems with the Negativity Constraint Applied to Identity]
[The application of the Negativity Constraint to identity would make certain formulations false, in correspondence with our intuitions. For example, it makes Sherlock Holmes = Pegasus false. However, it still produces counter-examples similar to ones normally found in negative free logics. So we would want Father Christmas = Santa Claus or Santa Claus = Santa Claus to be true, but the negativity constraint makes them false.]
[(ditto)]
It should be noted that applying the Negativity Constraint to identity gives rise to further apparent counter-examples of the kind that we have already met in 13.4.6. It would certainly seem to be false that Sherlock Holmes = Pegasus. But it would seem to be true that Father Christmas = Santa Claus – or even that Santa Claus = Santa Claus.
(299)
[Problems of Growth as Still Holding]
[The substitutivity of identicals remains valid regardless of which treatment of identity we choose, and thus we still have problems dealing with situations where something develops so much that it remains identical but is not substitutable with regard to its predications at the beginning and end of its development. For example, one same person begins as a baby and ends as an adult. But it seems odd that we are allowed then to say that the adult is a baby.]
[(ditto). See sections 12.6.5, 12.6.6, 12.6.7, and 12.6.8.]
It should also be noted that whichever treatment of identity one employs, the Substitutivity of Identicals is still valid. Hence, moving to a free logic does nothing to alleviate the problems about identity noted in 12.6.5–12.6.8.
[Similarities Between Classical Logic and Free Logic with Only Outer Quantifiers]
[“With just outer quantifiers, free logic is just classical logic plus a distinguished predicate for existence. And in positive free logic, even this predicate satisfies no special semantic conditions. The only difference is therefore simply one of informal interpretation” (299).]
[(ditto)]
Let me finish with a couple of observations about the relationship between classical logic and free logic. With just outer quantifiers, free logic is just classical logic plus a distinguished predicate for existence. And in positive free logic, even this predicate satisfies no special semantic conditions. The only difference is therefore simply one of informal interpretation.
(299)
[Similarities Between Classical Logic and Free Logic with Just Inner Quantifiers]
[“With just inner quantifiers, consider a free logic interpretation – positive or negative, with or without identity – where D = E; this is a classical interpretation. Hence, any inference (not involving ℭ) that is valid in the logic is valid in classical logic” (299).]
[(ditto)]
With just inner quantifiers, consider a free logic interpretation – positive or negative, with or without identity – where D = E; this is a classical interpretation. Hence, any inference (not involving ℭ) that is valid in the logic is valid in classical logic. (See 3.2.8.) The converse is not the case, as we have had several occasions to note.
(299)
[Dissimilarities in the Other Direction]
[But, “there is a limited relationship in the other direction;” for example, “∀xPx ⊭ ∃xPx […] but this is classically valid” (299).]
[(ditto)]
However, there is a limited relationship in the other direction. Let the inference with premises Σ and conclusion A be valid in classical logic. Let C be the set of constants that occur in A and all members of Σ, and let Π = { ℭc:c ∈ C} ∪ {∃xℭx}. (The quantified sentence is redundant if C = φ.) Then Π ∪ Σ ⊨ A. This is proved in 13.7.13. Note that the quantified member of Π is necessary. For ∀xPx ⊭ ∃xPx (as may be checked using tableaux), but this is classically valid.
(299)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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