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by Corry Shores
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[Priest, Introduction to Non-Classical Logic, entry directory]
[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
16.
Necessary Identity in Modal Logic
16.3
The Negativity Constraint
Brief summary:
(16.3.1) “In this section, we will see how the addition of the Negativity Constraint affects matters” (352). (16.3.2) “In the presence of the [negativity] constraint, non-existent objects cannot be in the extension of the identity predicate. Hence, vw(=) = {⟨d, d ⟩ : d ∈ vw(ℭ)}” (352). (16.3.3) Priest next gives the identity rules for our negatively constrained system. (16.3.4) Priest then provides an example tableau in VK(NI) with the negativity constraint for a valid formula. (16.3.5) When we have the negativity constraint, necessary identity is invalidated. (16.3.6) “To read off a counter-model from an open branch of a tableau when the Negativity Constraint is in operation, we give constants the same denotation provided they are said to be the same at some world. Thus, for | example, if we have a = b,i and b = c,j, we give a, b and c the same denotation” (354).
[Adding the Negativity Constraint]
[Non-Existent Objects as Not in the Identity Predicate Under the Negativity Constraint]
[The Identity Rules]
[Example Tableau]
[The Invalidity of Necessary Identity Under the Negativity Constraint]
[Counter-Models and Example]
Summary
[Adding the Negativity Constraint]
[“In this section, we will see how the addition of the Negativity Constraint affects matters” (352).]
[(ditto)]
In this section, we will see how the addition of the Negativity Constraint affects matters.
(352)
[Non-Existent Objects as Not in the Identity Predicate Under the Negativity Constraint]
[“In the presence of the [negativity] constraint, non-existent objects cannot be in the extension of the identity predicate. Hence, vw(=) = {⟨d, d ⟩ : d ∈ vw(ℭ)}” (352).]
[Recall the following about the Negativity Constraint 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 13.4.2, see p.293)
Now recall from section 13.6.2 that:
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.
(from the brief summary of section 13.6.2, see p.297)
Thus (and see section 13.6.3):]
In the presence of the constraint, non-existent objects cannot be in the extension of the identity predicate. Hence, vw(=) = {⟨d, d ⟩ : d ∈ vw(ℭ)}.
(352)
[The Identity Rules]
[Priest next gives the identity rules for our negatively constrained system.]
[(ditto)]
For the corresponding tableaux, the identity rules become:
Self-Identity of Existents (SIE)
ℭa,i
↓
a = a,i
(You can always add a line of the form a = a,i if you already have ℭa,i)
Intra-World Substitutivity of Identicals (ISI,D)
a = b,i
↓
Ax(a),i
↓
Ax(b),i
(where Ax(a) is any atomic formula except a = b)
Identity Invariance Rule (IIR,D)
a = b,i
.
ℭa,j (or ℭb,j)
↓
a = b,j
(where j is any world parameter on the branch distinct from i)
(353, with names and additional text at the bottom made by me. See p.350 section 16.2.3)
(where Ax(a) is any atomic formula except a = b). Note the comments of 13.6.3 about the tableau rules for identity in free logic, which apply equally here.
(353)
[Example Tableau]
[Priest then provides an example tableau in VK(NI) with the negativity constraint for a valid formula.]
[We will now make a tableau example in VK(NI). I tried to figure the rules for it in section 16.2.4. I am assuming that we should substitute those identity rules with the ones given above. Here I will place all of them, including the negativity constraint rule, but probably they are not right.
| Double Negation Development (¬¬D) |
| ¬¬A,i ↓ A,i |
| Conjunction Development (∧D) |
| A ∧ B,i ↓ A,i B,i |
| Negated Conjunction Development (¬∧D) |
| ¬(A ∧ B),i ↓ ¬A ∨ ¬B,i |
| Disjunction Development (∨D) |
| A ∨ B,i ↙ ↘ A,i B,i |
| Negated Disjunction Development (¬∨D) |
| ¬(A ∨ B),i ↓ ¬A,i ↓ ¬B,i |
| Conditional Development (⊃D) |
| A ⊃ B,i ↙ ↘ ¬A,i B,i |
| Negated Conditional Development (¬⊃D) |
| ¬(A ⊃ B),i ↓ A,i ↓ ¬B,i |
| Negated Necessity Development (¬□D) |
| ¬□A,i ↓ ◊¬A,i |
| Negated Possibility Development (¬◊D) |
| ¬◊A,i ↓ □¬A,i |
| Relative Necessity Development (□rD) |
| □A,i irj ↓ A,j
(both □A,i and irj must occur somewhere on the same branch, but in any order or location) |
| Relative Possibility Development (◊rD) |
| ◊A,i ↓ irj A,j
(j must be new: it cannot occur anywhere above on the branch) |
(p.24, section 2.4.4)
| Negated Existential Development (¬∃D) |
| ¬∃xA ↓ ∀x¬A |
| Negated Universal Development (¬∀D) |
| ¬∀xA ↓ ∃x¬A |
(p.266, section 12.4.1)
| Universal Instantiation Development (UI,D) |
| ∀xA,i ↙ ↘ ℭa,i Ax(a),i
where a is any constant on the branch. (If there are not any, we select one at will.) |
| Particular Instantiation Development (PI,D) |
| ∃xA,i ↓ ℭc,i ↓ Ax(c),i
where c is any constant that does not occur so far on the branch. |
(p.331, section 15.4.1)
| Self-Identity of Existents (SIE) |
| ℭa,i ↓ a = a,i
(You can always add a line of the form a = a,i if you already have ℭa,i) |
| Intra-World Substitutivity of Identicals (ISI,D) |
| a = b,i ↓ Ax(a),i ↓ Ax(b),i
(where Ax(a) is any atomic formula except a = b) |
| Identity Invariance Rule (IIR,D) |
| a = b,i . ℭa,j (or ℭb,j) ↓ a = b,j
(where j is any world parameter on the branch distinct from i) |
(353, with names and additional text at the bottom made by me. See p.350 section 16.2.3)
| Negativity Constraint Rule(NCR,D) |
| Pa1 ... an ↓ ℭa1 ⫶ ℭan |
(p.293, section 13.4.3, with name added at the top)
We will try these rules out below.]
Here is a tableau to show that ⊢a = b ⊃ □(ℭa ⊃ a = b) in VK(NI), the weakest normal quantified modal logic. (Clearly, a similar tableau works in VKt(NI), when □ is replaced by [F] or [P].)
| ⊢VK(NI) a = b ⊃ □(ℭa ⊃ a = b) | ||
| 1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .
9. .
10. .
11. .
12. .
13. | ¬(a = b ⊃ □(ℭa ⊃ a = b)),0 ↓ a = b,0 ↓ ¬□(ℭa ⊃ a = b),0 ↓ ℭa,0 ↓ ℭb,0 ↓ ◊¬(ℭa ⊃ a = b),0 ↓ 0r1 ↓ ¬(ℭa ⊃ a = b),1 ↓ ℭa,1 ↓ ¬a = b,1 ↓ 0r1 ↓ ¬a = b,1 ↓ a = b,1 ×
| P .
1¬⊃ .
1¬⊃ .
2NCR .
2NCR .
3¬□ .
6◊r .
6◊r .
8¬⊃ .
8¬⊃ .
10◊r .
10◊r .
2,7IRR (13×12)
valid |
(enumeration and step accounting are my own and are probably mistaken)
The last line follows from the appropriate applications of IIR.
(353)
[The Invalidity of Necessary Identity Under the Negativity Constraint]
[When we have the negativity constraint, necessary identity is invalidated.]
[Recall from section 16.2.4 that the formula for necessary identity is:
∀x∀y(x = y ⊃ □x = y)
(p.351, section 16.2.4)
We saw there that it was valid in VK(NI) without the negativity constraint, and its validity was proven with this table:
| ⊢VK(NI) ∀x∀y(x = y ⊃ □x = y) | ||
| 1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .
9. .
10. .
11. .
12. .
13. | ¬∀x∀y(x = y ⊃ □x = y),0 ↓ ∃x¬∀y(x = y ⊃ □x = y),0 ↓ ℭa,0 ↓ ¬∀y(a = y ⊃ □a = y),0 ↓ ∃y¬(a = y ⊃ □a = y),0 ↓ ℭb,0 ↓ ¬(a = b ⊃ □a = b),0 ↓ a = b,0 ↓ ¬□a = b,0 ↓ ◊¬a = b,0 ↓ 0r1 ↓ ¬a = b,1 ↓ a = b,1 ×
| P .
1¬∀ .
2PI .
2PI .
4¬∀ .
5PI .
5PI .
7¬⊃ .
7¬⊃ .
9¬□ .
10◊r .
10◊r .
8,11IRR (13×12)
valid |
(enumeration and step accounting are my own and are probably mistaken. See p.351, section 16.2.4)
But recall from section 16.3.3 above that our Identity Invariance Rule is different now that we have the negativity constraint.
| Identity Invariance Rule (IIR,D) |
| a = b,i . ℭa,j (or ℭb,j) ↓ a = b,j
(where j is any world parameter on the branch distinct from i) |
(353, with names and additional text at the bottom made by me)
In our tableau above, we do not have a line of the form ℭa,1 (or ℭb,1), even though we have one of the form a = b,0. So we cannot derive the final line, and thus with the negativity constrain, necessary identity fails.
| ⊬VK(NI) ∀x∀y(x = y ⊃ □x = y) | ||
| 1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .
9. .
10. .
11. .
12. .
13. | ¬∀x∀y(x = y ⊃ □x = y),0 ↓ ∃x¬∀y(x = y ⊃ □x = y),0 ↓ ℭa,0 ↓ ¬∀y(a = y ⊃ □a = y),0 ↓ ∃y¬(a = y ⊃ □a = y),0 ↓ ℭb,0 ↓ ¬(a = b ⊃ □a = b),0 ↓ a = b,0 ↓ ¬□a = b,0 ↓ ◊¬a = b,0 ↓ 0r1 ↓ ¬a = b,1 ↓ a = b,1 ×
| P .
1¬∀ .
2PI .
2PI .
4¬∀ .
5PI .
5PI .
7¬⊃ .
7¬⊃ .
9¬□ .
10◊r .
10◊r .
8,11IRR (13×12)
valid |
(enumeration and step accounting are my own and are probably mistaken. See p.351, section 16.2.4)
]
NI does not hold in VK(NI) (or VKt(NI)) with the Negativity Constraint. The tableau is as for the first one of 16.2.4, except that the last line is missing. We cannot infer a = b,1, since we have neither ℭa,1 nor ℭb,1.
(353)
[Counter-Models and Example]
[“To read off a counter-model from an open branch of a tableau when the Negativity Constraint is in operation, we give constants the same denotation provided they are said to be the same at some world. Thus, for | example, if we have a = b,i and b = c,j, we give a, b and c the same denotation” (354).]
[(ditto)]
To read off a counter-model from an open branch of a tableau when the Negativity Constraint is in operation, we give constants the same denotation provided they are said to be the same at some world. Thus, for | example, if we have a = b,i and b = c,j, we give a, b and c the same denotation.2 The first tableau of 16.2.4 (truncated before the last line) then gives the interpretation depicted as follows:
_________xxxxxxxxxxxxxxxxxxxxx________
|xxxx∂ax|xxxxxxxxxxxxxxxxxx|xxxx∂ax|x
|xℭxx√xx|xxxxxwoxx→xxw1xxxx|xℭxx×xx|
_________xxxxxxxxxxxxxxxxxxxxx________
Both a and b denote ∂a. I leave it as an exercise to show that this counter-model works.
(353-354)
2. In fact, if we have lines of the form a = b, i and b = c, j, then there is a line of the form ℭb,j (by the NCR) and a = b,j (by the IIR). Hence, the worlds at issue can always be taken to be the same.
(354)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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