14 Aug 2018

Priest (16.3) An Introduction to Non-Classical Logic, ‘The Negativity Constraint ,’ summary

 

 

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

 

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).

 

 

 

 

 

Contents

 

16.3.1

[Adding the Negativity Constraint]

 

16.3.2

[Non-Existent Objects as Not in the Identity Predicate Under the Negativity Constraint]

 

16.3.3

[The Identity Rules]

 

16.3.4

[Example Tableau]

 

16.3.5

[The Invalidity of Necessary Identity Under the Negativity Constraint]

 

16.3.6

[Counter-Models and Example]

 

 

 

 

 

 

Summary

 

16.3.1

[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)

[contents]

 

 

 

 

 

 

16.3.2

[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 d1v(ℭ), and …and dnv(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)

[contents]

 

 

 

 

 

 

16.3.3

[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)

[contents]

 

 

 

 

 

 

16.3.4

[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)

[contents]

 

 

 

 

 

 

16.3.5

[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:

xy(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

.

.

2PI

.

2PI

.

.

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

.

.

2PI

.

2PI

.

.

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)

[contents]

 

 

 

 

 

 

16.3.6

[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

|xxxxx|xxxxxwoxxxxw1xxxx|xxx×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 b,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)

[contents]

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

 

Priest (16.2) An Introduction to Non-Classical Logic, ‘Necessary Identity,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[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

 

16.2

Necessary Identity

 

 

 

 

Brief summary:

(16.2.1) We will now define the identity predicate in a quantified normal modal logic. (16.2.2) “The denotation of the identity predicate is the same in every world, w, of an interpretation: vw(=) = {⟨d, d⟩ : dD}. ” (350). (16.2.3) There are three tableau rules for identity (see below).

 

Principle of Identity

Development (=D)

.

a = a,i

 

(You can always add a line of the form a = a,i)

 

Substitutivity of Identicals (SI,D)

a = b,i

Ax(a),i

Ax(b),i

 

(where A is any atomic sentence distinct from a = b.)

(Note: the world index on every line is the same, so substitution is licensed only within a world.)

 

Identity Invariance Rule (IIR,D)

a = b,i

a = b,j

 

(where j is any world parameter on the branch distinct from i)

(350, with names and additional text at the bottom made by me)

 

(16.2.4) Priest next gives two example tableaux for valid formulas in VK(NI), which is a variable domain system with necessary identity. And, “For future reference, we will call the formula ∀xy(x = y ⊃ □x = y) NI (Necessary Identity)” (351). (16.2.5) Priest next gives an example tableau for an invalid formula. (16.2.6) “Counter-models are read off from open branches as usual. In particular, where there is a bunch of lines of the form a = b, 0, b = c, 0, etc., a single denotation is provided for all the constants” (352). (16.2.7) Priest next gives an example counter-model.

 

 

 

 

 

 

Contents

 

16.2.1

[The Identity Predicate in Quantified Normal Modal Logic]

 

16.2.2

[Defining the Identity Predicate]

 

16.2.3

[The Three Identity Tableau Rules]

 

16.2.4

[Example Tableaux 1 and 2. NI (Necessary Identity)]

 

16.2.5

[Example Tableau 3]

 

16.2.6

[Counter-Models]

 

16.2.7

[Example Counter-Model]

 

 

 

 

 

 

Summary

 

16.2.1

[The Identity Predicate in Quantified Normal Modal Logic]

 

[We will now define the identity predicate in a quantified normal modal logic.]

 

[(ditto). (See section 12.5.1).]

Assume that we are dealing with any quantified (constant or variable domain) normal modal logic (without the Negativity Constraint). As in the classical case (12.5.1), we now distinguish one of the binary predicates as the identity predicate.

(350)

[contents]

 

 

 

 

 

 

16.2.2

[Defining the Identity Predicate]

 

[“The denotation of the identity predicate is the same in every world, w, of an interpretation: vw(=) = {⟨d, d⟩ : dD}. ” (350).]

 

[(ditto). (See section 12.5.2).]

The denotation of the identity predicate is the same in every world, w, of an interpretation: vw(=) = {⟨d, d⟩ : dD}.

(350)

[contents]

 

 

 

 

 

 

16.2.3

[The Three Identity Tableau Rules]

 

[There are three tableau rules for identity (see below).]

 

[The first two of the three rules of identity are the same as in section 12.5.3, only now with world designations:

 

Principle of Identity

Development (=D)

.

a = a,i

 

(You can always add a line of the form a = a,i)

 

Substitutivity of Identicals (SI,D)

a = b,i

Ax(a),i

Ax(b),i

 

(where A is any atomic sentence distinct from a = b.)

(Note: the world index on every line is the same, so substitution is licensed only within a world.)

(350, with names and additional text at the bottom made by me)

 

The third rule is the Identity Invariance Rule (IIR):

 

Identity Invariance Rule (IIR,D)

a = b,i

a = b,j

 

(where j is any world parameter on the branch distinct from i)

(350, with names and additional text at the bottom made by me)

]

There are three tableau rules for identity. The first two are exactly as in the classical case (12.5.3), modulo an appropriate world parameter:

 

Principle of Identity

Development (=D)

.

a = a,i

 

(You can always add a line of the form a = a,i)

 

Substitutivity of Identicals (SI,D)

a = b,i

Ax(a),i

Ax(b),i

 

(where A is any atomic sentence distinct from a = b.)

(Note: the world index on every line is the same, so substitution is licensed only within a world.)

(350, with names and additional text at the bottom made by me)

 

– where, recall, A is any atomic sentence other than a = b. Note that in SI, the world index on every line is the same, so substitution is licensed only within a world. The third rule is the following:

 

Identity Invariance Rule (IIR,D)

a = b,i

a = b,j

 

(where j is any world parameter on the branch distinct from i)

(350, with names and additional text at the bottom made by me)

 

where j is any world parameter on the branch distinct from i. I will call this the Identity Invariance Rule (IIR).

(350)

[contents]

 

 

 

 

 

 

16.2.4

[Example Tableaux 1 and 2. NI (Necessary Identity)]

 

[Priest next gives two example tableaux for valid formulas in VK(NI), which is a variable domain system with necessary identity. And, “For future reference, we will call the formula ∀xy(x = y ⊃ □x = y) NI (Necessary Identity)” (351). ]

 

[Recall from section 16.1.2 that:

If S is any system of logic without identity, S(NI) will denote the system augmented by necessary identity, and S(CI) will denote the system of logic augmented by contingent identity.

(p.349, section 16.1.2)

We will be working now with a system called VK(NI). That means the system is augmented by necessary identity. I think that means it is identity as defined above in section 16.2.2 and especially section 16.2.3. The important idea there was that identity does not vary between worlds. The next thing to note is that the system in question is VK. Recall from section 15.3 that VK is a variable domain quantification modal logic system. What was notable about it is that the domain of quantification remains the same for all worlds, however, the valuation function v assigns for each world its own subset of the domain of existing things. We will now do tableaux in VK(NI), but the problem is that we have not yet summarized the sections where the rules are given for it. So let us try to reconstruct them for our purposes here, and later we may go back to do the whole sections in question. In section 15.4.1 (p.331), we learn that the tableau rules are modified from CK, which is a constant domain system. In section 14.3.1 (pp.309-310) we learn that the tableaux for CK are modified from those of K. We can start there. From what I understand from section 14.3.1, we will obtain the CK tableaux by  using the K ones in section 2.4.4 (p.24) and adding quantifier rules from section 14.3.1 (p.310), which are like those for classical logic, section 12.4.1 (p.266); however, instead of the two instantiation rules given there (in section 14.3.1), we swap them with the ones given in section 15.4.1 (p.331). I probably having something here wrong, but for now let us just lay out the rules as such, including the identity rules from section 16.2.3 above:

 

 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)

 

Principle of Identity

Development (=D)

.

a = a,i

 

(You can always add a line of the form a = a,i)

 

Substitutivity of Identicals (SI,D)

a = b,i

Ax(a),i

Ax(b),i

 

(where A is any atomic sentence distinct from a = b.)

(Note: the world index on every line is the same, so substitution is licensed only within a world.)

 

Identity Invariance Rule (IIR,D)

a = b,i

a = b,j

 

(where j is any world parameter on the branch distinct from i)

(350, with names and additional text at the bottom made by me)

 

With those as a possible set of rules we should use, see the example tableaux below for valid formulas where we try to apply them.]

Here are tableaux to demonstrate that ⊢VK(NI) xy(x = y ⊃ □x = y), and ⊢VK(NI) xy(xy ⊃ □x ≠ y). Clearly, the tableaux work in a similar way in VKt(NI), when □ is replaced by [F] or [P]. Since the variable domain logics are sub-logics of the corresponding constant domain logics (15.4.7, 15.5.5), these inferences are valid in all constant and variable domain quantified modal logics. It is the validity of these formulas that give this notion of identity its name: all true statements of identity or difference are necessarily | true (true for all future/past times). For future reference, we will call the formula ∀xy(x = y ⊃ □x = y) NI (Necessary Identity).

ditto)]

 

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

.

.

2PI

.

2PI

.

.

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)

 

The last line is obtained by applying the IIR from line eight.

 

VK(NI) ∀x∀y(x ≠ y ⊃ □x ≠ y)

1.

.

2.

.

3.

.

4.

.

5.

.

6.

.

7.

.

8.

.

9.

.

10.

.

11.

.

12.

.

13.

.

14.

¬∀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

a = b,0

×

                    

P

.

.

2PI

.

2PI

.

.

5PI

.

5PI

.

7¬⊃

.

7¬⊃

.

9¬

.

10◊r

.

10◊r

.

12¬¬

.

13,11

IRR

(14×8)

valid

(enumeration and step accounting are my own and are probably mistaken)

 

Again, the last line is obtained by applying the IIR

(350-351)

[contents]

 

 

 

 

 

 

16.2.5

[Example Tableau 3]

 

[Priest next gives an example tableau for an invalid formula.]

 

[We will now do an example tableau in CK(NI), which is a constant domain system with necessary identity. Given what we said above in section 16.2.4, I will guess that the rules we will use could be the following:

 

 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

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

Ax(c),i

 

where c is any constant that does not occur so far on the branch.

(p.331, section 15.4.1)

 

Principle of Identity

Development (=D)

.

a = a,i

 

(You can always add a line of the form a = a,i)

 

Substitutivity of Identicals (SI,D)

a = b,i

Ax(a),i

Ax(b),i

 

(where A is any atomic sentence distinct from a = b.)

(Note: the world index on every line is the same, so substitution is licensed only within a world.)

 

Identity Invariance Rule (IIR,D)

a = b,i

a = b,j

 

(where j is any world parameter on the branch distinct from i)

(350, with names and additional text at the bottom made by me)

 

Let us apply these rules for the tableau example below.]

Here is another tableau to show that ⊬CK(NI) □∀xy((Sax ∧ Say) ⊃ xy):

 

CK(NI) □∀x∀y((Sax ∧ Say) ⊃ x ≠ y)

1.

.

2.

.

3.

.

4.

.

5.

.

6.

.

7.

.

8.

.

9.

.

10.

.

11.

.

12.

.

13.

.

14.

¬□∀x∀y((Sax ∧ Say) ⊃ x ≠ y),0

¬∀x∀y((Sax ∧ Say) ⊃ x ≠ y),0

0r1

¬∀x∀y((Sax ∧ Say) ⊃ x ≠ y),1

∃x¬∀y((Sax ∧ Say) ⊃ x ≠ y),1

¬∀y((Sab ∧ Say) ⊃ b ≠ y),1

∃y¬((Sab ∧ Say) ⊃ b ≠ y),1

¬((Sab ∧ Sac) ⊃ b ≠ c),1

Sab ∧ Sac,1

¬b ≠ c,1

Sab,1

Sac,1

b = c,1

b = c,0

 

                    

P

.

.

2◊r

.

2◊r

.

.

5PI

.

.

7PI

.

8¬

.

8¬

.

9∧

.

9∧

.

10¬¬

.

3,13IRR

(open)

invalid

(enumeration and step accounting are my own and are probably mistaken)

(352)

[contents]

 

 

 

 

 

 

16.2.6

[Counter-Models]

 

[“Counter-models are read off from open branches as usual. In particular, where there is a bunch of lines of the form a = b, 0, b = c, 0, etc., a single denotation is provided for all the constants” (352).]

 

[We will now make a counter-model, which is done in the “usual way.” As we have not been summarizing the previous sections, I am not entirely sure I know what way that is. But for now, I will use our most recent method, from section 13.3.4:

To read off a counter-model from an open branch of a tableau, the procedure is exactly as for classical logic, and E = v(ℭ). Since every object in D has a name in the interpretation, and given the definition of E, 13.2.6 assures us that to check that v(∃xA) = 1, we just have to show that v(Ax(c)) = 1 for some c such that ℭc is on the branch; and to check that v(∀xA) = 1, we just have to show that v(Ax(c)) = 1 for every constant, c, such that ℭc is on the branch.

(292, section 13.3.4)

But as in section 12.5.9and 13.6.5, we only need a single denotation for all identical constants in a world.]

Counter-models are read off from open branches as usual. In particular, where there is a bunch of lines of the form a = b, 0, b = c, 0, etc., a single denotation is provided for all the constants, as in 12.5.9and 13.6.5. (The 0 could, in fact, be any line number, because of the IIR.)

(352)

[contents]

 

 

 

 

 

 

16.2.7

[Example Counter-Model]

 

[Priest next gives an example counter-model.]

 

[(ditto)]

Thus, in the counter-model given by the tableau of 16.2.5, W = {w0, w1}, w0Rw1, D = {∂a, ∂b}, v(a) = ∂a, v(b) = v(c) = ∂b, and vw1 (S) = {⟨∂a, ∂b ⟩}. In a picture:

_______________xxxxxxxxxxxxxxxx______________

|xSxxx∂axxx∂bx|xxxxxxxxxxxxxxx|xSxxx∂axxx∂bx|

|x∂axx×xxxx×xx|xxxwoxxxxw1xxx|x∂axx×xxxxxx|

|x∂bxx×xxxx×xx|xxxxxxxxxxxxxxx|x∂bxx×xxxx×xx|

_______________xxxxxxxxxxxxxxxx______________

 

I leave it as an exercise to check that this interpretation works.

(352)

[contents]

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.