Priest (21.8) An Introduction to Non-Classical Logic, ‘Identity,’ 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

 

21

Many-valued Logics

 

21.8

Identity

 

 

 

 

Brief summary:

(21.8.1) We define identity in our many-valued quantified logics as:

(=)(d₁, d₂) ∈ D iff d₁ = d₂

(21.8.2) Under this definition of identity, the following inferences are valid: ⊨ a=a and a=b, Pa ⊨ Pb. (21.8.3) Under this definition, the following inferences are also valid: a=b ⊨ b=a and a=b, b=c ⊨ a=c. (In other words, identity is reflexive (see above), symmetric, and transitive.) It is also substitutable: a =b, A_x(a) ⊨ A_x(b). This holds even when identity is valued i. (21.8.4) “If we are in a logic where i is thought of as neither true nor false, and we enforce the neutrality constraint, then the truth conditions for identity become: if v(a) ∈ E and v(b) ∈ E then v(=)(a, b) ∈ D iff v(a) = v(b) ; if v(a) ∉ E or v(b) ∉ E then v(=)(a, b) = i (which makes sense provided that i ∉ D). Or, if we dispense with the outer domain, and take the denotation function to be a partial function: if v(a) and v(b) are defined then v(=)(a, b) ∈ D iff v(a) = v(b) ; if either v(a) or v(b) is not defined then v(=)(a, b) = i ”(467). (21.8.5) But, if in our logic i is neither true nor false and we also enforce the neutrality constraint, then ⊨ a=a is no longer valid (for, if a is non-existent, then a=a is i, and thus not a designated value). However, a=b, Pa ⊨ Pb and more generally, a=b, A_x(a) ⊨ A_x(b) are valid. (21.8.6) Lastly, Priest notes that “given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a” (467).

 

 

 

 

 

Contents

 

21.8.1

[Defining Identity]

 

21.8.2

[Some Properties of Identity]

 

21.8.3

[Other Properties of Identity]

 

21.8.4

[Truth-Conditions for Neutral Gap Free Logics]

 

21.8.5

[Certain Valid and Invalid Statements in These Logics]

 

21.8.6

[Other Valid Formulas]

 

 

 

 

 

Summary

 

21.8.1

[Defining Identity]

 

[We define identity in our many-valued quantified logics as: (=)(d₁, d₂) ∈ D iff d₁ = d₂.]

 

[(ditto)]

If we now suppose that one of the predicates in the language is the identity predicate, then the natural truth conditions for this are:

v(=)(d₁, d₂) ∈ D iff d₁ = d₂

(467)

[contents]

 

 

 

 

 

 

21.8.2

[Some Properties of Identity]

 

[Under this definition of identity, the following inferences are valid: ⊨ a=a and a=b, Pa ⊨ Pb.]

 

[(ditto)]

It is not difficult to check that ⊨ a=a and a=b, Pa ⊨ Pb. Thus, for the second of these, suppose that in an interpretation a = b is designated. Then v(a) = v(b). So v(P)(v(a)) ∈ D iff v(P)(v(b)) ∈ D.

(467)

[contents]

 

 

 

 

 

 

21.8.3

[Other Properties of Identity]

 

[Under this definition, the following inferences are also valid: a=b ⊨ b=a and a=b, b=c ⊨ a=c. (In other words, identity is reflexive (see above), symmetric, and transitive.) It is also substitutable: a =b, A_x(a) ⊨ A_x(b). This holds even when identity is valued i.]

 

[(ditto)]

Similarly, it is not difficult to check that a=b ⊨ b=a and a=b, b=c ⊨ a=c. More generally, a = b, A_x(a) ⊨ A_x(b); for the proof of this, see 21.11.4. Note that this fact in no way depends on identities taking only classical values. Identities may well take the value i in LP or RM₃ (or b in FDE).

(467)

[contents]

 

 

 

 

 

 

21.8.4

[Truth-Conditions for Neutral Gap Free Logics]

 

[“If we are in a logic where i is thought of as neither true nor false, and we enforce the neutrality constraint, then the truth conditions for identity become: if v(a) ∈ E and v(b) ∈ E then v(=)(a, b) ∈ D iff v(a) = v(b) ; if v(a) ∉ E or v(b) ∉ E then v(=)(a, b) = i (which makes sense provided that i ∉ D). Or, if we dispense with the outer domain, and take the denotation function to be a partial function: if v(a) and v(b) are defined then v(=)(a, b) ∈ D iff v(a) = v(b) ; if either v(a) or v(b) is not defined then v(=)(a, b) = i ”(467).]

 

[(ditto)]

If we are in a logic where i is thought of as neither true nor false, and we enforce the neutrality constraint, then the truth conditions for identity become:

if v(a) ∈ E and v(b) ∈ E then v(=)(a, b) ∈ D iff v(a) = v(b)

if v(a) ∉ E or v(b) ∉ E then v(=)(a, b) = i

(which makes sense provided that i ∉ D). Or, if we dispense with the outer domain, and take the denotation function to be a partial function:

if v(a) and v(b) are defined then v(=)(a, b) ∈ D iff v(a) = v(b)

if either v(a) or v(b) is not defined then v(=)(a, b) = i

(467)

[contents]

 

 

 

 

 

 

21.8.5

[Certain Valid and Invalid Statements in These Logics]

 

[But, if in our logic i is neither true nor false and we also enforce the neutrality constraint, then ⊨ a=a is no longer valid (for, if a is non-existent, then a=a is i, and thus not a designated value). However, a=b, Pa ⊨ Pb and more generally, a=b, A_x(a) ⊨ A_x(b) are valid.]

 

[(ditto) (Note: I am assuming here, probably incorrectly, that in this logic with i as gap that i is not a designated value. We saw in section 7.3 that the logics with gaps have 1 as the only designated value, while, as we saw in section 7.4, the glut ones have i and 1 as the designated values.)]

It is clear that it will not now be the case that ⊨ a=a. (Take v(a) to be not in E, or undefined.) However it is still the case that a=b, Pa ⊨ Pb. If the first premise is true, then v(a) and v(b) are both in E (or defined), and the argument then proceeds as in 21.8.2. Indeed, more generally, a=b, A_x(a) ⊨ A_x(b). The proof is to be found in 21.11.4.

(467)

[contents]

 

 

 

 

 

 

21.8.6

[Other Valid Formulas]

 

[Lastly, Priest notes that “given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a” (467).]

 

[(ditto)]

Note that, given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a, as is easy to check.

(467)

[contents]

 

 

 

 

 

From:

 

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

 

 

 

 

Priest (21.7) An Introduction to Non-Classical Logic, ‘Neutral Free Logics,’ 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

 

21

Many-valued Logics

 

21.7

Neutral Free Logics

 

 

 

 

 

Brief summary:

(21.7.1) We will now examine neutral free logics, where applying a predicate to a non-existent object always results in the semantic value neither true nor false (i). (21.7.2) Free logics are rendered neutral free logics by the addition of the neutrality constraint: if, for some 1 ≤ j ≤ n, d_j ∉ E, then v(P)(d₁, . . . , d_n) = i. (In other words, formulas that predicate at least one non-existent object will be valued i, here understood as neither true nor false.) (21.7.3) Neutral free logics can alternatively be defined by using only a domain E of existents and by using the v denotation function for names as a partial function that leaves some values undefined, and so: 

if v(a₁) = d₁, . . . , v(a_n) = d_n then v(Pa₁ . . . a_n) = v(P)(d₁ . . . d_n)

if any of v(a₁), …, v(a_n) is undefined, v(Pa₁ . . . a_n) = i

(466)

(21.7.4) We can use this strategy also to give an alternative definition for negative free logics: “The denotation function for names is taken to be partial, and the truth conditions of atomic sentences are given as [in the section above], replacing ‘= i’ with ‘≠ 1’ ” (466). So (I presume, perhaps incorrectly):

if v(a₁) = d₁, . . . , v(a_n) = d_n then v(Pa₁ . . . a_n) = v(P)(d₁ . . . d_n)

if any of v(a₁), …, v(a_n) is undefined, v(Pa₁ . . . a_n) ≠ 1

(21.7.5) “The Neutrality Constraint gives rise to valid inferences that are not valid in a positive free logic. For example […], Pa₁ . . . a_n ⊨ ℭa₁ ∧ . . . ∧ ℭa_n and ¬ Pa₁ . . . a_n ⊨ ℭa₁ ∧ . . . ∧ ℭa_n. Negative free logics make the first of these valid, but not the second” (466). (21.7.6) In neutral free logics, we would say that statements with non-existent objects like “The greatest prime number is even” and “The King of France is bald” are valued i or neither true nor false. But we cannot say that all statements with non-existent objects are neither true nor false. “For it would seem that ‘The greatest prime number exists’ and ‘The King of France exists’ are both false, not neither true nor false” (466). But, that prevents there from being an obvious formal standard for determining which statements are exceptions. And so, by making one arbitrary exception for existence statements of non-existent objects, what stops us from making other exceptions, like saying that certain statements regarding non-existents are true, like “Homer worshipped Zeus” and “I am thinking about Sherlock Holmes”? (21.7.7) “Hence, though some sentences with non-denoting terms may be neither true nor false, not all would seem to be; the most appropriate free logic, even in a many-valued context, would appear to be a positive one” (467).

 

 

 

 

 

 

Contents

 

21.7.1

[Non-Existent Objects as Creating Gaps, in Neutral Free Logics]

 

21.7.2

[The Neutrality Constraint]

 

21.7.3

[An Alternative Definition of Neutral Free Logics Using a Partial v Function]

 

21.7.4

[Using This Strategy to Alternatively Define Negative Free Logics]

 

21.7.5

[Additional Valid Inferences in Neutral Free Logics]

 

21.7.6

[A Problem with Neutral Free Logics: Arbitrary Exceptions]

 

21.7.7

[Positive Free Logics as the Best for Many Values]

 

 

 

 

 

 

Summary

 

21.7.1

[Non-Existent Objects as Creating Gaps, in Neutral Free Logics]

 

[We will now examine neutral free logics, where applying a predicate to a non-existent object always results in the semantic value neither true nor false (i). ]

 

[In chapter 13 we dealt with free logics. They involve distinguishing a subset of the domain that is the set of existent things, with the remainder being non-existent ones (see section 13.2.2). In section 13.4 we distinguished positive, negative, and neutral free logics. And in section 13.4.1 we saw that this distinction between existent and non-existent domain members eliminated certain problematic inferences, like the inference that anything that can be predicated must exist (A_x(a) ⊨ ∃xA) and that it is impossible for nothing to exist (∃x(Px ∨ ¬Px).) But in section 13.4.2, we learned that:

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, . . . , d_n⟩ ∈ v(P) then d₁ ∈ v(ℭ), and …and d_n ∈ 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)

And finally, in section 13.4.7 we noted that:

An alternative to negative free logics would be neutral free logics, which say that sentences containing names that do not refer to existent objects would be neither true nor false. We deal with this in ch.21.

(From the brief summary of section 13.4.7)

Priest notes now that  “In positive free logics, applying a predicate to a non-existent object can result in any semantic value. In negative logics, it always results in the value false (0). In a neutral logic it is always neither true nor false (i);” and our particular focus now is on neutral logics now, as we covered the other two types before and postponed the neutral ones for this section.]

In 13.4 we noted that free logics can be classified as positive, negative, or neutral. In positive free logics, applying a predicate to a non-existent object can result in any semantic value. In negative logics, it always results in the value false (0). In a neutral logic it is always neither true nor false (i). We looked at positive and negative free logics in chapter 13. We are now in a position to see what a neutral free logic is like.

(465)

[contents]

 

 

 

 

 

 

21.7.2

[The Neutrality Constraint]

 

[Free logics are rendered neutral free logics by the addition of the neutrality constraint: if, for some 1 ≤ j ≤ n, d_j ∉ E, then v(P)(d₁, . . . , d_n) = i. (In other words, formulas that predicate at least one non-existent object will be valued i, here understood as neither true nor false.)]

 

[We next examine the neutrality constraint, which when added to a free logic will generate a neutral free logic. I am not certain about how this formulation works, so please see the quotation below. It says specifically:

if, for some 1 ≤ j ≤ n, d_j ∉ E, then v(P)(d₁, . . . , d_n) = i

I will guess at the meaning. Overall, it seems to be saying that formulas (which in quantified logic contain predicates) that predicate at least one non-existent object will be valued i, which is presumably here neither true nor false.]

A neutral free logic is a logic with a value which may be thought of as neither true nor false, such as i in K₃ or Ł₃ (or the value n in FDE – see the next chapter), which satisfies the condition that for any n-place predicate:

if, for some 1 ≤ j ≤ n, d_j ∉ E, then v(P)(d₁, . . . , d_n) = i.

Call this the Neutrality Constraint. (Depending on the context, the converse condition might also be plausible: if v(P)(d₁, . . . , d_n) = i then, for some 1 ≤ j ≤ n, d_j ∉ E. Only non-existent objects give rise to truth value gaps.) Note that the Negativity Constraint can be added just as much to a many-valued logic as it can be to a two-valued logic, giving rise to a many-valued negative free logic.

(465)

[contents]

 

 

 

 

 

 

21.7.3

[An Alternative Definition of Neutral Free Logics Using a Partial v Function]

 

[Neutral free logics can alternatively be defined by using only a domain E of existents and by using the v denotation function for names as a partial function that leaves some values undefined, and so:  “if v(a₁) = d₁, . . . , v(a_n) = d_n then v(Pa₁ . . . a_n) = v(P)(d₁ . . . d_n) ; if any of v(a₁), …, v(a_n) is undefined, v(Pa₁ . . . a_n) = i” (466).]

 

[(ditto)]

Neutral free logics can be formulated in a different, but equivalent, way. We may dispense with the ‘outer domain’ altogether. The only domain we need is E. Instead of taking the denotation function for names, v, to be a total function, we let it be partial. That is, for some inputs the output may just not be defined – just as division is not defined if the divisor is zero. (Division is, in fact, a partial function.) The appropriate truth conditions for atomic sentences are then:

if v(a₁) = d₁, . . . , v(a_n) = d_n then v(Pa₁ . . . a_n) = v(P)(d₁ . . . d_n)

if any of v(a₁), …, v(a_n) is undefined, v(Pa₁ . . . a_n) = i.

It is not difficult to see that the truth value of any sentence comes out the same under this policy. (The truth conditions make this clear for atomic sentences. For other formulas, this follows by a simple induction.)

(466)

[contents]

 

 

 

 

 

 

21.7.4

[Using This Strategy to Alternatively Define Negative Free Logics]

 

[We can use this strategy also to give an alternative definition for negative free logics: “The denotation function for names is taken to be partial, and the truth conditions of atomic sentences are given as [in the section above], replacing ‘= i’ with ‘≠ 1’ ” (466). So: if v(a₁) = d₁, . . . , v(a_n) = d_n then v(Pa₁ . . . a_n) = v(P)(d₁ . . . d_n) ; if any of v(a₁), …, v(a_n) is undefined, v(Pa₁ . . . a_n) ≠ 1 ]

 

[(ditto)]

Note that we can follow the same strategy with respect to negative free logics as well. The denotation function for names is taken to be partial, and the truth conditions of atomic sentences are given as in 21.7.3, replacing ‘= i’ with ‘≠ 1’.²

(466)

2. An even stronger constraint replaces ‘= i’ with ‘= 0’. But this constraint, equivalent in a classical context, is less natural in a many-valued context. The intuition behind the Negativity Constraint is simply that atomic sentences containing names that do not refer to (existent) objects cannot be true.

(466)

[contents]

 

 

 

 

 

 

21.7.5

[Additional Valid Inferences in Neutral Free Logics]

 

[“The Neutrality Constraint gives rise to valid inferences that are not valid in a positive free logic. For example […], Pa₁ . . . a_n ⊨ ℭa₁ ∧ . . . ∧ ℭa_n and ¬ Pa₁ . . . a_n ⊨ ℭa₁ ∧ . . . ∧ ℭa_n. Negative free logics make the first of these valid, but not the second” (466).]

 

[(ditto)]

The Neutrality Constraint gives rise to valid inferences that are not valid in a positive free logic. For example, as is easy to check, Pa₁ . . . a_n ⊨ ℭa₁ ∧ . . . ∧ ℭa_n and ¬ Pa₁ . . . a_n ⊨ ℭa₁ ∧ . . . ∧ ℭa_n. Negative free logics make the first of these valid, but not the second.

(466)

[contents]

 

 

 

 

 

 

21.7.6

[A Problem with Neutral Free Logics: Arbitrary Exceptions]

 

[In neutral free logics, we would say that statements with non-existent objects like “The greatest prime number is even” and “The King of France is bald” are valued i or neither true nor false. But we cannot say that all statements with non-existent objects are neither true nor false. “For it would seem that ‘The greatest prime number exists’ and ‘The King of France exists’ are both false, not neither true nor false” (466). But, that prevents there from being an obvious formal standard for determining which statements are exceptions. And so, by making one arbitrary exception for existence statements of non-existent objects, what stops us from making other exceptions, like saying that certain statements regarding non-existents are true, like “Homer worshipped Zeus” and “I am thinking about Sherlock Holmes”?]

 

[(ditto) (Note: I am not entirely sure I grasp the problem with exceptions. Is it simply that we have no way to formally determine which exceptions there should be? Is it that by having these arbitrary exceptions, we somehow make it needless to have the neutrality constraint to begin with?)]

Neutral free logics are usually motivated by examples such as ‘The greatest prime number is even’ and ‘The King of France is bald’. But note that one would seem to have to make exceptions for the existence predicate itself. For it would seem that ‘The greatest prime number exists’ and ‘The King of France exists’ are both false, not neither true nor false. And once one has made an exception for one predicate, it seems somewhat arbitrary not to admit other exceptions, such as those we noted in connection with negative free logics in 13.4.6.

(466)

[contents]

 

 

 

 

 

 

21.7.7

[Positive Free Logics as the Best for Many Values]

 

[“Hence, though some sentences with non-denoting terms may be neither true nor false, not all would seem to be; the most appropriate free logic, even in a many-valued context, would appear to be a positive one” (467).]

 

[(ditto)]

Hence, though some sentences with non-denoting terms may be neither true nor false, not all would seem to be; the most appropriate free logic, even in a many-valued context, would appear to be a positive one.

(467)

[contents]

 

 

 

 

 

 

 

From:

 

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

 

 

 

 

Priest (21.5) An Introduction to Non-Classical Logic, ‘Their Free Versions,’ 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

 

21

Many-valued Logics

 

21.5

Their Free Versions

 

 

 

 

Brief summary:

(21.5.1) Quantified many-valued logics still have the problematic inferences Pa ⊨ ∃xPx and ∀xPx ⊨ Pa, and they can be solved using free logics. (21.5.2) Our quantified 3-valued logics are structured in the following way: “We take the language to contain an existence predicate, ℭ. An interpretation is a triple ⟨D, E, v⟩. D is the domain of all objects, and E ⊆ D contains those that are thought of as existent. For every constant, c, v(c) ∈ D. For every n-place predicate, P, v(P) is a function such that if d₁, . . . , d_n ∈ D, v(P)(d₁, . . . , d_n) ∈ V. v(ℭ) is such that: v(ℭ)(d) ∈ D iff d ∈ E . Truth conditions are as in the non-free case, except that for the quantifiers v(∀xA) = Min({A_x(k_d): d ∈ E}) (not D), and v(∃xA) = Max({A_x(k_d): d ∈ E})” (461-462). (21.5.3) Using these semantics, counter-models can be constructed for the above problematic inferences. (21.5.4) In our free version of many-valued logics, we establish validity and invalidity in the same way as the non-free versions. When D = E, then “anything valid in any many-valued free logic is valid in the corresponding non-free logic.” “Conversely, suppose that the inference with premises Σ and conclusion A is valid in one of our 3-valued logics. Let C be the set of constants that occur in A and all members of Σ, and let Π = {ℭc: c ∈ C} ∪ {∃x ℭx}. (The quantified sentence is redundant if C ≠ φ.) Then Π∪Σ ⊨ A in the corresponding free logic (where quantifiers are inner)” (462).

 

 

 

 

 

 

Contents

 

21.5.1

[Using Free Logics to Solve Problematic Inferences in Many-Valued Quantified Logics]

 

21.5.2

[The Structure and Truth-Conditions of Our 3-Valued Logics]

 

21.5.3

[The Invalidity of the Problematic Inferences Showable Using Counter-Models]

 

21.5.4

[Validity and Invalidity in Free Many-Valued Logics. Comparisons with Non-Free Versions.]

 

 

 

 

 

Summary

 

21.5.1

[Using Free Logics to Solve Problematic Inferences in Many-Valued Quantified Logics]

 

[Quantified many-valued logics still have the problematic inferences Pa ⊨ ∃xPx and ∀xPx ⊨ Pa, and they can be solved using free logics.]

 

[Recall from section 12.6.3 a problem in quantified logic where anything predicated must exist (thus Pegasus, which is predicated as a mythological figure, as having wings, etc., must exist), on account of the validity of Pa ⊨ ∃xPx. Priest says that this is valid in 3-valued logics too. Priest also notes that ∀xPx ⊨ Pa is also valid and is problematic, but I am not sure where we have seen it before (if it is in section 12.6, I have not found it yet. I came across something similar in section 15.2.3. So maybe the problem with it comes about when a is non-existent. Maybe furthermore this is problematic if P is taken to be the existence predicate. But I am just guessing very poorly here, sorry.) Now recall from from chapter 13 how free logics deal with these problems by using free logics. Regarding free logics, recall now from the brief summary of section 13.2 that in

(13.2.1) […] free logics we have the one-place existence predicate ℭ. We can think of ℭa as meaning ‘a exists’. (13.2.2) In free logics, we have our main domain of all objects, D, and we have the “inner domain” E. It is a a subset of D that we think of as being the set of all existent objects. (“An interpretation for the language is a triple ⟨D, E, v⟩, where D is a non-empty set, and E (the ‘inner domain’) is a (possibly empty) subset of D. One can think of D as the set of all objects, and E as the set of all existent objects” (290).) So suppose D contains Sherlock Holmes, the Pegasus, and Julius Caesar. Here, although all of them are in D, only Caesar is in E. (13.2.3) “As in classical logic, v assigns every constant in the language a member of D, and every n-place predicate a subset of Dⁿ. In any interpretation, v(ℭ) = E” (290).

(from the brief summary of section 13.2, quotation is Priest’s)

In section 13.4 and  section 13.5 we examined these solutions more closely. The important point now is that we can use free logic in a similar way to solve these problems in many-valued quantified logics.]

It is not difficult to check that in all the 3-valued logics in our compass

Pa ⊨ ∃xPx

∀xPx ⊨ Pa

Thus, for the first, if Pa is designated in an interpretation then v(P)(v(a)) ∈ D, in which case v(∃xPx) ∈ D. But one might well have reservations about these inferences, as we have already observed in 12.6. And just as one can formulate a free version of classical logic, as we did in chapter 13, one can formulate free versions of many-valued logics.

(461)

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21.5.2

[The Structure and Truth-Conditions of Our 3-Valued Logics]

 

[Our quantified 3-valued logics are structured in the following way: “We take the language to contain an existence predicate, ℭ. An interpretation is a triple ⟨D, E, v⟩. D is the domain of all objects, and E ⊆ D contains those that are thought of as existent. For every constant, c, v(c) ∈ D. For every n-place predicate, P, v(P) is a function such that if d₁, . . . , d_n ∈ D, v(P)(d₁, . . . , d_n) ∈ V. v(ℭ) is such that: v(ℭ)(d) ∈ D iff d ∈ E . Truth conditions are as in the non-free case, except that for the quantifiers v(∀xA) = Min({A_x(k_d): d ∈ E}) (not D), and v(∃xA) = Max({A_x(k_d): d ∈ E})” (461-462).]

 

[We will now structure the quantified many-valued free logic similarly to what we saw above in section 21.5.1 in reference to section 13.2. We will have the existence predicate ℭ, whose denotation is the set of existent things in the domain, which is the subset E of the domain D, with the remainder being the set of non-existent things in the domain. Every constant is assigned a member in the domain. I am not entirely sure I understand the predicate evaluation, but it seems to be that the v function assigns a truth-value to predicates when its assigned constituents are in the domain (see the quotation below). I also may not follow the evaluation of the existence predicate, but it seems to be saying that the value of an existence predicate is a designated value only if the predicated thing is in the set of existent things. But I am not sure why we are using the designated values here. Finally, the truth conditions are the same as in the non-free cases, except for how the quantifiers are truth-evaluated. In section 21.2.3, Priest discusses the evaluation of formulas:

Given this structure, an evaluation, v, assigns every constant a member of D and every n-place predicate an n-place function from the domain into the truth values. (So if P is any predicate, v(P) is a function with inputs in D and an output in V.) Given an evaluation, every formula, A, is then assigned a value, v(A), in V recursively, as follows. If P is any n-place predicate:

v(Pa₁ . . . a_n) = v(P)(v(a₁), . . . , v(a_n))

For each n-place propositional connective, c:

v(c(A₁, . . . , A_n)) = f_c(v(A₁), . . . , v(A_n))

as in the propositional case. And for each quantifier, q:

v(qxA) = f_q({v(A_x(k_d)): d ∈ D})

(In a free many-valued logic, ‘D’ is replaced by ‘E’.) For example, v(∀xA) = f_∀({v(A_x(k_d)): d ∈ D}). Thus, the value of qxA is determined by the set of the values of substitution instances of A formed using the names of all members of the domain of quantification.

(p.457, section 21.2.3)

I am not exactly sure how the connectives are evaluated, but I will guess it is in accordance with the tables given in section 7.3 and section 7.4. The quantifiers are evaluated as:

v(∀xA) = Min({A_x(k_d): d ∈ E}) (not D)

v(∃xA) = Max({A_x(k_d): d ∈ E}).

]

We take the language to contain an existence predicate, ℭ. An interpretation is a triple ⟨D, E, v⟩. D is the domain of all objects, and E ⊆ D contains those that are thought of as existent. For every constant, c, v(c) ∈ D. For every n-place predicate, P, v(P) is a function such that if d₁, . . . , d_n ∈ D, v(P)(d₁, . . . , d_n) ∈ V. v(ℭ) is such that:

v(ℭ)(d) ∈ D iff d ∈ E

| Truth conditions are as in the non-free case, except that for the quantifiers v(∀xA) = Min({A_x(k_d): d ∈ E}) (not D), and v(∃xA) = Max({A_x(k_d): d ∈ E}).

(461-462)

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21.5.3

[The Invalidity of the Problematic Inferences Showable Using Counter-Models]

 

[Using these semantics, counter-models can be constructed for the above problematic inferences.]

 

[(ditto). (Note: I have not yet carried out these exercises, but I would imagine they would be done using the trial and error method shown in section 21.4.6.)]

It is now not difficult to construct counter-models to the inferences of 21.5.1. Details are left as an exercise.

(462)

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21.5.4

[Validity and Invalidity in Free Many-Valued Logics. Comparisons with Non-Free Versions.]

 

[In our free version of many-valued logics, we establish validity and invalidity in the same way as the non-free versions. When D = E, then “anything valid in any many-valued free logic is valid in the corresponding non-free logic.” “Conversely, suppose that the inference with premises Σ and conclusion A is valid in one of our 3-valued logics. Let C be the set of constants that occur in A and all members of Σ, and let Π = {ℭc: c ∈ C} ∪ {∃x ℭx}. (The quantified sentence is redundant if C ≠ φ.) Then Π∪Σ ⊨ A in the corresponding free logic (where quantifiers are inner)” (462)]

 

[(This last part is a bit complex, so see the quotation below. But I will stumble through it a bit up to then.) We establish validity and invalidity in the same manner as with the non-free versions of these logics. As we saw in section 21.4, we use argumentation techniques like reductio (section 21.4.4) and contraposition (section 21.4.5) to show validity. To make counter-models to show invalidity, we use a trial and error method (section 21.4.6). (I cannot say much more on this right now, as I did not try to learn those techniques yet.) Priest now has us note “the special case of a free interpretation where D = E is a non-free interpretation. Hence, anything valid in any many-valued free logic is valid in the corresponding non-free logic” (462). I do not quite grasp what that means yet. I suppose we are talking about cases of free logic interpretations where the set of existents is identical to the whole domain. And so, it would seem to me, to be structured at its basis as a non-free logic, with the E set being redundant to the domain. The final point in this paragraph I am not following, but I will stumble through it a bit. We think of an argument I think in a non-free 3-valued logic with a set of premises that we call Σ  and the conclusion called A. In the premises and conclusion there may be constants. Now we form another set. First recall from section 0.1.8 that a union is defined in the following way:

The union of two sets, X, Y, is the set containing just those things that are in X or Y (or both). This is written as X ∪ Y. So a ∈ X ∪ Y if and only if a ∈ X or a ∈ Y.

(page xxviii, section 0.1.8)

So this other set, called Π, is defined as

{ℭc: c ∈ C} ∪ {∃x ℭx}

But I may not get that well. It seems to be the union of two sets, with the first set being the set of all the existing constants in the premises and conclusion and another set which I cannot grasp, but is maybe the set of existentially quantifiable members of the domain. I probably have that wrong, but even if it is correct, it would seem to be redundant with the first set. In fact, the next sentence reads, “(The quantified sentence is redundant if C ≠ φ.)” But I am not sure really how all this works. Next Priest writes, “Then Π∪Σ ⊨ A in the corresponding free logic (where quantifiers are inner)” (462). I do not grasp this either, but it seems to be saying that on the basis of the union of the set of existing things mentioned in the premises and conclusion of the argument along with those premises themselves, we can infer the conclusion. So I am really guessing here, but maybe that is like saying we start with a non-free logic, and we take the premises, and then we affirm the existence of items they are predicating as well as the existence of the conclusion’s items, and we can infer the conclusion itself in the free logic version of the 3-valued logic. Please see the quotation, as I am guessing poorly here.]

To establish the validity or invalidity of inferences in the free version of a many-valued logic, we may proceed as in the non-free case. But note the special case of a free interpretation where D = E is a non-free interpretation. Hence, anything valid in any many-valued free logic is valid in the corresponding non-free logic. Conversely, suppose that the inference with premises Σ and conclusion A is valid in one of our 3-valued logics. Let C be the set of constants that occur in A and all members of Σ, and let Π = {ℭc: c ∈ C} ∪ {∃x ℭx}. (The quantified sentence is redundant if C ≠ φ.) Then Π∪Σ ⊨ A in the corresponding free logic (where quantifiers are inner). (This is true even when the language contains the identity predicate, and is proved in 21.11.6.)

(462)

[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

 

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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

 

16.2

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: v_w(=) = {⟨d, d⟩ : d ∈ D}. ” (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 ∀x∀y(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)

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16.2.2

[Defining the Identity Predicate]

 

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

 

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

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

(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)

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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 ∀x∀y(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) ∀x∀y(x = y ⊃ □x = y), and ⊢_VK(NI) ∀x∀y(x ≠ y ⊃ □x ≠ y). Clearly, the tableaux work in a similar way in VK^t(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 ∀x∀y(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 . 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)

 

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 . 1¬∀ . 2PI . 2PI . 4¬∀ . 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) □∀x∀y((Sax ∧ Say) ⊃ x ≠ y):

 

⊬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 . 1¬□ . 2◊r . 2◊r . 4¬∀ . 5PI . 6¬∀ . 7PI . 8¬⊃ . 8¬⊃ . 9∧ . 9∧ . 10¬¬ . 3,13IRR (open) invalid

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

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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(A_x(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(A_x(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.)

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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 v_w1 (S) = {⟨∂a, ∂b ⟩}. In a picture:

_______________xxxxxxxxxxxxxxxx______________

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

|x∂axx×xxxx×xx|xxxwoxx→xxw1xxx|x∂axx×xxxx√xx|

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

_______________xxxxxxxxxxxxxxxx______________

 

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

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From:

 

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