Pirates & Revolutionaries: Priest (9.5) Introduction to Non-Classical Logic, ‘Tableaux for N4,’ summary

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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 I:

Propositional Logic

Logics with Gaps, Gluts and Worlds

9.5

Tableaux for N₄

Brief summary:

(9.5.1) The tableau rules for N₄ are the same as for K₄, except the rules for → will apply only at world 0.

Double Negation

Development, True (¬¬D,+)

¬¬A,+i

↓

A,+i

Double Negation

Development, False (¬¬D,−)

¬¬A,−i

↓

A,−i

Conjunction

Development, True (∧D,+)

A ∧ B,+i

↓

A,+i

B,+i

Conjunction

Development, False (∧D,−)

A ∧ B,−i

↙ ↘

A,−i B,−i

Negated Conjunction

Development, True (¬∧D,+)

¬(A ∧ B),+i

↓

¬A ∨ ¬B,+i

Negated Conjunction

Development, False (¬∧D,−)

¬(A ∧ B),−i

↓

¬A ∨ ¬B,−i

Disjunction

Development, True (∨D,+)

A ∨ B,+i

↙ ↘

A,+i B,+i

Disjunction

Development, False (∨D,−)

A ∨ B,-i

↓

A,-i

B,-i

Negated Disjunction

Development, True (¬∨D, +)

¬(A ∨ B),+i

↓

¬A ∧ ¬B,+i

Negated Disjunction

Development, False(¬∨D, -)

¬(A ∨ B),-i

↓

¬A ∧ ¬B,-i

Conditional

Development, True (→D,+)

A → B,+i

↙ ↘

A,-j B,+j

j is every number that occurs on the branch (and this rule applies only to world 0)

Conditional

Development, False (→D,−)

A → B,-i

↓

A,+j

B,-j

j is a new number. (Here i will always be 0 and j will be 1)

Negated Conditional

Development, True (¬→D, +)

¬(A → B),+i

↓

A,+j

¬B,+j

j is a new number. (Here i will always be 0 and j will be 1)

Negated Conditional

Development, False(¬→D, -)

¬(A → B),-i

↙ ↘

A,-j ¬B,-j

j is every number that occurs on the branch (and this rule applies only to world 0. So i will always be 0)

(165, titles for the rules are my own additions. Note that the rules for double negation and disjunction are not in the text and are probably mistaken. Also, I am guessing about the conditionals, too.)

(9.5.2) Priest then gives an example of a formula that is valid in N₄ but not in K₄. (9.5.3) We construct counter-models from open branches in the following way. There is a world w_ifor each i on the branch. For all propositional parameters, p, in every world (normal or not) and for conditionals, A → B, at non-normal worlds only, if p,+i or A → B,+i occurs on the branch, set pρ_wi1 or A → Bρ_wi1; if ¬p,+i or ¬(A → B),+i occurs on the branch, set pρ_wi0 or A → Bρ_wio. There are no other facts about ρ. (9.5.4) “N₄ is a sub-logic of K₄, but not the other way around,” because all valid formulas of N₄ are valid in K₄, but not all valid formulas of K₄ are valid in N₄. (9.5.5) “The tableaux for N₄ are sound and complete with respect to the semantics” (169).

Contents

9.5.1

[The Tableau Rules for N₄]

9.5.2

[Tableau Example for N₄]

9.5.3

[Counter-Models]

9.5.4

[N₄ as a Proper Sub-Logic of K₄,]

9.5.5

[The Soundness and Completeness of N₄]

Summary

9.5.1

[The Tableau Rules for N₄]

[The tableau rules for N₄ are the same as for K₄, except the rules for → will apply only at world 0.]

[Recall from section 9.2 that K₄ is a possible worlds First Degree Entailment (and thus four value-situationed) system. In section 9.4 we noted that we would want our conditional to be able to express statements that suppose certain laws of logic be suspended and then say what might follow from that suspension. For this we appealed to non-normal worlds, which are worlds where the normal laws of logic might fail (see section 4). In section 9.4, Priest gave the semantic rules for such a system. They are the same rules for K₄ except in non-normal worlds, the conditional is assigned its value not recursively but in advance by the ρ relation (see section 9.4.8). This new non-normal worlds First Degree Entailment system is called N₄. We now wonder how we construct tableaux in N₄. Priest says we use the same rules as for K₄, but the rules for → will apply only at world 0. As such, I will guess that the rules would be the following, but I am not certain this is correct.

Double Negation

Development, True (¬¬D,+)

¬¬A,+i

↓

A,+i

Double Negation

Development, False (¬¬D,−)

¬¬A,−i

↓

A,−i

Conjunction

Development, True (∧D,+)

A ∧ B,+i

↓

A,+i

B,+i

Conjunction

Development, False (∧D,−)

A ∧ B,−i

↙ ↘

A,−i B,−i

Negated Conjunction

Development, True (¬∧D,+)

¬(A ∧ B),+i

↓

¬A ∨ ¬B,+i

Negated Conjunction

Development, False (¬∧D,−)

¬(A ∧ B),−i

↓

¬A ∨ ¬B,−i

Disjunction

Development, True (∨D,+)

A ∨ B,+i

↙ ↘

A,+i B,+i

Disjunction

Development, False (∨D,−)

A ∨ B,-i

↓

A,-i

B,-i

Negated Disjunction

Development, True (¬∨D, +)

¬(A ∨ B),+i

↓

¬A ∧ ¬B,+i

Negated Disjunction

Development, False(¬∨D, -)

¬(A ∨ B),-i

↓

¬A ∧ ¬B,-i

Conditional

Development, True (→D,+)

A → B,+i

↙ ↘

A,-j B,+j

j is every number that occurs on the branch (and this rule applies only to world 0)

Conditional

Development, False (→D,−)

A → B,-i

↓

A,+j

B,-j

j is a new number. (Here i will always be 0 and j will be 1)

Negated Conditional

Development, True (¬→D, +)

¬(A → B),+i

↓

A,+j

¬B,+j

j is a new number. (Here i will always be 0 and j will be 1)

Negated Conditional

Development, False(¬→D, -)

¬(A → B),-i

↙ ↘

A,-j ¬B,-j

j is every number that occurs on the branch (and this rule applies only to world 0. So i will always be 0)

(165, titles for the rules are my own additions. Note that the rules for double negation and disjunction are not in the text and are probably mistaken.)

Priest notes also that although the rules for → apply only at world 0, we will never need to assume that there is more than one normal world in a counter-model (but I am not sure how that works yet. It could be that 0 is always the normal world and any others will be the non-normal ones. See section 9.5.3 below.)]

Tableaux for N₄ can be obtained by modifying those for K₄. Specifically, the rules are exactly the same as those for K₄, except that the rules for → apply at world 0 only. (It turns out that we never need to assume that there is more than one normal world in a counter-model.)

(168)

[contents]

9.5.2

[Tableau Example for N₄]

[Priest then gives an example of a formula that is valid in N₄ but not in K₄.]

[Priest next gives an example of one that is valid in N₄ but not in K₄. Let us do both to compare them.

⊢_K4 ¬(p → p) → (q → q)

¬(p → p) → (q → q),0–

↓

¬(p → p),1+

↓

(q → q),1–

↓

q,2+

↓

q,2–

1→–

3→–

(5×6)

valid

(not in the text, and probably mistaken)

Compare this with the formula in N₄, given as quotation below.]

For example: ⊬ ¬(p → p) → (q → q):

⊬_N4 ¬(p → p) → (q → q)

1.

.

2.

.

3.

.

.

¬(p → p) → (q → q),0–

↓

¬(p → p),1+

↓

(q → q),1–

P

.

1→–

.

1→–

(open)

invalid

(p.168, enumeration and step accounting are my own and are probably mistaken)

The tableau finishes there! (In K₄ an application of the rule for untrue → to the last line would immediately close it.)

(168)

[contents]

9.5.3

[Counter-Models]

[We construct counter-models from open branches in the following way. There is a world w_ifor each i on the branch. For all propositional parameters, p, in every world (normal or not) and for conditionals, A → B, at non-normal worlds only, if p,+i or A → B,+i occurs on the branch, set pρ_wi1 or A → Bρ_wi1; if ¬p,+i or ¬(A → B),+i occurs on the branch, set pρ_wi0 or A → Bρ_wio. There are no other facts about ρ.]

[Recall from section 9.3.7 that in K₄ we construct counter models from open branches in the following way: “There is a world w_ifor each i on the branch; for propositional parameters, p, if p,+i occurs on the branch, set pρ_wi1; if ¬p,+i occurs on the branch, set pρ_wi0. ρ relates no parameter to anything else” (p.166, section 9.3.7). Priest says we will construct a counter-model for N₄ in the same way, except now only world 0 is normal, with the remainder being non-normal. Also, the stipulations above for the ρ relations hold for propositional parameters of all worlds and also for formulas of the form A → B at non-normal worlds. As we can see from our tableau above, there are no propositional parameters. So no ρ relation is given to them. However, we do have a conditional formula that is true, namely, ¬(p → p) . That means we set p → pρ_w₁0. (From this we can infer that the negation ¬(p → p) is true at world 1. But since we have not stipulated a value for (q → q), that means it is neither true nor false in world 1. That means that in ¬(p → p) → (q → q) the antecedent is true but the consequent is not true in world 1. And thus the whole formula is not true at world 0. For, recall the rule for the conditional:

A → Bρ_w1 iff for all w′ ∈ W such that Aρ_w_′1, Bρ_w_′1

A → Bρ_w0 iff for some w′ ∈ W, Aρ_w_′1 and Bρ_w_′0

(p.164, section 9.2.4)

Since world 1 is one of the worlds, despite being non-normal, it determines the value of the formula in the normal world as false. I am not sure I have this right, so please see the quotation below.)]

We read off a counter-model from an open branch exactly as for K₄ (9.3.7), except that the only normal world is w₀ – all others are non-normal – and the recipe for determining ρ is applied to propositional parameters at all worlds, and to any formula of the form A → B at non-normal worlds. Thus, in the tableau of the previous paragraph, W = {w₀,w₁}; N = {w₀} and p → pρ_w₁0, there being no other facts about ρ. Since ¬(p → p) is true at w₁, and q → q is not true at w₁, ¬(p → p) → (q → q) is not true at w₀.

(168)

[contents]

9.5.4

[N₄ as a Proper Sub-Logic of K₄,]

[“N₄ is a sub-logic of K₄, but not the other way around,” because all valid formulas of N₄ are valid in K₄, but not all valid formulas of K₄ are valid in N₄.]

[Recall from section 8.4.13 that if all valid inferences of system A are valid in system B, then system A is a sub-logic of system B. And if in addition to that not all formulas of system B are valid in A, then system A is a proper sub-logic of system B. We saw above a formula in K₄ that is not valid in N₄. Thus “N₄ is a sub-logic of K₄, but not the other way around.”]

Since interpretations for K₄ are special cases of interpretations for N₄ (namely, when W − N = φ), N₄ is a sub-logic of K₄, but not the other way around, as this example shows.

(168)