22 Jun 2018

Priest (9.5) Introduction to Non-Classical Logic, ‘Tableaux for N4,’ 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 I:

Propositional Logic

 

9.

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 wi for 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 pρwi 1 or A Bρwi 1; if ¬p,+i or ¬(A B),+i occurs on the branch, set pρwi0 or A Bρwi o. 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)

1.

.

2.

.

3.

.

4.

.

5.

.

¬(p → p) → (q → q),0–

¬(p → p),1+

(q → q),1

q,2+

q,2

×

P

.

1→–

.

1→–

.

3→–

.

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) → (qq):

 

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 wi for 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 pρwi 1 or A Bρwi 1; if ¬p,+i or ¬(A B),+i occurs on the branch, set pρwi0 or A Bρwi o. 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 wi for each i on the branch; for propositional parameters, p, if p,+i occurs on the branch, set pρwi 1; if ¬p,+i occurs on the branch, set pρwi0. 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 pρw10. (From this we can infer that the negation ¬(p p) is true at world 1. But since we have not stipulated a value for (qq), that means it is neither true nor false in world 1. That means that in ¬(p p) → (qq) 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 Bρw1 iff for all w′ ∈ W such that Aρw1, Bρw1

A Bρw0 iff for some w′ ∈ W, Aρw1 and Bρw0

(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 pρw10, there being no other facts about ρ. Since ¬(p p) is true at w1, and q q is not true at w1, ¬(p p) → (qq) 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)

[contents]

 

 

 

 

9.5.5

[The Soundness and Completeness of N4]

 

[“The tableaux for N4 are sound and complete with respect to the semantics” (169).]

 

[Priest ends by noting that:]

The tableaux for N4 are sound and complete with respect to the semantics. This is proved in 9.8.8–9.8.9.

(169)

[contents]

 

 

 

 

 

From:

 

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

 

 

.

 

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