by Corry Shores
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[Priest, Introduction to NonClassical 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 NonClassical Logic: From If to Is
Part I:
Propositional Logic
9.
Logics with Gaps, Gluts and Worlds
9.5
Tableaux for N_{4}
Brief summary:
(9.5.1) The tableau rules for N_{4} are the same as for K_{4}, 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 N_{4} but not in K_{4}. (9.5.3) We construct countermodels from open branches in the following way. There is a world w_{i }for each i on the branch. For all propositional parameters, p, in every world (normal or not) and for conditionals, A → B, at nonnormal worlds only, if p,+i or A → B,+i occurs on the branch, set pρ_{wi }1 or A → Bρ_{wi }1; if ¬p,+i or ¬(A → B),+i occurs on the branch, set pρ_{wi}0 or A → Bρ_{wi }o. There are no other facts about ρ. (9.5.4) “N_{4} is a sublogic of K_{4}, but not the other way around,” because all valid formulas of N_{4} are valid in K_{4}, but not all valid formulas of K_{4} are valid in N_{4}. (9.5.5) “The tableaux for N_{4} are sound and complete with respect to the semantics” (169).
[The Tableau Rules for N_{4}]
[Tableau Example for N_{4}]
[CounterModels]
[N_{4} as a Proper SubLogic of K_{4},]
[The Soundness and Completeness of N_{4}]
Summary
[The Tableau Rules for N_{4}]
[The tableau rules for N_{4} are the same as for K_{4}, except the rules for → will apply only at world 0.]
[Recall from section 9.2 that K_{4} is a possible worlds First Degree Entailment (and thus four valuesituationed) 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 nonnormal 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 K_{4} except in nonnormal worlds, the conditional is assigned its value not recursively but in advance by the ρ relation (see section 9.4.8). This new nonnormal worlds First Degree Entailment system is called N_{4}. We now wonder how we construct tableaux in N_{4}. Priest says we use the same rules as for K_{4}, 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 countermodel (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 nonnormal ones. See section 9.5.3 below.)]
Tableaux for N_{4} can be obtained by modifying those for K_{4}. Specifically, the rules are exactly the same as those for K_{4}, 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 countermodel.)
(168)
[Tableau Example for N_{4}]
[Priest then gives an example of a formula that is valid in N_{4} but not in K_{4}.]
[Priest next gives an example of one that is valid in N_{4} but not in K_{4}. 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 N_{4}, 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 K_{4} an application of the rule for untrue → to the last line would immediately close it.)
(168)
[CounterModels]
[We construct countermodels from open branches in the following way. There is a world w_{i }for each i on the branch. For all propositional parameters, p, in every world (normal or not) and for conditionals, A → B, at nonnormal worlds only, if p,+i or A → B,+i occurs on the branch, set pρ_{wi }1 or A → Bρ_{wi }1; if ¬p,+i or ¬(A → B),+i occurs on the branch, set pρ_{wi}0 or A → Bρ_{wi }o. There are no other facts about ρ.]
[Recall from section 9.3.7 that in K_{4} we construct counter models from open branches in the following way: “There is a world w_{i }for each i on the branch; for propositional parameters, p, if p,+i occurs on the branch, set pρ_{wi }1; if ¬p,+i occurs on the branch, set pρ_{wi}0. ρ relates no parameter to anything else” (p.166, section 9.3.7). Priest says we will construct a countermodel for N_{4} in the same way, except now only world 0 is normal, with the remainder being nonnormal. 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 nonnormal 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_{w}_{1}0. (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_{w}1 iff for all w′ ∈ W such that Aρ_{w}_{′}1, Bρ_{w}_{′}1
A → Bρ_{w}0 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 nonnormal, 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 countermodel from an open branch exactly as for K_{4} (9.3.7), except that the only normal world is w_{0} – all others are nonnormal – and the recipe for determining ρ is applied to propositional parameters at all worlds, and to any formula of the form A → B at nonnormal worlds. Thus, in the tableau of the previous paragraph, W = {w_{0},w_{1}}; N = {w_{0}} and p → pρ_{w}_{1}0, there being no other facts about ρ. Since ¬(p → p) is true at w_{1}, and q → q is not true at w_{1}, ¬(p → p) → (q → q) is not true at w_{0}.
(168)
[N_{4} as a Proper SubLogic of K_{4},]
[“N_{4} is a sublogic of K_{4}, but not the other way around,” because all valid formulas of N_{4} are valid in K_{4}, but not all valid formulas of K_{4} are valid in N_{4}.]
[Recall from section 8.4.13 that if all valid inferences of system A are valid in system B, then system A is a sublogic 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 sublogic of system B. We saw above a formula in K_{4} that is not valid in N_{4}. Thus “N_{4} is a sublogic of K_{4}, but not the other way around.”]
Since interpretations for K_{4} are special cases of interpretations for N_{4} (namely, when W − N = φ), N_{4} is a sublogic of K_{4}, but not the other way around, as this example shows.
(168)
[The Soundness and Completeness of N_{4}]
[“The tableaux for N_{4} are sound and complete with respect to the semantics” (169).]
[Priest ends by noting that:]
The tableaux for N_{4} are sound and complete with respect to the semantics. This is proved in 9.8.8–9.8.9.
(169)
From:
Priest, Graham. 2008 [2001]. An Introduction to NonClassical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
.
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