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
4.
Non-Normal Modal Logics; Strict Conditionals
4.3
Tableaux for Non-Normal Modal Logics
Brief summary:
(4.3.1) The tableau rules for non-normal modal logics N are mostly the same as for normal modal logics K. We need however to add the following exception: “If world i occurs on a branch of a tableau, call it □-inhabited if there is some node of the form □B,i on the branch. The rule for ◊A,i is activated only when i = 0 or i is □-inhabited” (65). (4.3.2) The ◊-rule (that if you have ◊A,i on one node you can obtain that there is an accessible world where A holds) applies only to normal worlds, because possibility in non-normal worlds does not require it holding in an accessible world; rather, simply all possibilities are true regardless of other worlds. (4.3.3) Priest gives an example showing how the ◊-rule is applied when dealing with world 0. (4.3.4) Priest gives another example where we see that as world 1 is not □-inhabited, we do not apply the ◊-rule to a case in world 1 where there is the possibility operator. (4.3.5) We form counter-examples while keeping in mind which worlds are non-normal. We assign worlds in accordance with the i numbers. We assign R relations in accordance with irj formulations. And nodes of the form p, i we assign vwi(p) = 1. And for nodes of the form ¬p, i, we assign vwi(p) = 0. If there are neither of these two, then vwi(p) can be given any value we want . (4.3.6) Priest gives an example of a counter-model. When depicting non-normal worlds, we place the world designator in a box but write the true formulas in that world above the box. (4.3.7) Tableaux for Nρ, Nρτ, etc. use the same additional rules as for Kρ, Kρτ,etc. (4.3.8) “The tableaux for N and its extensions are sound and complete” (67).
[Additional Notions and Rules for Non-Normal Modal Logic Tableaux]
[The Rationale for the ◊-Rule]
[Example 1]
[Example 2]
[Counter-Model Formation]
[Counter-Model Example]
[Rules for N Extensions]
[The Soundness and Completeness of N]
Summary
[Additional Notions and Rules for Non-Normal Modal Logic Tableaux]
[The tableau rules for non-normal modal logics N are mostly the same as for normal modal logics K. We need however to add the following exception: “If world i occurs on a branch of a tableau, call it □-inhabited if there is some node of the form □B,i on the branch. The rule for ◊A,i (2.4.4) is activated only when i = 0 or i is □-inhabited.”]
[We will now learn the tableau procedures for non-normal modal logics, N, whose semantics we learned in the previous section, 4.2. Let us first review the tableau rules for normal modal logics from section 2.4, as we will be building from them. Our tableaux’s nodes have one of two structures: A,i, where A is a formula and i is a natural number indicating the world in which the formula holds, or {2} irj, where i is a natural number for a world that accesses world j, also given as a natural number (the r stays as r) (section 2.4.1). We test for validity by setting the premises to true in world 0 and the negation of the conclusion to true in world 0 (section 2.4.2). We indicate worlds on our tableaux, and the branches inherit the world indicators from above (section 2.4.3). Here are the rules (note, the names are my own. I made them following David Agler (see section 4.2 of Agler’s Symbolic Logic text. Their purpose is so I can follow the reasoning behind the steps of the trees.):
| 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)
Branches close when there are contradictions in the same world (section 2.4.5). An inference is valid if it makes a completed closed tree, and invalid if it makes a completed open tree. We make counter-models using completed open branches. We assign worlds in accordance with the i numbers. We assign R relations in accordance with irj formulations. And nodes of the form p,i we assign vwi(p) = 1. And for nodes of the form ¬p,i, we assign vwi(p) = 0. If there are neither of these two, then vwi(p) can be given any value we want (section 2.4.7). Priest says that the tableau procedures are the same except for some other complications that we will see illustrated later. We now have the notion of a branch being “□-inhabited.” It seems that if we have a node of the form □B,i then it is □-inhabited. It seems we need this designation in order to know when to use the rule for ◊A,i
| 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)
Priest says that this rule “is activated only when i = 0 or i is □-inhabited.”]
A tableau technique for N is obtained by modifying the technique for K as follows. If world i occurs on a branch of a tableau, call it □-inhabited if there is some node of the form □B,i on the branch. The rule for ◊A,i (2.4.4) | is activated only when i = 0 or i is □-inhabited. Otherwise, details are the same as for K.
(65-66)
[The Rationale for the ◊-Rule]
[The ◊-rule (that if you have ◊A,i on one node you can obtain that there is an accessible world where A holds) applies only to normal worlds, because possibility in non-normal worlds does not require it holding in an accessible world; rather, simply all possibilities are true regardless of other worlds.]
[Priest next gives the rationale for the ◊-rule stated above in section 4.3.1. I will not summarize these ideas well, so just skip to the quotation. My best guess is the following, but I will revise it later when I understand it better. In section 4.2.5, we defined semantic validity for non-normal modal logics in the following way:
Logical validity is defined in terms of truth preservation at normal worlds, thus:
∑ ⊨ A iff for all interpretations ⟨W, N, R, v⟩ and all w ∈ N: if vw(B) = 1 for all B ∈ ∑ then vw(A) = 1.
⊨ A iff φ ⊨ A, i.e., iff for all ⟨W, N, R, v⟩ and all w ∈ N, vw(A) = 1.
(p.65, section 4.2.5)
As we see, even though we are dealing with non-normality, validity is a matter of truth preservation in just normal worlds. Let us try to comment now on the current paragraph. “the tableau is a search for a normal world where the premises are true and the conclusion is false.” So our notion of proof-theoretic validity (see section 1.3.3) seems to match what we said about semantic validity, with respect to normal worlds. He also says that for this reason, “If i = 0, i must be a normal world”. I did not follow that, but maybe the idea is that when testing for validity, there is this idea that the truth preservation holds for all interpretations. So maybe the idea is that we are free to think of an inference under the conditions that it holds in some normal world, in order to ultimately determine if it is valid in some logical system. I do not know. I am not sure if this is possible for what we are doing, but what if there are no normal worlds in a particular interpretation? Is it irrelevant because we are supposed to also include other interpretations with normal worlds? Or would we instead say that validity should still hold for that situation where there are only non-normal worlds? If so, I wonder then if the point would be that the formula would have to be valid, but vacuously so, because it is the case that there are no normal worlds where the premises are true and the conclusion false (since there are no normal worlds to begin with). At any rate, his point is that when we set up our tableau, the original 0 world for one reason or another will designate a normal world. And recall from section 4.3.1 above that “The rule for ◊A,i (2.4.4) is activated only when i = 0 or i is □-inhabited.” So since we set our original formulas to world 0, which is normal, that can activate the ◊-rule. The next idea is “If i > 0, it can be assumed to be non-normal as long as the branch of the tableau is not □-inhabited.” This is also tricky, and I will guess. We said that in non-normal worlds, nothing is necessary. And “If world i occurs on a branch of a tableau, call it □-inhabited if there is some node of the form □B,i on the branch.” So for some reason, if there is no indication that there is a true necessity in a world, then we can assume it is non-normal. I am not sure why, but maybe a lack of such an indication is enough to know there are no other such cases of necessity, or maybe it is enough for us to arbitrarily stipulate that there are no other such cases, but I am not sure. So again recall the ◊-rule: “The rule for ◊A,i is activated only when i = 0 or i is □-inhabited.” So when i > 0, it fulfills neither criteria (being equal to zero or being necessity inhabited), then this rule is not activated. But I am still not sure why all this is the rationale for the rule itself. Maybe the idea is the following, but my guesses are getting wilder. It might be that we need to explain why is it that the ◊-rule is only applied when “i = 0 or i is □-inhabited.” We said that in both cases, they are normal worlds. So what would be the problem with the ◊-rule being used in non-normal worlds? The rule is the following:
| Relative Possibility Development (◊rD) |
| ◊A,i ↓ irj A,j
(j must be new: it cannot occur anywhere above on the branch) |
(24)
It is saying that if something is possible in some world, then there must be another world where it is the case. Maybe we next suppose that were something possible in a non-normal world, that would not mean there is another world where it is the case. My best guess for this is that we have rejected the normal semantic rule for evaluating possibility in non-normal worlds. In normal worlds, if something is possible, it must be the case in some accessible possible world. But we do not have that standard of evaluation for non-normal worlds, thus our tableau rule should only apply to normal worlds. Sorry for not helping here; this is the quotation:]
The rationale for the new ◊-rule is, roughly, as follows. If i = 0, i must be a normal world (since the tableau is a search for a normal world where the premises are true and the conclusion is false), and so the ◊-rule is applied in the usual way. If i > 0, it can be assumed to be non-normal as long as the branch of the tableau is not □-inhabited. Nothing, then, needs to be done. But as soon as i is □-inhabited, it can no longer be non-normal (since nothing of the form □A is true at a non-normal world), and so the standard rule for ◊ must be applied. The next two subsections give example tableaux for N.
(66)
[Example 1]
[Priest gives an example showing how the ◊-rule is applied when dealing with world 0.]
[Priest will give some examples. Here is the first one.]
⊢N □(A ⊃ B) ⊃ (□A ⊃ □B):
| ⊢N □(A ⊃ B) ⊃ (□A ⊃ □B) | ||
| 1. . 2. . 3. . 4. . 5. . 6. . 7a. 7b. . 8. . 9. . 10. . . | ¬(□(A ⊃ B) ⊃ (□A ⊃ □B)),0
↓ □(A ⊃ B),0 ↓ ¬(□A ⊃ □B),0 ↓ □A,0 ↓ ¬□B,0 ↓ ◊¬B,0 ↓ 0r1 ¬B,1 ↓ A ⊃ B,1 ↓ A,1 ↙ ↘ ¬A,1 B,1 × × | P . 1¬⊃D . 1¬⊃D . 3¬⊃D . 3¬⊃D . 5¬□D . 6◊rD 6◊rD . 2,7a□rD . 4,7a□rD . 8⊃D (9)(7b)
|
The ◊-rule is applied to ◊¬B,0, because we are dealing with world 0.
(66, enumeration and step-accounting are my additions)
[Example 2]
[Priest gives another example where we see that as world 1 is not □-inhabited, we do not apply the ◊-rule to a case in world 1 where there is the possibility operator.]
[Here is the second example.]
⊬N □(p ⊃ □(q ⊃ q)):
| ⊬N □(p ⊃ □(q ⊃ q)) | ||
| 1. . 2. . 3a. 3b. . 4. . 5. . 6. . . | ¬□(p ⊃ □(q ⊃ q)),0
↓ ◊¬(p ⊃ □(q ⊃ q)),0 ↓ 0r1 ¬(p ⊃ □(q ⊃ q)),1 ↓ p,1 ↓ ¬□(q ⊃ q),1 ↓ ◊¬(q ⊃ q),1
| P . 1¬□D . 2◊rD 2◊rD . 3b¬⊃D . 3b¬⊃D . 5¬□D (open) |
| On the (only) branch of the tableau, world 1 is not □-inhabited. Consequently, the ◊-rule is not applied to the last line, and the tableau ends open.
(67-68, enumeration and step-accounting are my additions)
[Counter-Model Formation]
[We form counter-examples while keeping in mind which worlds are non-normal. We assign worlds in accordance with the i numbers. We assign R relations in accordance with irj formulations. And nodes of the form p, i we assign vwi(p) = 1. And for nodes of the form ¬p, i, we assign vwi(p) = 0. If there are neither of these two, then vwi(p) can be given any value we want .]
[Recall from section 4.3.2 that our tableau seeks a normal world where the premises are true and the conclusion false. (Priest has us bear in mind our comments from that section, but I am not sure exactly what is most relevant here. It may not be what I note above.) Next recall the counter-model method for modal logics from section 2.4.7 :
Counter-models can be read off from an open branch of a tableau in a natural way. For each number, i, that occurs on the branch, there is a world, wi; wiRwj iff irj occurs on the branch; for every propositional parameter, p, if p, i occurs on the branch, vwi(p) = 1, if ¬p, i occurs on the branch, vwi(p) = 0 (and if neither, vwi(p) can be anything one wishes).
(p.27, section 2.4.7)
]
Bearing in mind the comments of 4.3.2, it is easy to see how a countermodel for an inference can be read off from an open tableau branch. The method is exactly the same as for K, except that world 0 is always normal, and all other worlds are non-normal, unless they are □-inhabited.
(67)
[Counter-Model Example]
[Priest gives an example of a counter-model. When depicting non-normal worlds, we place the world designator in a box but write the true formulas in that world above the box.]
[Priest now gives an example where in the non-normal world 1 from the example above, we assign p as 1. Unfortunately I have not yet figured out how this makes □(p ⊃ □(q ⊃ q)) be false. My best wild guess at the moment is that it has something to do with all necessity formulations being false in world 1 and that somehow leading to the original formula being false in the normal world 0. If I learn the reasoning here, I will revise this section.]
Thus, in the counter-model determined by the tableau of 4.3.4, W = {w0, w1}; N = {w0}; w0Rw1; and v is such that vwi(p) = 1. If we indicate that a world is non-normal by putting it in a box, the interpretation can be depicted thus:
w0xxxx→xxxx__p__
w0xxxx→xxxx|xw1x|
w0xxxx→xxxx|____|
(p.67, in the original, there is closed box around w1.)
[Rules for N Extensions]
[Tableaux for Nρ, Nρτ, etc. use the same additional rules as for Kρ, Kρτ,etc.]
[Recall from section 3.3 (especially section 3.3.2) the extra tableau rules for the extensions of K. The will be the same for N.]
Tableaux for Nρ, Nρτ, etc. are obtained by adding the extra tableau rules for ρ, ρτ, etc., as for K (3.3).
(67)
[The Soundness and Completeness of N]
[“The tableaux for N and its extensions are sound and complete”.]
Priest ends by noting that:
The tableaux for N and its extensions are sound and complete with respect to their respective semantics. The proof can be found in 4.10.
(67)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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