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
3.
Normal Modal Logics
3.5
S5
Brief summary:
(3.5.1) The normal modal logic S5 has the universal or υ (upsilon) constraint, meaning that every world relates to every other world: for all w1 and w2, w1Rw2. (3.5.2) Given that under an υ-interpretation, all worlds access all others, we need not be concerned with the parts of our semantic evaluation rules that mention the R relation. As such, we evaluate necessity and possibility operators in the following way:
vw(□A) = 1 iff for all w′ ∈ W, vw′ (A) = 1
vw(◊A) = 1 iff for some w′ ∈ W , vw′(A) = 1
(3.5.3) We make our tableaux for S5 using the tableau rules for modal logic, but eliminating the r designations; and: “Applying the ◊-rule to ◊A,i gives a new line of the form A,j (new j); and in applying the □-rule to □A,i, we add A,j for every j” (45).
| S5 Relative Necessity Development (□rD) |
| □A,i ↓ A,j
(for every j) |
| S5 Relative Possibility Development (◊rD) |
| ◊A,i ↓ A,j
(j must be new: it cannot occur anywhere above on the branch) |
(modified from p.24, section 2.4.4)
(3.5.4) Kρστ and Kυ are equivalent logical systems, because whatever is semantically valid in the one is semantically valid in the other. (3.5.5) S5 stands for both Kυ and Kρστ, on account of their logical equivalence. (3.5.6) S numbering indicates the system’s relative strength.
[S5 and the υ (Upsilon) Constraint]
[The Lack of a Need for the R Relation in υ-Interpretations]
[Tableau Rules for S5]
[The Logical Equivalence of Kρστ and Kυ]
[Kυ and Kρστ as Both S5]
[S Numbering as Strength Indicator]
Summary
[S5 and the υ (Upsilon) Constraint]
[The normal modal logic S5 has the universal or υ (upsilon) constraint, meaning that every world relates to every other world: for all w1 and w2, w1Rw2.]
[Priest will now show us the normal logic S5. Recall first from section 3.2.3 the various constraints we saw for the accessibility relation R that generate different types of modal logics:
ρ (rho), reflexivity: for all w, wRw.
σ (sigma), symmetry: for all w1, w2, if w1Rw2, then w2Rw1.
τ (tau), transitivity: for all w1, w2, w3, if w1Rw2 and w2Rw3, then w1Rw3.
η (eta), extendability: for all w1, there is a w2 such that w1Rw2.
(p.36, section 3.2.3)
It seems, but I am not sure, that S5 is defined primarily by a new constraint, namely upsilon, υ, meaning “universal.” Here, every world relates to every other world.]
The system S5 is special. To see how, let an υ-interpretation – ‘υ’ (upsilon) for universal – be an interpretation in which R satisfies the following condition: for all w1 and w2, w1Rw2 – everything relates to everything.
(45)
[The Lack of a Need for the R Relation in υ-Interpretations]
[Given that under an υ-interpretation, all worlds access all others, we need not be concerned with the parts of our semantic evaluation rules that mention the R relation. As such, we evaluate necessity and possibility operators in the following way: vw(□A) = 1 iff for all w′ ∈ W, vw′ (A) = 1; vw(◊A) = 1 iff for some w′ ∈ W , vw′(A) = 1.]
[Recall from section 2.3.5 the semantic evaluation for necessity and possibility:
For any world w ∈ W:
vw(◊A) = 1 if, for some w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
vw(□A) = 1 if, for all w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
(Priest p.22, section 2.3.5)
Priest’s next point is that under our new “υ-interpretation, R drops out of the picture altogether, in effect” (45). I am not certain, but the idea might be that under this constraint, we can be sure that all worlds access all others, and thus it makes no difference really whether or not we keep the stipulation above, “... such that wRw′”. It would seem rather that we would formulate the two in the following way, but please trust the quotation below. (Note we also use “iff” instead of “if...otherwise”.)
vw(□A) = 1 iff for all w′ ∈ W, vw′ (A) = 1
vw(◊A) = 1 iff for some w′ ∈ W , vw′(A) = 1
]
In an υ-interpretation, R drops out of the picture altogether, in effect. We can just as well define an υ-interpretation to be a pair ⟨W, v⟩, where the truth conditions for □ are simply: vw(□A) = 1 iff for all w′ ∈ W, vw′ (A) = 1; and similarly for ◊.
(45)
[Tableau Rules for S5]
[We make our tableaux for S5 using the tableau rules for modal logic, but eliminating the r designations; and: “Applying the ◊-rule to ◊A,i gives a new line of the form A,j (new j); and in applying the □-rule to □A,i, we add A,j for every j” (45).]
[Recall from section 2.4 the tableau rules for modal logic. We use the same rules for the truth functional operators. Our rules for the modal operators will be slightly different, because they normally specify world relativities, which we no longer need to do. So let us modify those rules from section 2.4.4.
| S5 Relative Necessity Development (□rD) |
| □A,i ↓ A,j
(for every j) |
| S5 Relative Possibility Development (◊rD) |
| ◊A,i ↓ A,j
(j must be new: it cannot occur anywhere above on the branch) |
(modified from p.24, section 2.4.4)
We of course do not need to use the tableau rules for Kρ, Kσ, and Kτ (see section 3.3.2), because we can assume that all these constraints are already in operation.]
Tableaux for Kυ can also be formulated very simply: r is never mentioned. Applying the ◊-rule to ◊A,i gives a new line of the form A,j (new j); and in applying the □-rule to □A,i, we add A,j for every j. For example,
⊢Kυ ◊A ⊃ □◊A:
| ⊢Kυ ◊A ⊃ □◊A | ||
| 1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . 9. . 10. . | ¬(◊A ⊃ □◊A),0
↓ ◊A,0 ↓ ¬□◊A,0 ↓ ◊¬◊A,0 ↓ A,1 ↓ ¬◊A,2 ↓ □¬A,2 ↓ ¬A,0 ↓ ¬A,1 ↓ ¬A,2 × | P . 1¬⊃D . 1¬⊃D . 3◊¬□D . 2◊rD . 4◊rD . 6¬◊D . 7□rD . 7□rD . 7□rD (9×5)
|
(45, enumerations and step accounting are my own and are not to be trusted)
[The Logical Equivalence of Kρστ and Kυ]
[Kρστ and Kυ are equivalent logical systems, because whatever is semantically valid in the one is semantically valid in the other.]
[Priest now notes that Kρστ and Kυ are equivalent logical systems, because whatever is semantically valid in the one is semantically valid in the other.]
Now, Kρστ and Kυ are, in fact, equivalent, in the sense that Σ ⊨Kρστ A iff Σ ⊨Kυ A. Half of this fact is obvious. It is easy to check that if a relationship satisfies the condition υ it satisfies the conditions ρ, σ and τ. Hence, if truth is preserved at all worlds of all ρστ-interpretations, it is preserved at all worlds of all υ-interpretations. Hence, if Σ ⊨Kρστ A, then Σ ⊨Kυ A. The converse is not so obvious. (A proof can be found in 3.7.5.)
(45)
[Kυ and Kρστ as Both S5]
[S5 stands for both Kυ and Kρστ, on account of their logical equivalence.]
[Priest also notes that often times the name S5 is used for both Kυ and Kρστ, on account of their logical equivalence.]
Because of the equivalence between Kυ and Kρστ, the name S5 tends to be used, indifferently, for either of these systems.
(45)
[S Numbering as Strength Indicator]
[S numbering indicates the system’s relative strength.]
[It seems that the number in the S system nomenclature is often based on their relative strengths.]
There are many other normal modal logics. Some of these glorify in names such as S4.2. The number indicates that the system is between S4 and S5 in strength, but otherwise is not to be taken too seriously.
(46)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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