20 Jun 2018

Priest (9.3) Introduction to Non-Classical Logic, ‘Tableaux for K4’, 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

 

9.3

Tableaux for K4

 

 

 

 

Brief summary:

(9.3.1) We will formulate the tableau procedures for K4 by modifying those for FDE. (9.3.2) Nodes in our possible worlds FDE tableaux take the “form A,+i or A,−i, where i is a natural number.” To test for validity, we compose our initial list by formulating our premise nodes as “B,+0” and our conclusion as “A,−0”.  “A branch closes if it contains a pair of the form A,+i and A,−i” (164). (9.3.3) The tableau rules for the extensional connectives (∧, ∨ and ¬) in K4 are the same as for FDE except “i is carried through each rule.”[Below I include the conditional rules from the next section. Note that the rules for double negation and disjunction are not in the text and are probably mistaken.]

 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

 

 Conditional

Development, False (→D,−)

  A B,-i

A,+j

B,-j

.

j is a new number

 

 Negated Conditional

Development, True (¬D, +)

¬(A B),+i

A,+j

¬B,+j

.

j is a new number

 

 Negated Conditional

Development, False(¬D, -)

¬(A B),-i

↙     ↘

A,-j     ¬B,-j

.

j is every number that occurs on the branch

(165, titles for the rules are my own additions)

(9.3.4) To the above rules we add those for the conditional [see the rules just above, where they were moved to.] (9.3.5) Priest then gives a tableau example showing a valid inference. (9.3.6) Priest next gives an example of an inference that is not valid. (9.3.7) We make countermodels 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” (166). [Note, I was not able to get that to work for the example. Given how the example is formulated in the working way that it is, I wonder if we should instead follow these following rules, but please trust the text over this: For propositional parameters, p, if p,+i occurs on the branch, set pρwi 1; if ¬p,+i occurs on the branch, set ¬pρwi1; if p,i occurs on the branch, set pρwi 0; and if ¬p,i occurs on the branch, set ¬pρwi0. In the counter-model representation, place the formulas under their proper world designator, and use + for formulas that are 1 and − for formulas that are 0 (not in the text and probably mistaken).] (9.3.8) (It can be proven that the possible worlds FDE tableaux are sound and complete with respect to the semantics.)

 

 

 

 

 

Contents

 

9.3.1

[Sources of the Tableaux for K4]

 

9.3.2

[Tableau Formulation]

 

9.3.3

[Tableau Rules for the Extensional Connectives (∧, ∨ and ¬)]

 

9.3.4

[Tableau Rules for the Conditional (→)]

 

9.3.5

[Example 1: Valid]

 

9.3.6

[Example 2: Invalid]

 

9.3.7

[Countermodels]

 

9.3.8

[The Soundness and Completeness of the Tableaux]

 

 

 

 

 

Summary

 

9.3.1

[Sources of the Tableaux for K4]

 

[We will formulate the tableau procedures for K4 by modifying those for FDE.]

 

[In the previous section (9.2), Priest gave the semantics for K4, which is a possible worlds semantics combined with First Degree Entailment (FDE). Now our task is the tableau rules. Recall from section 3.5.5 that normal modal logic S5 is another name for Kυ, and in section 3.5.3 Priest gives the tableau rules for S5 by making slight modifications to those for modal logic. Priest says now that in the same was as this, we can formulate the tableau system for K4 by modifing the tableau system for FDE that we used in section 8.3.]

A tableau system for K4 can be obtained by modifying the system for FDE of 8.3, in the same way that the tableau system for classical propositional logic is modified in order to obtain one for (3.5.3).

(164)

[contents]

 

 

 

 

9.3.2

[Tableau Formulation]

 

[Nodes in our possible worlds FDE tableaux take the “form A,+i or A,−i, where i is a natural number.” To test for validity, we compose our initial list by formulating our premise nodes as “B,+0” and our conclusion as “A,−0”.  “A branch closes if it contains a pair of the form A,+i and A,−i” (164).]

 

[Recall from section 8.3.2 that FDE tableaux have the symbols + and – coming after the formulas to indicate whether the formula is true or not. And recall from section 2.4.1 that after the formulas we write numbers (or when as variables, as lowercase letters starting with i) indicating the world that the formula holds in. We will now combine these conventions such that we indicate the world and whether or not it is true in that world, by combining the + or the – (truth value) symbols with the lower-case letter (world) symbols. Thus if we write A,+i, that intuitively means that formula A is true in world i. The tableaux will test for validity, of course. So recall from section 2.4.2 that for modal tableaux, “the initial list for the tableau comprises A,0, for every premise, A (if there are any), and ¬B,0, where B is the conclusion” (p. 24). Intuitively, we are setting it up so that all the premises are set for true in world 0 and the negation of the conclusion is set for true in world 0 also. And in FDE tableaux (recall from section 8.3.3), we set the premises to true and the conclusion false, using the plus/minus symbols:

To test the claim that A1, ... , AnB , we start with an initial list of the form:

A1,+

.

.

.

An,+

B,−

(p.144, section 8.3.3)

In our possible worlds FDE, we will combine these conventions. We set premises to +0 (true in world 0) and the conclusion to −0 (not true in world 0). A branch in a tableaux normally closes when there is something like a contradiction along it. So in our case, a branch closes when along it is a pair of nodes of the form A,+i and A,−i (in other words, if we find a formula that is both true and not true in the same world).]

A node now has the form A,+i or A,−i, where i is a natural number. The initial list comprises a node of the form B,+0 for every premise, B, and A,−0, where A is the conclusion. A branch closes if it contains a pair of the form A,+i and A,−i.

(164)

[contents]

 

 

 

 

9.3.3

[Tableau Rules for the Extensional Connectives (∧, ∨ and ¬)]

 

[The tableau rules for the extensional connectives (∧, ∨ and ¬) in K4 are the same as for FDE except “i is carried through each rule.”]

 

[Recall from section 8.3.4 the FDE tableau rules for double negation, conjunction, and disjunction. Priest says that they are the same as for possible worlds FDE (K4), except “i is carried through each rule.” Priest provides the rules for conjunction, but I will guess them for double negation and  disjunction too. Now, more specifically, Priest says that the rules are the same for the “extensional connectives,” which in section 9.2.3 he says are ∧, ∨ and ¬. So I am not entirely sure if there is a rule here for double negation; but I am guessing it is ok, because the FDE semantics rule for negation in section 8.2.6 toggles the 1/0 value.

Conjunction

Development, True (D,+)

A ∧ B,+i

A,+i

B,+i

Conjunction

Development, False (D,−)

A ∧ B,−i

↙   ↘

A,−i      B,−i

(165)

 Negated Conjunction

Development, True (¬D,+)

¬(A ∧ B),+i

¬A ¬B,+i

 Negated Conjunction

Development, False (¬D,−)

¬(A ∧ B),−i

¬A ¬B,−i

 Double Negation

Development, True (¬¬D,+)

¬¬A,+i

A,+i

 Double Negation

Development, False (¬¬D,−)

¬¬A,−i

A,−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

]

The rules for the extensional connectives are exactly the same as those of 8.3.4 for FDE, except that i is carried through each rule. | Thus, for example, the rules for ∧ are:

Conjunction

Development, True (D,+)

A ∧ B,+i

A,+i

B,+i

Conjunction

Development, False (D,−)

A ∧ B,−i

↙     ↘

A,−i     B,−i

(164-165, titles for the rules are my own additions)

[contents]

 

 

 

 

9.3.4

[Tableau Rules for the Conditional (→)]

 

[To the above rules we add those for the conditional.]

 

[Priest then gives the rules for the conditional.]

The rules for the conditional are as follows:

 Conditional

Development, True (→D,+)

A → B,+i

↙   ↘

A,-j      B,+j

.

j is every number that occurs on the branch

 Conditional

Development, False (→D,−)

  A B,-i

A,+j

B,-j

.

j is a new number

 Negated Conditional

Development, True (¬D, +)

¬(A B),+i

A,+j

¬B,+j

.

j is a new number

 Negated Conditional

Development, False(¬D, -)

¬(A B),-i

↙     ↘

A,-j     ¬B,-j

.

j is every number that occurs on the branch

(165, titles for the rules are my own additions)

In the rules that split the branch, j is every number that occurs on the branch. In the other two rules, j is a new number.

(165)

[contents]

 

 

 

 

 

9.3.5

[Example 1: Valid]

 

[Priest then gives a tableau example showing a valid inference.]

 

[Priest gives an example that involves transitivity, which is valid.]

Example: A B, B C A C:

A → B, B → C ⊢ A → C

1.

.

2.

.

3.

.

4.

.

5.

.

6.

.

7.

A → B,+0

B → C,+0

A → C,–0

A,+1

C,–1

↙     ↘

A,–1         B,+1

    ×          ↓   

                ¬B,–1     C,+1

                 ×         ×

 

P

.

P

.

P

.

3→–

.

3→–

.

1→+

(6×4)

2→+

(7×6)

(7×5)

Valid

(enumeration and step accounting are not my own and are probably mistaken)

The fourth and fifth lines are obtained by applying the rule for untrue → to the third line. The two splits are then obtained by applying the rule for true → to the first and second lines respectively.

(165)

[contents]

 

 

 

 

 

9.3.6

[Example 2: Invalid]

 

[Priest next gives an example of an inference that is not valid.]

 

[The next tableau example is invalid. For this one, we need to keep in mind that for the conditional, “In the rules that split the branch, j is every number that occurs on the branch. In the other two rules, j is a new number” (165, section 9.3.4 above). See lines 5 and 6 where where we must repeat the rule for both worlds.]

Example: p → q ¬q → ¬p:

p → q ⊬ ¬q → ¬p

1.

.

2.

.

3.

.

4.

.

5.

.

6.

p → q,+0

¬q → ¬p,–0

¬q,+1

¬p,–1

↙     ↘

p,–0        q,+0

↙   ↓      ↓  

  p,–1  q,+1    p,–1   q,+1

P

.

P

.

2→–

.

2→–

.

1→+

.

1→+

(open)

Invalid

(166, enumeration and step accounting are not my own and are probably mistaken)

[contents]

 

 

 

 

 

9.3.7

[Countermodels]

 

[We make countermodels 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” (166). (Note, I was not able to get that to work for the example. Given how the example is formulated in the working way that it is, I wonder if we should instead follow these following rules, but please trust the text over this: For propositional parameters, p, if p,+i occurs on the branch, set pρwi 1; if ¬p,+i occurs on the branch, set ¬pρwi1; if p,i occurs on the branch, set pρwi 0; and if ¬p,i occurs on the branch, set ¬pρwi0. In the counter-model representation, place the formulas under their proper world designator, and use + for formulas that are 1 and − for formulas that are 0 (not in the text and probably mistaken).)]

 

[We make counter-models from open branches in the following way. For every i number, we designate a world with that number. We then look for propositional parameters. Ones that are non-negated and true in a world we set them in the countermodel to true in that world. Likewise, for any negated propositional parameter that is true in a world, we set that unnegated parameter to zero in that world. So let us consider the propositional parameters in the left-most branch: p,–1; p,–0; ¬p,–1; ¬q,+1. (Note, the rules as they are given in the text did not work for me in this example. Priest writes, “for propositional parameters, p, if p,+i occurs on the branch, set pρwi 1; if ¬p,+i occurs on the branch, set pρwi0.” Suppose we just follow the rules as they are here.  Then our counter-model would be simply be that q is false in world 1, because ¬q,+1 is the only one on the branch with a + sign, and the rules as they are given only stipulate for + sign formulations. Rather, to get to the counter-model example as Priest has it, which, as he shows, works properly, I would think we need further stipuations. Probably they are implied already in Priest’s stipulations, and I just cannot infer them correctly. But in my simplistic way of thinking, I would need the following set of stipulations to obtain the working counter-model Priest produces: For propositional parameters, p, if p,+i occurs on the branch, set pρwi 1; if ¬p,+i occurs on the branch, set ¬pρwi1; if p,i occurs on the branch, set pρwi 0; and if ¬p,i occurs on the branch, set ¬pρwi0. In the counter-model representation, place the formulas under their proper world designator, and use + for formulas that are 1 and − for formulas that are 0.)]

Counter-models are read off from open branches of tableaux in the natural 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. Thus, the counter-model defined by the leftmost branch of the tableau of 9.3.6 may be depicted thus:

xxxw0xxxxxxxw1

xxxpxxxxxxp

xxxxxxxxxxx−¬p

xxxxxxxxxxxq

(+A indicates that A is true; −A indicates that it is untrue.) At every world, p is untrue. Hence, p q is true at w0. But ¬q is true at w1, and ¬p is not true there. Hence, ¬q → ¬p is not true at w0.

(166)

[contents]

 

 

 

 

 

9.3.8

[The Soundness and Completeness of the Tableaux]

 

[It can be proven that the possible worlds FDE tableaux are sound and complete with respect to the semantics.]

 

[Priest ends by noting that:]

The tableaux are sound and complete with respect to the semantics. This is proved in 9.8.1–9.8.7.

[contents]

 

 

 

 

 

 

 

 

 

 

From:

 

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

 

 

.

 

18 Jun 2018

Priest (9.2) An Introduction to Non-Classical Logic, ‘Introduction [to ch.9, “Logics with Gaps, Gluts and Worlds”],’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[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

 

9.

Logics with Gaps, Gluts and Worlds

 

9.2

Adding →

 

 

 

 

Brief summary:

(9.2.1) In order to introduce a well-functioning conditional into FDE, we could build a possible world semantics upon it. “To effect this, let us add a new binary connective, →, to the language of FDE to represent the conditional. By analogy with, a relational | interpretation for such a language is a pair ⟨W, ρ⟩, where W is a set of worlds, and for every w W, ρw is a relation between propositional parameters and the values 1 and 0” (163-164). (9.2.2) We will use the symbol → for the conditional operator in our possible worlds FDE semantics. We still use the ρ relation to assign truth-values. But we also will specify the worlds in which that value holds. (9.2.3) The evaluation rules for ∧, ∨ and ¬ and just like those for FDE, only now with worlds specified.

A Bρw1 iff Aρw1 and Bρw1

A Bρw0 iff Aρw0 or Bρw0

(164)

 

Aw1 iff w1 or w1

Aw0 iff w0 and w0

 

¬w1 iff w0

¬w0 iff w1

(not in the text)

(9.2.4) In our possible worlds FDE, a conditional is true if in all worlds, whenever the antecedent is true, so is the consequent. And it is false if there is at least one world where the antecedent is true and the consequent false.

A Bρw1 iff for all w′ ∈ W such that Aρw1, Bρw1

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

(9.2.5) In our possible worlds FDE, “semantic consequence is defined in terms of truth preservation at all worlds of all interpretations:

Σ ⊨ A iff for every interpretation, ⟨W, ρ⟩, and all w W: if Bρw1 for all B ∈ Σ, Aρw1

(164)

(9.2.6) “A natural name for this logic would be 4. We will call it, more simply, K4” (164).

 

 

 

 

 

 

Contents

 

9.2.1

[Introducing the Conditional into FDE with Possible Worlds Semantics]

 

9.2.2

[Notational Conventions: →, ρ, etc.]

 

9.2.3

[The Evaluation Rules for ∧, ∨ and ¬.]

 

9.2.4

[The Evaluation Rule for the Conditional →]

 

9.2.5

[Semantic Consequence as Truth Preservation at All Worlds]

 

9.2.6

[Naming this Logic 4 or K4]

 

 

 

 

 

Summary

 

9.2.1

[Introducing the Conditional into FDE with Possible Worlds Semantics]

 

[In order to introduce a well-functioning conditional into FDE, we could build a possible world semantics upon it.]

 

[Let us review some things about First Degree Entailment. First recall from section 1.3.1 the notion of interpretation in classical logic:

An interpretation of the language is a function, v, which assigns to each propositional parameter either 1 (true), or 0 (false). Thus, we write things such as v(p) = 1 and v(q) = 0.

(5)

In section 8.1 we learned how in FDE our interpretations – rather than being functions that assign values  as in the other cases –are instead formulated as relations between formulas and standard truth values. In section 8.2, we noted the following in our brief summary:

In our semantics for First Degree Entailment (FDE), our only connectives are ∧, ∨ and ¬ (with A ⊃ B being defined as ¬A ∨ B.) FDE uses relations rather than functions to evaluate truth. So a truth-valuing interpretation in FDE is a relation ρ between propositional parameters and the values 1 and 0. We write 1 for p relates to 1, and 0 for p relates to 0. This allows a formula to have one of the following four value-assignment situations: just true (1, e.g.: 1), just false (0, e.g.: 0), both true and false (1 and 0, e.g.: 1, o), and neither true nor false (no such valuing formulations). In FDE, being false (that is, relating to 0) does not automatically mean being untrue (that is, not relating to 1), because it can still be related to 1 along with 0. For formulas built up with connectives, we use the same criteria as in classical logic to evaluate them, only here we can have formulas taking both values.

(from our brief summary of section 8.2)

As we can see, there is no conditional operator in FDE. Now recall from section 8.6.5 that modus ponens fails for the conditional operator in FDE (this has to do with the fact that disjunctive syllogism fails in FDE.) Priest next notes that “In any case, as we have seen, using possible-world semantics provides a much more promising approach to the logic of conditional operators.” I am not certain, but perhaps he is referring to the strict conditional. (That is a guess, because we have found problems with the strict conditional, like explosion. See section 4.8.) So, to better incorporate the conditional into FDE, we might combine FDE with possible-world semantics.]

9.2.1 FDE has no conditional operator. The material conditional, AB, does not even satisfy modus ponens, as we saw in 8.6.5. In any case, as we have seen, using possible-world semantics provides a much more promising approach to the logic of conditional operators. Thus, an obvious thing to do is to build a possible-world semantics on top of the relational semantics of FDE.

(163)

[contents]

 

 

 

 

9.2.2

[Notational Conventions: →, ρ, etc.]

 

[We will use the symbol → for the conditional operator in our possible worlds FDE semantics. We still use the ρ relation to assign truth-values. But we also will specify the worlds in which that value holds.]

 

[We will now use → for the conditional in our possible worlds FDE. It is a binary connective (connecting the antecedent to the consequent). Since we are dealing with possible worlds, that means a conditional can have a different truth value depending on which world it is said to hold (or not hold) in. So suppose we have an A B formula, and it is true in world 1 but false in world 2. Recall that ρ is our truth-assigning relation. So we would have A Bρw11 and A Bρw20.]

To effect this, let us add a new binary connective, →, to the language of FDE to represent the conditional. By analogy with, a relational | interpretation for such a language is a pair ⟨W, ρ⟩, where W is a set of worlds, and for every w W, ρw is a relation between propositional parameters and the values 1 and 0.

(163-164)

[contents]

 

 

 

 

9.2.3

[The Evaluation Rules for ∧, ∨ and ¬.]

 

[The evaluation rules for ∧, ∨ and ¬ and just like those for FDE, only now with worlds specified.]

 

[Recall from section 8.2.6 the evaluation rules for the connectives ∧, ∨ and ¬. Now we will relativize them for worlds. Priest gives the one for conjunction, and I will guess the formulations for disjunction and conjunction.

A Bρw1 iff Aρw1 and Bρw1

A Bρw0 iff Aρw0 or Bρw0

(164)

 

Aw1 iff w1 or w1

Aw0 iff w0 and w0

 

¬w1 iff w0

¬w0 iff w1

(not in the text)

]

The truth and falsity conditions for the extensional connectives (∧, ∨ and ¬) are exactly those of 8.2.6, except that they are relativised to each world, w. Thus, for example, the truth and falsity conditions for conjunction are:

A Bρw1 iff Aρw1 and Bρw1

A Bρw0 iff Aρw0 or Bρw0

(164)

[contents]

 

 

 

 

9.2.4

[The Evaluation Rule for the Conditional →]

 

[In our possible worlds FDE, a conditional is true if in all worlds, whenever the antecedent is true, so is the consequent. And it is false if there is at least one world where the antecedent is true and the consequent false.]

 

[Recall from section 4.5.4 and 5.2.8 that I tried to formulate the rule for evaluating the strict conditional. We now get the correct formulation for the strict conditional:

vw(AB) = 1 if for all w′ such that vw (A) = 1, vw (B) = 1;

vw(A⥽B) = 0 if for some w′, vw (A) = 1 and vw(B) = 0.

(164)

The formulation for → will be similar, only now using the ρ relation. A conditional is true if in all worlds, whenever the antecedent is true, so is the consequent. And it is false if there is at least one world where the antecedent is true ant the consequent false.]

For the truth and falsity conditions for →, recall that the truth and falsity conditions for ⥽ in come to this:

vw(AB) = 1 if for all w′ such that vw (A) = 1, vw (B) = 1; and vw(A⥽B) = 0 if for some w′, vw (A) = 1 and vw(B) = 0. Making the obvious generalisation:

A Bρw1 iff for all w′ ∈ W such that Aρw1, Bρw1

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

(164)

[contents]

 

 

 

 

9.2.5

[Semantic Consequence as Truth Preservation at All Worlds]

 

[In our possible worlds FDE, “semantic consequence is defined in terms of truth preservation at all worlds of all interpretations: Σ ⊨ A iff for every interpretation, ⟨W, ρ⟩, and all w W: if Bρw1 for all B ∈ Σ, Aρw1” (164).]

 

[Recall from section 8.2.8 that semantic consequence in FDE is defined as:

Σ ⊨ A iff for every interpretation, ρ, if 1 for all B ∈ Σ then 1

(p. 144, section 8.2.8)

and for modal logics (section 2.3.11):

Σ ⊨ A iff for all interpretations ⟨W, R, v⟩ and all w W: if vw(B) = 1 for all B ∈ Σ, then vw(A) = 1.

(p.23, section 2.3.11)

We combine them for our definition of semantic validity in possible worlds FDE.]

Semantic consequence is defined in terms of truth preservation at all worlds of all interpretations:

Σ ⊨ A iff for every interpretation, ⟨W, ρ⟩, and all w W: if Bρw1 for all B ∈ Σ, Aρw1

(164)

[contents]

 

 

 

 

9.2.6

[Naming this Logic 4 or K4]

 

[“A natural name for this logic would be 4. We will call it, more simply, K4” (164).]

 

[Priest will now say that “A natural name for this logic would be 4. We will call it, more simply, K4.” I do not understand the naming conventions, so should not comment. K is the name for normal modal logics (section 2.1.2). We can place constraints on the accessibility relation R like:

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.

(p.36, section 3.2.3)

to get more versions of K, like Kρ or Kρσ.  Another restriction is υ: “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” (p.45, section  3.5.1). In section  3.5.4, Priest explains that Kρστ and Kυ are equivalent logical systems. So we have already a sense for Kυ. Perhaps the idea is that our possible worlds FDE will be (so far) a normal modal logic with the universal constraint, meaning that every world has an accessibility relation to every other world (and thus also they have reflexivity, symmetry, and transitivity), but I am guessing. Yet, what about the subscript “3”? I will guess further. Recall from section 7.3 strong Kleene three-valued logic, written as K3. Just as a guess, I wonder if the subscript there means three-valued, and so here Priest calls our possible worlds 4-value situation semantics Kυ4 and more simply, K4.]

A natural name for this logic would be Kυ4. We will call it, more simply, K4.

(164)

[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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