by Corry Shores

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[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* Kυ*, a relational | interpretation for such a language is a pair ⟨*W*, ρ⟩, where* W *is a set of worlds, and for every* w *∈ *W*, ρ_{w} 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∧Bρ1 iff_{w}Aρ1 and_{w}Bρ1_{w}

A∧Bρ_{w}0 iffAρ_{w}0 orBρ_{w}0(164)

A∨Bρ1 iff_{w}Aρ1 or_{w}Bρ1_{w}

A∨Bρ0 iff_{w}Aρ0 and_{w}Bρ0_{w}

¬

Aρ1 iff_{w}Aρ0_{w}¬

Aρ0 iff_{w}Aρ1_{w}(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→Bρ1 iff for all_{w}w′ ∈Wsuch thatAρ_{w}_{′}1,Bρ_{w}_{′}1

A→Bρ0 iff for some_{w}w′ ∈W,Aρ_{w}_{′}1 andBρ_{w}_{′}0

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

Σ ⊨

Aiff for every interpretation, ⟨W, ρ⟩, and allw∈W: ifBρ1 for all_{w}B∈ Σ,Aρ1_{w}

(164)

(9.2.6) “A natural name for this logic would be *Kυ*_{4}. We will call it, more simply, *K*_{4}” (164).

[Introducing the Conditional into *FDE* with Possible Worlds Semantics]

[Notational Conventions: →, ρ, etc.]

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

[The Evaluation Rule for the Conditional →]

[Semantic Consequence as Truth Preservation at All Worlds]

[Naming this Logic *Kυ*_{4} or *K*_{4}]

Summary

[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

interpretationof the language is a function,v, which assigns to each propositional parameter either 1 (true), or 0 (false). Thus, we write things such asv(p) = 1 andv(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

ρbetween propositional parameters and the values 1 and 0. We writepρ1 forprelates to 1, andpρ0 forprelates to 0. This allows a formula to have one of the following four value-assignment situations: just true (1, e.g.:pρ1), just false (0, e.g.:pρ0), both true and false (1 and 0, e.g.:pρ1,pρo), 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

FDEhas no conditional operator. The material conditional,A⊃B, does not even satisfymodus 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 ofFDE.(163)

[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 *→* B*ρ_{w1}1 and *A *→* B*ρ_{w2}0.]

To effect this, let us add a new binary connective, →, to the language of

FDEto represent the conditional. By analogy withKυ, a relational | interpretation for such a language is a pair ⟨W, ρ⟩, whereWis a set of worlds, and for everyw∈W, ρ_{w}is a relation between propositional parameters and the values 1 and 0.(163-164)

[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∧Bρ1 iff_{w}Aρ1 and_{w}Bρ1_{w}

A∧Bρ_{w}0 iffAρ_{w}0 orBρ_{w}0(164)

A∨Bρ1 iff_{w}Aρ1 or_{w}Bρ1_{w}

A∨Bρ0 iff_{w}Aρ0 and_{w}Bρ0_{w}

¬

Aρ1 iff_{w}Aρ0_{w}¬

Aρ0 iff_{w}Aρ1_{w}(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∧Bρ1 iff_{w}Aρ1 and_{w}Bρ1_{w}

A∧Bρ_{w}0 iffAρ_{w}0 orBρ_{w}0(164)

[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:

v(_{w}A⥽B) = 1 if for allw′ such thatv_{w}_{′ }(A) = 1,v_{w}_{′ }(B) = 1;

v(A⥽_{w}B) = 0 if for somew′,v_{w}_{′}(A) = 1 andv_{w}_{′ }(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

Kυcome to this:

v(_{w}A⥽B) = 1 if for allw′ such thatv_{w}_{′ }(A) = 1,v_{w}_{′ }(B) = 1; andv(A⥽_{w}B) = 0 if for somew′,v_{w}_{′}(A) = 1 andv_{w}_{′ }(B) = 0. Making the obvious generalisation:

A→Bρ1 iff for all_{w}w′ ∈Wsuch thatAρ_{w}_{′}1,Bρ_{w}_{′}1

A→Bρ0 iff for some_{w}w′ ∈W,Aρ_{w}_{′}1 andBρ_{w}_{′}0(164)

[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 *B*ρ* _{w}*1 for all

*B*∈ Σ,

*A*ρ

*1” (164).]*

_{w}

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

Σ ⊨

Aiff for every interpretation,ρ, ifBρ1 for allB∈ Σ thenAρ1(p. 144, section 8.2.8)

and for modal logics (section 2.3.11):

Σ ⊨

Aiff for all interpretations ⟨W,R,v⟩ and allw∈W: ifv(_{w}B) = 1 for allB∈ Σ, thenv(_{w}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 ⊨

Aiff for every interpretation, ⟨W, ρ⟩, and allw∈W: ifBρ1 for all_{w}B∈ Σ,Aρ1_{w}(164)

[Naming this Logic *Kυ*_{4} or *K*_{4}]

[“A natural name for this logic would be *Kυ*_{4}. We will call it, more simply, *K*_{4}” (164).]

[Priest will now say that “A natural name for this logic would be *Kυ*_{4}. We will call it, more simply, *K*_{4}.” 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

w_{1},w_{2}, ifw_{1}Rw_{2}, thenw_{2}Rw_{1}.τ (tau), transitivity: for all

w_{1},w_{2},w_{3}, ifw_{1}Rw_{2}andw_{2}Rw_{3}, thenw_{1}Rw_{3}.(p.36, section 3.2.3)

to get more versions of *K*, like *K*ρ or *K*ρσ. Another restriction is υ: “let an υ-interpretation – ‘υ’ (upsilon) for universal – be an interpretation in which *R *satisfies the following condition: for all *w*_{1} and *w*_{2}, *w*_{1}*R**w*_{2} – everything relates to everything” (p.45, section 3.5.1). In section 3.5.4, Priest explains that *K*ρστ and *K*υ are equivalent logical systems. So we have already a sense for *K*υ. 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 *K*_{3}. 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 *K*υ_{4} and more simply, *K*_{4}.]

A natural name for this logic would be

Kυ_{4}. We will call it, more simply,K_{4}.(164)

From:

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

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