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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, 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)
A ∨ Bρw1 iff Aρw1 or Bρw1
A ∨ Bρw0 iff Aρw0 and Bρw0
¬Aρw1 iff Aρw0
¬Aρw0 iff Aρw1
(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 Aρw′1, Bρw′1
A → Bρw0 iff for some w′ ∈ W, Aρw′1 and Bρw′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 ⊨ 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 Kυ4. We will call it, more simply, K4” (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 Kυ4 or K4]
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 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 pρ1 for p relates to 1, and pρ0 for p relates 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 FDE has no conditional operator. The material conditional, A ⊃ B, 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)
[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 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, ρw 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 ∧ Bρw1 iff Aρw1 and Bρw1
A ∧ Bρw0 iff Aρw0 or Bρw0
(164)
A ∨ Bρw1 iff Aρw1 or Bρw1
A ∨ Bρw0 iff Aρw0 and Bρw0
¬Aρw1 iff Aρw0
¬Aρw0 iff Aρw1
(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)
[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(A⥽B) = 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 Kυ come to this:
vw(A⥽B) = 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 Aρw′1, Bρw′1
A → Bρw0 iff for some w′ ∈ W, Aρw′1 and Bρw′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 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 Bρ1 for all B ∈ Σ then Aρ1
(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)
[Naming this Logic Kυ4 or K4]
[“A natural name for this logic would be Kυ4. We will call it, more simply, K4” (164).]
[Priest will now say that “A natural name for this logic would be Kυ4. 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)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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