Priest (9.7a) Introduction to Non-Classical Logic, ‘Logics of Constructible Negation,’ 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 (in this entry the boldface on Aristotle and Boethius are in the original). 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

 

9.7a

Logics of Constructible Negation

 

 

 

 

Brief summary:

(9.7a.1) We will now examine logics of constructible negation, which add an account of negation to the negationless part of intuitionistic logic (or positive intuitionistic logic). The important feature of these logics is that “unlike intuitionist logic, they treat truth and falsity even-handedly” (175). (9.7a.2) We will “Consider interpretations of the form ⟨W, R, ρ⟩, where W is the usual set of worlds, R is a reflexive and transitive binary relation on W, and for every w ∈ W, and propositional parameter, p, ρ_w relates p to 1, 0, both or neither, subject to the heredity constraints: if pρ_w1 and wRw′, then pρ_w′1 ; if pρ_w0 and wRw′, then pρ_w′0 ” (175). (9.7a.3) Priest next provides the truth/falsity conditions for the connectives in our logic of constructible negation.

 

A ∧ Bρ_w1 iff Aρ_w1 and Bρ_w1

A ∧ Bρ_w0 iff Aρ_w0 or Bρ_w0

 

A ∨ Bρ_w1 iff Aρ_w1 or Bρ_w1

A ∨ Bρ_w0 iff Aρ_w0 and Bρ_w0

 

¬Aρ_w1 iff Aρ_w0

¬Aρ_w0 iff Aρ_w1

 

A ⊐ Bρ_w1 iff for all w′ such that wRw′, either it is not the case that Aρ_w′1 or Bρ_w′1

A ⊐ Bρ_w0 iff Aρ_w1 and Bρ_w0

(175)

Validity is truth preservation in all worlds of all interpretations. (9.7a.4) Priest next gives the tableau rules for I₄.

 

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

(p.165, section 9.3.3; titles for the rules are my own additions)

 

Conditional Development, True (⊐D, +)

A ⊐ B,+i irj ↙     ↘ A,-j       B,+j . j is any number on the branch

 

Conditional Development, False (⊐D,-)

A ⊐ B,-i ↓ irj A,+j B,-j . j is new to the branch

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

 

Negated Conditional Development, True (¬⊐D,+)

¬(A ⊐ B),+i ↓ A,+i ¬B,+i

 

Negated Conditional Development, False (¬⊐D,-)

¬(A ⊐ B),-i ↙     ↘ A,-i      ¬B,-i

 

ρ, Reflexivity (ρrD)

ρ . ↓ iri

 

τ, Transitivity (τrD)

τ irj jrk ↓ .irk

(see p.38, section 3.3.2; with my naming additions)

 

Heredity, Unnegated, True (hD)

p,+i .irj ↓ p,+j . p is any propositional parameter

 

Heredity, Negated, True (¬hD)

¬p,+i .irj ↓ ¬p,+j . p is any propositional parameter

(176, with my naming additions)

(9.7a.5) Priest then does some example tableaux to show that ⊢_I4 ¬¬A ⊐ A, and ⊬_I4 (p ∧ ¬p) ⊐ q. (9.7a.6) Counter-models are formed in the following way. “There is a world w_i for each i on the branch; for propositional parameters, p, if p,+i occurs on the branch, set pρ_wi1; if ¬p,+i occurs on the branch, set pρ_wi0. ρ relates no parameter to anything else” (p.166). “w_iRw_j  iff irj occurs on the branch” (p.27). (9.7a.7) We obtain I₃ by adding the Exclusion Constraint to I₄: “for no p and W, pρ_w1 and pρ_w0.” (This makes it similar to  K₃.) Our tableaux for I₃ have the additional branch closure rule that a branch closes when both a propositional parameter and its negation are true in some same world. (9.7a.8) Formulas lacking negation that are valid in I are also valid in I₄ and I₃. (9.7a.9) But negation behaves differently in I than it does in I₄ and I₃. (9.7a.10) By changing I₄’s conditional rule for falsity and the tableau rule for negated conditional, we get a logic called W.

A ⊐ Bρ_w0 iff A ⊐ ¬Bρ_w1 (i.e., for all w′ such that wRw′, either it is not the case that Aρ_w′1 or Bρ_w′ 0).

 

Negated Conditional Development, True (¬⊐D,+)

¬(A ⊐ B),±i ↓ A ⊐ ¬B,±i

(178, naming is my own)

 

(9.7a.11) W is a connexive logic, meaning that there are two particular inferences it has: Aristotle ¬(A ⊐ ¬A) and Boethius (A ⊐ B) ⊐ ¬(A ⊐ ¬B). (9.7a.12) Unlike all other logics we deal with, connexive logics are not sub-logics of classical logic; for, not all the inferences that are valid in connexive logics are also valid in classical logic, here especially: Aristotle and Boethius. (9.7a.13) Aristotle and Boethius have intuitive appeal, despite being heterodox principles of conditionality. (9.7a.14) Another feature that W has that the others of this book lack is that its class of logical truths is inconsistent, namely, (p ∧ ¬p) ⊐ ¬(p ∧ ¬p) contradicts Aristotle ¬(A ⊐ ¬A). (9.7a.15) The tableaux for I₄, I₃, and W are sound and complete.

 

 

 

 

 

 

Contents

 

9.7a.1

[Logics of Constructible Negation]

 

9.7a.2

[The Structure of Logics of Constructible Negation]

 

9.7a.3

[Semantic Rules for Connectives. Validity.]

 

9.7a.4

[The Tableau Rules for I₄]

 

9.7a.5

[Example Tableaux]

 

9.7a.6

[Counter-Models]

 

9.7a.7

[I₃ and the Exclusion Constraint]

 

9.7a.8

[The Equivalence of I and I₄ (and I₃) Without Negation in Play]

 

9.7a.9

[The Non-Equivalence of I and I₄ (and I₃) With Negation in Play]

 

9.7a.10

[W]

 

9.7a.11

[Connexive Logic. Aristotle and Boethius Inferences]

 

9.7a.12

[Connexive Logics as Not Sub-Logics of Classical Logic]

 

9.7a.13

[The Intuitive Appeal of Aristotle and Boethius ]

 

9.7a.14

[The Inconsistent Logical Truths of W]

 

9.7a.15

[The Soundness and Completeness of I₄, I₃, and W]

 

 

 

 

 

 

 

Summary

 

9.7a.1

[Logics of Constructible Negation]

 

[We will now examine logics of constructible negation, which add an account of negation to the negationless part of intuitionistic logic (or positive intuitionistic logic). The important feature of these logics is that “unlike intuitionist logic, they treat truth and falsity even-handedly” (175).]

 

[Priest will now discuss a number of logics that are similar to the non-normal words FDE ones we have been examining. We fashion them by beginning with the negation free part of intuitionistic logic (called positive intuitionistic logic) and adding into it an account of negation. We call these “logics of constructible negation.” The important feature of theses logics is that “unlike intuitionist logic, they treat truth and falsity even-handedly” (175).]

Let us end this chapter with a brief look at a few other notable logics in the same ballpark as the ones we have already considered. These are obtained, essentially, by taking positive intuitionist logic – that is, the negation-free part of intuitionist logic – and grafting on a different account of negation. The logics are often called logics of constructible negation. The mark of these logics is that, unlike intuitionist logic, they treat truth and falsity even-handedly.

(175)

[contents]

 

 

 

 

 

 

9.7a.2

[The Structure of Logics of Constructible Negation]

 

 

[We will “Consider interpretations of the form ⟨W, R, ρ⟩, where W is the usual set of worlds, R is a reflexive and transitive binary relation on W, and for every w ∈ W, and propositional parameter, p, ρ_w relates p to 1, 0, both or neither, subject to the heredity constraints: if pρ_w1 and wRw′, then pρ_w′1 ; if pρ_w0 and wRw′, then pρ_w′0 ” (175).]

 

[We begin with interpretations in which we have worlds, an accessibility relation between them that is reflexive and transitive but apparently not reciprocal. Recall the three main sorts of world relativity constraints from section 3.2.3):

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)

There is also a heredity constraint, do when something is true (or false) in some world, it is also true (or false) in all worlds that it accesses (to recall the accessibility relation, see section 2.3.3.]

Consider interpretations of the form ⟨W, R, ρ⟩, where W is the usual set of worlds, R is a reflexive and transitive binary relation on W, and for every w ∈ W, and propositional parameter, p, ρ_w relates p to 1, 0, both or neither, subject to the heredity constraints:

if pρ_w1 and wRw′, then pρ_w′1

if pρ_w0 and wRw′, then pρ_w′0

The truth conditions in 9.7a.3 then ensure that these conditions hold for all formulas, not just propositional parameters. (See 9.11, problem 9.)

(175)

[contents]

 

 

 

 

 

 

9.7a.3

[Semantic Rules for Connectives. Validity.]

 

[Priest next provides the truth/falsity conditions for the connectives in our logic of constructible negation. Validity is truth preservation in all worlds of all interpretations.]

 

[Priest continues the semantics and provides the truth/falsity conditions for connectives. We use ⊐ for the conditional, as we are using a modified intuitionistic system (see section 6.3.2). And just like in K₄, an “inference is valid if it is truth-preserving in all worlds of all interpretations” (see section 9.2.5) (176).]

The truth/falsity conditions for the connectives are as follows. I write the conditional as ⊐, to make the connection with intuitionist logic clear.

 

A ∧ Bρ_w1 iff Aρ_w1 and Bρ_w1

A ∧ Bρ_w0 iff Aρ_w0 or Bρ_w0

 

A ∨ Bρ_w1 iff Aρ_w1 or Bρ_w1

A ∨ Bρ_w0 iff Aρ_w0 and Bρ_w0

 

¬Aρ_w1 iff Aρ_w0

¬Aρ_w0 iff Aρ_w1

 

A ⊐ Bρ_w1 iff for all w′ such that wRw′, either it is not the case that Aρ_w′1 or Bρ_w′1

A ⊐ Bρ_w0 iff Aρ_w1 and Bρ_w0

|

An inference is valid if it is truth-preserving in all worlds of all interpretations, as in K₄. Call this logic I₄.¹⁰

(175-176)

10 The reason that the logic is called one of constructible negation is that – unlike intuitionist logic – for a conditional to be false, its antecedent must be true and its consequent must be false. That is, we must be able to construct a counter-example to it.

(176)

[contents]

 

 

 

 

 

 

9.7a.4

[The Tableau Rules for I₄]

 

[Priest next gives the tableau rules for I₄.]

 

[Priest says that the rules for I₄ are the same as those for K₄, (see section 9.3.3) only we have different rules for the conditional. We also have rules to model the reflexivity and transitivity and also the heredity constraints (see 9.7a.2 above). And finally, “A tableau closes if we have lines of the form A,+i and A,−i” (176). (In the following, I will add the additional rules that are mentioned, taken from section 9.3.3 and 3.3.2. Note that the rules for double negation and disjunction are not in the text and are probably mistaken.)]

Tableaux for I₄ are the same as those for K₄, except that the rules for the conditional are:

 

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

(p.165, section 9.3.3; titles for the rules are my own additions)

 

Conditional Development, True (⊐D, +)

A ⊐ B,+i irj ↙     ↘ A,-j       B,+j . j is any number on the branch

 

Conditional Development, False (⊐D,-)

A ⊐ B,-i ↓ irj A,+j B,-j . j is new to the branch

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

 

In the first rule, j is any number on the branch. In the second, j is new to the branch.

 

Negated Conditional Development, True (¬⊐D,+)

¬(A ⊐ B),+i ↓ A,+i ¬B,+i

 

Negated Conditional Development, False (¬⊐D,-)

¬(A ⊐ B),-i ↙     ↘ A,-i      ¬B,-i

 

We also have the rules for reflexivity and transitivity of r (3.3.2),

 

ρ, Reflexivity (ρrD)

ρ . ↓ iri

 

τ, Transitivity (τrD)

τ irj jrk ↓ .irk

(see p.38, section 3.3.2; with my naming additions)

 

and the heredity rules:

 

Heredity, Unnegated, True (hD)

p,+i .irj ↓ p,+j . p is any propositional parameter

 

Heredity, Negated, True (¬hD)

¬p,+i .irj ↓ ¬p,+j . p is any propositional parameter

(176, with my naming additions)

 

where p is any propositional parameter.

A tableau closes if we have lines of the form A,+i and A,−i.

(176)

[contents]

 

 

 

 

 

 

9.7a.5

[Example Tableaux]

 

[Priest then does some example tableaux to show that ⊢_I4 ¬¬A ⊐ A, and ⊬_I4 (p ∧ ¬p) ⊐ q.]

 

[(ditto).]

Here are tableaux to show that ⊢ ¬¬A ⊐ A, and ⊬ (p ∧ ¬p) ⊐ q:

⊢ ¬¬A ⊐ A
1. . 2. . 3. . 4. . 5. . 6.	¬¬A ⊐ A,-0 ↓ 0r0 ↓ 0r1, 1r1 ↓ ¬¬A,+1 ↓ A,-1 ↓ A,+1 ×	P . 1ρr . 1⊐-;2,3τ . 1⊐- . 1⊐- . 4¬¬ (6×5) valid

(enumeration and step accounting are my own and are probably mistaken)

|

The last line is obtained by the rule for double negation.

 

⊬ (p ∧ ¬p) ⊐ q
1. . 2. . 3. . 4. . 5. . 6. . 7.	(p ∧ ¬p) ⊐ q,-0 ↓ 0r0 ↓ 0r1, 1r1 ↓ p ∧ ¬p,+1 ↓ q,-1 ↓ p,+1 ↓ ¬p,+1	P . 1ρr . 1⊐-;2,3τ . 1⊐- . 1⊐- . 4∧+ . 4∧+ (open) invalid

(enumeration and step accounting are my own and are probably mistaken)

(176-177)

[contents]

 

 

 

 

 

 

9.7a.6

[Counter-Models]

 

[Counter-models are formed in the following way. “There is a world w_i for each i on the branch; for propositional parameters, p, if p,+i occurs on the branch, set pρ_wi1; if ¬p,+i occurs on the branch, set pρ_wi0. ρ relates no parameter to anything else” (p.166). “w_iRw_j  iff irj occurs on the branch” (p.27).]

 

[Priest now will explain how to make counter-models from open branches. He says that we form them like we do for K₄ (see section 9.3.7). There he wrote:

Counter-models are read off from open branches of tableaux in the natural way. There is a world w_i for each i on the branch; for propositional parameters, p, if p,+i occurs on the branch, set pρ_wi1; if ¬p,+i occurs on the branch, set pρ_wi0. ρ relates no parameter to anything else.

(p.166, section 9.3.7)

And we must also include information about R like we do for modal logics. In section 2.4.7 he wrote:

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, w_i; w_iRw_j  iff irj occurs on the branch; for every propositional parameter, p, if p, i occurs on the branch, v_wi(p) = 1, if ¬p, i occurs on the branch, v_wi(p) = 0 (and if neither, v_wi(p) can be anything one wishes).

(p.27, section 2.4.7)

]

Counter-models are read off from open branches as for K₄ 9.3.7), except that details about R are read off as in tableaux for modal logics. Thus, the counter-model given by the tableau of 9.7a.5 is as follows:

x

xxx↷₀xxx→xxx↷

xxxw₀xxx→xxx+¬p

xxxw₀xxx→xxx+p

xxxw₀xxx→xxx-q

x

(177)

[contents]

 

 

 

 

 

 

9.7a.7

[I₃ and the Exclusion Constraint]

 

[We obtain I₃ by adding the Exclusion Constraint to I₄: “for no p and W, pρ_w1 and pρ_w0.” (This makes it similar to  K₃.) Our tableaux for I₃ have the additional branch closure rule that a branch closes when both a propositional parameter and its negation are true in some same world.]

 

[Recall the exclusion constraint from section 8.4.6:

Exclusion: for no p, pρ1 and pρ0

| i.e., no propositional parameter is both true and false.

(p.147-148, section 8.4.6.)

In section 8.4.7, Priest says that this produces the equivalent of K₃(see section 7.3.8). Priest now notes that we can obtain a standard variant of I₄ by adding the Exclusion Constraint, here as: “for no p and W, pρ_w1 and pρ_w0.” This variant we call I₃. It has an additional closure rule, namely, that branches close when both a propositional parameter and its negation are are true in some same world. Now recall the open tableau of 9.7a.5 that showed (p ∧ ¬p) ⊐ q is invalid. We see that in line 6 that p is true in world 1 and ¬p is also true in world one. Thus the branch would close, making this this formula valid in I₃.]

A standard variant of I₄ is obtained by adding the appropriate version of the Exclusion Constraint of 8.4.6:

for no p and W, pρ_w1 and pρ_w0

This ensures the corresponding statement for all formulas.¹¹ Call the logic I₃. Appropriate tableaux are obtained by adding the extra closure rule:

A,+i

¬A,+i

×

Clearly, the open tableau of 9.7a.5 closes in I₃, so (p ∧ ¬p) ⊐ q.

(177)

11. The proof is essentially as in the footnote of 8.4.6, except for the case for ⊐, which goes as follows. Suppose that A ⊐ Bρ_w1 and A ⊐ Bρ_w0. Then, by the second, Aρ_w1 and Bρ_w0. Moreover, by the first, Bρ_w1. This is impossible, by induction hypothesis.

[contents]

 

 

 

 

 

 

9.7a.8

[The Equivalence of I and I₄ (and I₃) Without Negation in Play]

 

[Formulas lacking negation that are valid in I are also valid in I₄ and I₃.]

 

[Priest next explains how inferences with formulas lacking negation that are valid in I are also valid in I₄ and I₃. ]

It is not difficult to see that for sentences that do not contain negation, an inference is valid in I₄ (and I₃) iff it is valid in intuitionist logic, I. To see this, note that any intuitionist interpretation, ⟨W, R, v⟩, corresponds to an I₄ (or I₃) interpretation ⟨W, R, ρ⟩, where v_w(p) = 1 iff pρ_w1; and vice | versa. A short argument by induction (for connectives other than negation) then shows that for every formula, A, v_w(A) = 1 iff Aρ_w1. (Details are left as an exercise.) In other words, the two sorts of interpretation are essentially the same.

(177-178)

[contents]

 

 

 

 

 

 

9.7a.9

[The Non-Equivalence of I and I₄ (and I₃) With Negation in Play]

 

[But negation behaves differently in I than it does in I₄ and I₃.]

 

[Above in section 9.7a.8 we noted that inferences with formulas lacking negation that are valid in I are also valid in I₄ and I₃. Priest now notes that for ones that do have negation, they are not equivalent. He cites as an example section 9.7a.5. We have not yet summarized the section on tableaux for intuitionistic logic. But consider comparing the first example from 9.7a.5 above with the example from 6.4.11, p.111:

⊬_I⇁⇁ p ⊐ p

⊢_I4 ¬¬A ⊐ A

(p.111 section 6.4.11; p.176 section 9.7a.5).

]

Clearly, I₄ (and I₃) differ from I in the behaviour of negation, however, as 9.7a.5 shows.

(178)

[contents]

 

 

 

 

 

 

9.7a.10

[W]

 

[By changing I₄’s conditional rule for falsity and the tableau rule for negated conditional, we get a logic called W.]

 

[Recall from section 9.7a.3 how we evaluate the conditional:

A ⊐ Bρ_w1 iff for all w′ such that wRw′, either it is not the case that Aρ_w′1 or Bρ_w′1

A ⊐ Bρ_w0 iff Aρ_w1 and Bρ_w0

(p.175, section 9.7a.3)

Priest next has us consider changing the falsity condition above to:

A ⊐ Bρ_w0 iff A ⊐ ¬Bρ_w1 (i.e., for all w′ such that wRw′, either it is not the case that Aρ_w′1 or Bρ_w′ 0).

After that, Priest gives the corresponding tableau rule for negated conditionals. It says that when a negated conditional is either true or false in some world, it is converted into an unnegated conditional in that world, only now with the consequent negated. This logic is called W.]

In the context of a discussion of conditionals, a further variation is worth noting. Suppose that in I₄ we change the falsity conditions for ⊐ to:

A ⊐ Bρ_w0 iff A ⊐ ¬Bρ_w1 (i.e., for all w′ such that wRw′, either it is not the case that Aρ_w′1 or Bρ_w′ 0).

The corresponding tableau rule for negated conditionals is simply:

 

Negated Conditional Development, True (¬⊐D,+)

¬(A ⊐ B),±i ↓ A ⊐ ¬B,±i

(178, naming is my own)

where the ± can be disambiguated consistently either way. Call this logic W (for Wansing).¹²

(178)

12. A similar modification of I₃ does not quite work. The Exclusion Constraint of 9.7a.7 is not sufficient to ensure that all formulas are not both true and false. A ⊐ B may be so, even though A and B are not (for example, if A is true at no worlds).

(178)

[contents]

 

 

 

 

 

 

9.7a.11

[Connexive Logic. Aristotle and Boethius Inferences]

 

[W is a connexive logic, meaning that there are two particular inferences it has: Aristotle ¬(A ⊐ ¬A) and Boethius (A ⊐ B) ⊐ ¬(A ⊐ ¬B).]

 

[This can change inference with negation, but it does not affect the negation free inferences. There are two new important valid inferences. One is called “Aristotle”: ¬(A ⊐ ¬A); the other is “Boethius”: (A ⊐ B) ⊐ ¬(A ⊐ ¬B). They characterize connexive logic, and W is connexive.]

The change makes no difference to the negation-free inferences, but it does affect the inferences involving negation. In particular, it is not difficult to check that both of the following are valid:

Aristotle ¬(A ⊐ ¬A)

Boethius (A ⊐ B) ⊐ ¬(A ⊐ ¬B)

The principles are so named because they are endorsed, arguably, by the philosophers in question. In modern logic, their holding characterises a logic as a connexive logic. There are many such logics. W is one of the simplest and most natural.

(178)

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9.7a.12

[Connexive Logics as Not Sub-Logics of Classical Logic]

 

[Unlike all other logics we deal with, connexive logics are not sub-logics of classical logic; for, not all the inferences that are valid in connexive logics are also valid in classical logic, here especially: Aristotle and Boethius.]

 

[Recall from section 8.4.13 that if all valid inferences of system A are valid in system B, then system A is a sub-logic of system B. And if in addition to that not all formulas of system B are valid in A, then system A is a proper sub-logic of system B. For all the logics we discuss in this book (except connexive logics), all inferences that are valid in the logic in question are also valid in classical logic. That means all the logics are sub-logics of classical logic. But connexive logic is the exception. For, “Aristotle and Boethius are not valid in classical logic”. ]

One reason why connexive logics are important is the following. All the propositional logics we will meet in this book, other than connexive logics, are sub-logics of classical logic (when the various negation | and conditional symbols are identified): any inference valid in the logic is valid in classical logic. Aristotle and Boethius are not valid in classical logic (when ⊐ is identified with ⊃). Indeed, they have instances that are classical contradictions. For example, ⊨ (p ∧ ¬p) ⊃ ¬(p ∧ ¬p) in classical logic (and even in most relevant logics).¹³ So connexive logics are very distinctive.

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13. In such logics, ⊨ (p ∧¬p) → p. By contraposition, ⊨ ¬p→¬(p ∧¬p), so ⊨ (p ∧¬p) → ¬(p ∧¬p).

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9.7a.13

[The Intuitive Appeal of Aristotle and Boethius ]

 

[Aristotle and Boethius have intuitive appeal, despite being heterodox principles of conditionality.]

 

[Priest next notes that although Aristotle and Boethius are highly heterodox principles of conditionality, they also have an intuitive appeal.]

Aristotle and Boethius are highly heterodox principles of conditionality. However, they do have a certain intuitive appeal. This makes connexive logics particularly interesting in the context of discussions of the conditional.

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9.7a.14

[The Inconsistent Logical Truths of W]

 

[Another feature that W has that the others of this book lack is that its class of logical truths is inconsistent, namely, (p ∧ ¬p) ⊐ ¬(p ∧ ¬p) contradicts Aristotle ¬(A ⊐ ¬A).]

 

[(ditto)]

Another notable feature of W is that its class of logical truths is inconsistent. It is not difficult to show that (p ∧ ¬p) ⊐ ¬(p ∧ ¬p) is valid. (Details are left as an exercise.) This contradicts Aristotle. W is the only propositional logic we will meet in this book with this property.¹⁴

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14, Most connexive logics in the literature are, in fact, consistent. This is because conjunction is usually taken to behave in a non-standard fashion.

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9.7a.15

[The Soundness and Completeness of I₄, I₃, and W]

 

[The tableaux for I₄, I₃, and W are sound and complete.]

 

[Priest ends by noting that we can prove that I₄, I₃, and W are sound and complete.]

The tableaux of this section are sound and complete with respect to their semantics. The proofs of this can be found in 9.8.

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From:

 

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

 

 

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