## 26 Jun 2018

### Priest (9.6) Introduction to Non-Classical Logic, ‘Star Again,’ summary

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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

9.6

Star Again

Brief summary:

(9.6.1) We can apply the N4 constructions to Routley Star ∗ semantics. (9.6.2) To the Routley semantics that we have seen before, we now add the rule for the conditional →, which gives us K. Here is the formalization:

Formally, a Routley interpretation is a structure ⟨W, ∗, v⟩, where W is a set of worlds, ∗ is a function from worlds to worlds such that w∗∗ = w, and v assigns each propositional parameter either the value 1 or the value 0 at each world. v is extended to an assignment of truth values for all formulas by the conditions:

vw(AB) = 1 if vw(A) = vw (B) = 1, otherwise it is 0. .

vw(AB) = 1 if vw(A) = 1 or vw (B) = 1, otherwise it is 0.

vwA) = 1 if vw*(A) = 0, otherwise it is 0.

| Note that vw*A) = 1 iff vw**(A) = 0 iff vw(A) = 0. In other words, given a pair of worlds, w and w* each of A and ¬A is true exactly once. Validity is defined in terms of truth preservation over all worlds of all interpretations.

(p.151-152, section 8.5.3)

Let ⟨W, ∗, v⟩ be any Routley interpretation (8.5.3). This becomes an interpretation for the augmented language when we add the following truth condition for →:

vw(A B) = 1 iff for all w′ ∈ W such that vw (A) = 1, vw(B) = 1

Call the logic that this generates, K.

(169)

(9.6.3) Priest then supplies the tableau rules for K.

 Conjunction Development, True (∧D,+x) A ∧ B,+x ↓ A,+x B,+x

 Conjunction Development, False (∧D,−x) A ∧ B,−x ↙      ↘ A,−x       B,−x

 Disjunction Development, True (∨D,+x) A ∨ B,+x ↙      ↘ A,+x        B,+x

 Disjunction Development, False (∨D,−x) A ∨ B,-x ↓ A,-x B,-x

 Negation Development, True (¬D,+x) ¬A,+x ↓ A,-x̄

 Negation Development, False (¬D,−x) ¬A,-x ↓ A,+x̄

 Conditional Development, True (→D,+x) A → B,+x ↙      ↘ A,-y      B,+y . where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch

 Conditional Development, False (→D,−x) A → B,-x ↓ A,+j B,-j . where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch. j must be new.

(last two are based on p.169, and those above from p.152, section 8.5.4, with names and bottom text added, possibly mistakenly; please consult the original text)

(9.6.4) Priest then gives an example tableau of an invalid formula: that p ¬q ⊬ ¬(p q). (9.6.5) We make counter-models using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world. When there is negation, however, we need to use values in the star-companion world. “W is the set of worlds which contains wx for every x and x̄ that occurs on the branch. For all i, w*i = wi# and w*i#= wi. v is such that if p,+x occurs on the branch, vx(p) = 1, and if p,−x occurs on the branch, vx(p) = 0” (170). (9.6.6) In K, we still have the problematic valid formula: ⊨ p → (qq). We can remedy this by adding non-normal worlds to get N. “An interpretation is a structure ⟨W, N, ∗, v⟩, where N W; for all w W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A B at every non-normal world. The truth conditions are exactly the same as for K, except that the truth conditions for → apply only at normal worlds; at non-normal worlds, they are already given by v. Validity is defined in terms of truth preservation at normal worlds. Call this logic N” (170). (9.6.7) We make our tableau for Nthe same way as for for K, only now the rules for the conditional → only apply for world 0. We generate counter-models the same way too. (9.6.8) The tableaux for K and N are sound and complete. (9.6.9) K4 and N4 are not equivalent to K and N. For example,  K and N validate  contraposition: p q ⊨ ¬q → ¬p, but K4 and N4 do not. (9.6.10) Additionally, K4 and N4 verify p ∧ ¬q ⊨ ¬(p q), but K and N do not.

Contents

9.6.1

[N4 and Routley ∗ ]

9.6.2

[K Semantics]

9.6.3

[K Tableau Rules]

9.6.4

[An Example Tableau: Invalid]

9.6.5

[Counter-Models]

9.6.6

[Non-Normal Worlds Routley ∗ :  N]

9.6.7

[Tableaux and Counter-Models for N]

9.6.8

[The Soundness and Completeness of N]

9.6.9

[The Non-Equivalence of K4 / N4 to K / N]

9.6.10

[More Non-Equivalence of K4 / N4 to K / N]

Summary

9.6.1

[N4 and Routley ∗ ]

[We can apply the N4 constructions to Routley Star ∗ semantics.]

[We should first review the basic ideas of Routley Star ∗ semantics, from section 8.5. The following is our brief summary of that section.

FDE can be given an equivalent, two-valued, possible world semantics in which the negation is an intensional operator, meaning that it is defined by means of related possible worlds. In this case, we use Routley’s star worlds. We have a star function, *, which maps a world to is “star” or “reverse” world (and back again to the first one, if applied yet another time. That bringing back to the first is what defines the function). So for any world w, the * function gives us its companion star world w. We evaluate conjunctions and disjunctions based on values in the given world. But what is notable in Routley Star semantics is that a negated formula in a world w is valued true in that world not on the basis of it unnegated form being false in that world w, but rather on the basis of its unnegated form being false in the star world w (so a negated formula is 1 in w if its unnegated form is 0 in w*.) Validity is defined as truth preservation for all worlds and interpretations. For constructing tableaux in Routley Star logic, we designate not just the truth-value for a formula but also the world in which that formula has that value. This is especially important for negation, where the derived formulas are found in the companion star world. Here is how the tableau rules work for Routley Star logic [quoting Priest, except for the rules tables, where I add my own names and abbreviations, following David Agler]:

Nodes are now of the form A,+x or A,−x, where x is either i or i#, i being a natural number. (In fact, i will always be 0, but we set things up in a slightly more general way for reasons to do with later chapters.) Intuitively, i# represents the star world of i. Closure occurs if we have a pair of the form A,+x and A,−x. The initial list comprises a node B,+0 for every premise, B, and A,−0, where A is the conclusion. The tableau rules are as follows, where x is either i or i#, and whichever of these it is, is the other.

 Conjunction Development, True (∧D,+x) A ∧ B,+x ↓ A,+x B,+x

 Conjunction Development, False (∧D,−x) A ∧ B,−x ↙      ↘ A,−x       B,−x

 Disjunction Development, True (∨D,+x) A ∨ B,+x ↙       ↘ A,+x        B,+x

 Disjunction Development, False (∨D,−x) A ∨ B,-x ↓ A,-x B,-x

 Negation Development, True (¬D,+x) ¬A,+x ↓ A,-x̄

 Negation Development, False (¬D,−x) ¬A,-x ↓ A,+x̄

(p.152, section 8.5.4, quoting. Note, names and abbreviations are my own and are not in the text. )

We test for validity first [as noted above] by setting every premise to true in the non-star world and the conclusion to false in the non-star world. We then apply all the rules possible, and if all the branches are closed [recall from above that closure occurs if we have a pair of the form A,+x and A,−x] then it is valid, and invalid otherwise [so it is invalid if any branches are open]. We then can make counter-models using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world (that is to say, “ if p,+x occurs on the branch, vwx(p) = 1, and if p,−x occurs on the branch, vwx(p) = 0.”) The equivalence between Routley Star semantics and FDE becomes apparent when we make the following translation: vw(p) = 1 iff 1; vw(p) = 0 iff 0.

(Brief summary for 8.5)

Next we need to recall from section 9.5 how Priest provided us the tableau rules for N4, which is the non-normal worlds variation of K4, which is a possible worlds First Degree Entailment (and thus four value-situationed) system. Priest says now that we can perform the same sorts of constructions for Routley ∗ semantics.

]

Before we move on to consider some of the implications of the preceding, let us pause to note that exactly the same sorts of construction can be performed with respect to the ∗ semantics.

(169)

[contents]

9.6.2

[K Semantics]

[To the Routley semantics that we have seen before, we now add the rule for the conditional →, which gives us K.]

[In section 8.5.3, Priest wrote:

Formally, a Routley interpretation is a structure ⟨W, ∗, v⟩, where W is a set of worlds, ∗ is a function from worlds to worlds such that w∗∗ = w, and v assigns each propositional parameter either the value 1 or the value 0 at each world. v is extended to an assignment of truth values for all formulas by the conditions:

vw(AB) = 1 if vw(A) = vw (B) = 1, otherwise it is 0. .

vw(AB) = 1 if vw(A) = 1 or vw (B) = 1, otherwise it is 0.

vwA) = 1 if vw*(A) = 0, otherwise it is 0.

| Note that vw*A) = 1 iff vw**(A) = 0 iff vw(A) = 0. In other words, given a pair of worlds, w and w* each of A and ¬A is true exactly once. Validity is defined in terms of truth preservation over all worlds of all interpretations.

(p.151-152, section 8.5.3)

Now Priest will add the rule for the conditional operator →, which will generate in our augmented language: K.]

Let ⟨W, ∗, v⟩ be any Routley interpretation (8.5.3). This becomes an interpretation for the augmented language when we add the following truth condition for →:

vw(A B) = 1 iff for all w′ ∈ W such that vw (A) = 1, vw(B) = 1

Call the logic that this generates, K.

(169)

[contents]

9.6.3

[K Tableau Rules]

[Priest then supplies the tableau rules for K.]

[Recall from section 9.6.1 above that we listed from section 8.5.4 the tableau rules for Routley ∗ logic. Priest says that we can obtain the full tableau rules for K by adding to those the rules for →. In the quotation]

Tableaux for K can be obtained by adding to the rules of 8.5.4, these rules for →:

 Conjunction Development, True (∧D,+x) A ∧ B,+x ↓ A,+x B,+x

 Conjunction Development, False (∧D,−x) A ∧ B,−x ↙      ↘ A,−x       B,−x

 Disjunction Development, True (∨D,+x) A ∨ B,+x ↙      ↘ A,+x        B,+x

 Disjunction Development, False (∨D,−x) A ∨ B,-x ↓ A,-x B,-x

 Negation Development, True (¬D,+x) ¬A,+x ↓ A,-x̄

 Negation Development, False (¬D,−x) ¬A,-x ↓ A,+x̄

 Conditional Development, True (→D,+x) A → B,+x ↙      ↘ A,-y      B,+y . where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch

 Conditional Development, False (→D,−x) A → B,-x ↓ A,+j B,-j . where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch. j must be new.

(last two are based on p.169, and those above from p.152, section 8.5.4, with names and bottom text added, possibly mistakenly; please consult the original text)

where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch;4 and in the second rule, j must be new. (Note that we do not need rules for negated →. The ∗ rules take care of that.)

(169)

4. So for a completed tableau, if either j or j# occurs on the branch, the rule needs to be applied to both j and j#.

(169)

[contents]

9.6.4

[An Example Tableau: Invalid]

[Priest then gives an example tableau of an invalid formula: that p ¬q ⊬ ¬(p q).]

[Priest then provides an example to show how to do a tableau in K. By the way, note for line 8 that the left branch closes, but the same rule could be still applied there as is applied on the right (because the development rule for true conditionals says it applies to all worlds on the branch), but it is unnecessary to do so.]

Here is a tableau to show that p ¬q ⊬ ¬(p q):

 p ∧ ¬q ⊬ ¬(p → q) 1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . p ∧ ¬q,+0 ↓ ¬(p → q),−0 ↓. p,+0 ↓ ¬q,+0 ↓ q,−0# ↓ p → q,+0# ↙             ↘ p,−0            q,+0      ×           ↙        ↘                 p,−0#     q,+0#                          ×   . P . P . 1∧+ . 1∧+ .  4¬− .  2¬→−   . 6→+ (7×4) 6→+  (8×5) open invalid

[based on p.153. Enumeration and step accounting are my own and are possibly mistaken.]

The splits are caused by applying the rule for true → to the line immediately before the first split. There are two worlds, 0 and 0#, so the rule has to be applied to both of them.

(169-170)

[contents]

9.6.5

[Counter-Models]

[We make counter-models using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world. When there is negation, however, we need to use values in the star-companion world. “W is the set of worlds which contains wx for every x and x̄ that occurs on the branch. For all i, w*i = wi# and w*i#= wi. v is such that if p,+x occurs on the branch, vx(p) = 1, and if p,−x occurs on the branch, vx(p) = 0” (170).]

[Recall from section 8.5.6 how we construct counter-models. In summary we said: “We make counter-models using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world. When there is negation, however, we need to use values in the star-companion world.” And Priest himself writes:

To read off a counter-model from an open branch: W = {w0,w0# } (there are only ever two worlds);  w*0 = w0# and (w0#)* = w0. (W and ∗ are always the same, no matter what the tableau.) v is such that if p,+x occurs on the branch, vwx(p) = 1, and if p,−x occurs on the branch, vwx(p) = 0. Thus, the counter-model defined by the righthand open branch of the second tableau of 8.5.5 has vwo(p) = 1, vwo(r) = 0 and vwo#(q) = 0. It is easy to check directly that this interpretation does the job. Since q is false at w0# , ¬q is true at w0, as, therefore, is q∨¬q; but p is true at w0, hence p ∧ (q ∨ ¬q) is true at w0. But r is false at w0, as required.

(p.153, section 8.5.6)

]

Counter-models are read off as is done without → (8.5.6), except that there may be more than two worlds now. Thus, W is the set of worlds which contains wx for every x and x̄ that occurs on the branch. For all i, w*i = wi# and w*i#= wi. v is such that if p,+x occurs on the branch, vx(p) = 1, and if p,−x occurs on the branch, vx(p) = 0. Thus, the counter-model from the open branch of the tableau of 9.6.4 may be depicted thus:

x

xxx+pxxx−p

xxx+qxxx−q

xxxw0xxxw*0

x

Since q is not true at w*0, ¬q is true at w0, as, then, is p ∧ ¬q. But at every world where p is true, q is true. Hence, p q is true at w*0, and so ¬(p q) is false (untrue) at w0.

(170)

[contents]

9.6.6

[Non-Normal Worlds Routley ∗ :  N]

[In K, we still have the problematic valid formula: ⊨ p → (qq). We can remedy this by adding non-normal worlds to get N. “An interpretation is a structure ⟨W, N, ∗, v⟩, where N W; for all w W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A B at every non-normal world. The truth conditions are exactly the same as for K, except that the truth conditions for → apply only at normal worlds; at non-normal worlds, they are already given by v. Validity is defined in terms of truth preservation at normal worlds. Call this logic N” (170).]

[Recall the following issue from section 9.4.

(9.4.2) In K4, if ⊨ A then ⊨ B A. That means ⊨ p → (qq) is valid, because ⊨ q q is valid. (9.4.3) Even though ⊨ p → (qq) , which contains the law of identity, is valid, we can think of a paradoxical instance of this that shows how the law of identity can fail: “if every instance of the law of identity failed, then, if cows were black, cows would be black. If every instance of the law failed, then it would precisely not be the case that if cows were black, they would be black” (167). (9.4.4) As we noted, the conditional should be able to express things that go against the laws of logic, like the law of identity. We should be able to formulate sentences in which the antecedent supposes some law of logic not to hold, and then the consequent would express what sorts of things would follow from that. Non-normal worlds are ones where the normal laws of logic may fail; so we should implement non-normal worlds: “we need to countenance worlds where the laws of logic are different, and so where laws of logic, like the law of identity, may fail. This is exactly what non-normal worlds are” (167). (9.4.5) We thus need to consider non-normal worlds where the laws of logic fail and, given how the conditionals express those laws, where the conditional takes on values different than it would in normal worlds (K4). (9.4.6) At a non-normal world, A B might be able to take on any sort of value, because the laws of logic may change in that world. (9.4.7) We “take an interpretation to be a structure ⟨W, N, ρ⟩, where W is a set of worlds, N W is the set of normal worlds (so that W N is the set of non-normal worlds), and ρ does two things. For every w, ρw is a relation between propositional parameters and the truth values 1 and 0, in the usual way. But also, for every non-normal world, w, ρw is a relation between formulas of the form A B and truth values” (167). (9.4.8) The truth conditions for connectives in our non-normal worlds K4 are the same as for K4, except in non-normal worlds, the conditional is assigned its value not recursively but in advance by the ρ relation.  Here are the truth conditions for normal worlds:

A Bρw1 iff Aρw1 and Bρw1

A Bρw0 iff Aρw0 or Bρw0

(p.164, section 9.2.3)

Aw1 iff w1 or w1

Aw0 iff w0 and w0

¬w1 iff Aρw0

¬w0 iff w1

(not in the text)

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

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

(p.164, section 9.2.4)

(9.4.9) Our non-normal worlds FDE system will be called N4, and it will define validity in the same way as for K4, namely, as truth preservation at all normal worlds of all interpretations.

(from the brief summary of section 9.4)

Priest notes now that in K, ⊨ p → (qq). We can fixe that problem like we did in in section 9.4 by adding non-normal worlds, giving us N.]

As in K4, in K, ⊨ p → (qq), as may easily be checked. To change this, we may add non-normal worlds in the same way. An interpretation is a structure ⟨W, N, ∗, v⟩, where N W; for all w W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A B at every non-normal world. The truth conditions are exactly the same as for K, except that the truth conditions for → apply only at normal worlds; at non-normal worlds, they are already given by v. Validity is defined in terms of truth preservation at normal worlds. Call this logic N.

(170)

[contents]

9.6.7

[Tableaux and Counter-Models for N]

[We make our tableau for Nthe same way as for for K, only now the rules for the conditional → only apply for world 0. We generate counter-models the same way too.]

[Priest then notes that we use the same tableau rules for Nas for K, except now the rules for the conditional only apply for world 0 (the only normal world). And counter-models are constructed the same way.]

The tableaux for Nare the same as those for K, except that the rules for → (9.6.3) are applied only at 0. Counter-models are also read off in the same way. Again, only w0 is normal.

(170)

[contents]

9.6.8

[The Soundness and Completeness of N]

[The tableaux for K and N are sound and complete.]

[Priest then notes that.]

Soundness and completeness for the tableaux for K and N are proved in 9.8.10–9.8.13.

[contents]

9.6.9

[The Non-Equivalence of K4 / N4 to K / N]

[K4 and N4 are not equivalent to K and N. For example,  K and N validate  contraposition: p q ⊨ ¬q → ¬p, but K4 and N4 do not.]

[Recall from section 8.5.8 that Routley Star semantics and FDE are equivalent when we make the translation: vw(p) = 1 iff 1; vw(p) = 0 iff 0. However, we lose this equivalence when we add the conditional →. The relational systems (I am guessing: K4 and N4) do not validate contraposition: p q ⊨ ¬q → ¬p. But the ∗ systems (K and N, I am guessing) do validate it.]

It should be noted that although the relational semantics and the ∗ semantics are equivalent for FDE, as we saw in 8.5.8, this equivalence no longer obtains once we add →. For a start, the ∗ systems (K and N) validate contraposition: p q ⊨ ¬q → ¬p. (Details are left as an exercise.) The relational systems do not. (We saw that this is not valid in K4, and a fortiori N4, in 9.3.6.)5

(171)

5. This may be changed by redefining the truth conditions of → (at normal worlds) in the relational semantics, as:

A Bρw1 iff for all w′ ∈ W (if Aρw1 then Bρw 1, and if Bρw 0 then Aρw 0).

Or, more simply, and equivalently, defining a new conditional AB as (A B) ∧ (¬B ¬A), and working with this.

(171)

[contents]

9.6.10

[More Non-Equivalence of K4 / N4 to K / N]

[Additionally, K4 and N4 verify p ∧ ¬q ⊨ ¬(p q), but K and N do not.]

[Also, K4 and N4 verify p ∧ ¬q ⊨ ¬(p q), but K and N do not.]

More fundamentally, because of the falsity conditions for →, the relation semantics (normal and non-normal) verify p ∧ ¬q ⊨ ¬(p q). (Details are left as an exercise.) But this inference fails in K (and a fortiori N), as we saw in 9.6.4.

(171)

[contents]

From:

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

.