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 distracting mistakes, because I have not finished proofreading.]

Summary of

Graham Priest

*An Introduction to Non-Classical Logic: From If to Is*

Part I. Propositional Logic

7. Many-valued Logics

7.4. *LP *and *RM*_{3}

Brief summary:

Last time we examined *K*_{3}, which was defined as:

V= {1,i, 0}

D= {1}

f;_{c}c∈C= {f,_{¬}f_{∧},f_{∨},f_{⊃}}(note,

A≡Bwe are defining as (A⊃B) ∧ (B⊃A) )

f_{¬}10 ii01

f_{∧}1i011 i0 iii0 00 0 0

f_{∨}1i011 1 1 i1 ii01 i0

f_{⊃}1i011 i0 i1 ii01 1 1

Here *i* means *neither true nor false*. *LP* has the same structure as *K*_{3}, except in *LP* we have *D* = {1, *i*}. And in *LP*, the 1 is understood to mean *true and true only*, 0 to mean *false and false only*, and *i* to mean *both true and false*. The connective functions then follow our intuition regarding this alternate sense for the *V* values. Suppose for *A*∧*B* that *A *is 1 and *B* is *i*. Since *B *is at least true, then *A*∧*B* is at least true. And since *B* is also at least false, *A*∧*B* is also at least false. So *A*∧*B* is both true and false, or *i*, which is what the truth tables calculate it to be. There are two notable advantages of *LP *over *K*_{3}. {1} In *LP*, unlike in *K*_{3}, the law of excluded middle holds:

⊭

_{K3 }p∨ ¬p⊨

∨ ¬_{LP}_{ }pp

And the principle of explosion, or the inference rule *ex falso quodlibet*, are not valid in *LP*, unlike in *K*_{3}:

p∧ ¬p⊨_{K3 }q

p∧ ¬p⊭_{LP }q

But there is one disadvantage of *LP *compared with *K*_{3}. In *LP*, *modus ponens* is not valid:

p,p⊃q⊨_{K3 }q

p,p⊃q⊭_{LP }q

This can be solved by changing the evaluation for the conditional connective in the following way, thereby creating *RM*_{3}:

f_{⊃} | 1 | i | 0 |

1 | 1 | 0 | 0 |

i | 1 | i | o |

0 | 1 | 1 | 1 |

Summary

7.4.1

[The only difference between *LP *and *K*_{3} is that *D* = {1, *i*}.]

[Recall from section 7.2 the structure for many-valued logics:

⟨

V,D, {f;_{c}c∈C}⟩

*V* is the set of assignable truth values. *D *is the set of designated values, which are those that are preserved in valid inferences (like 1 for classical bivalent logic). *C* is the set of connectives. *c* is some particular connective. And *f _{c}* is the truth function corresponding to some connective, and it operates on the truth values of the formula in question. And recall from section 7.3 the three-valued logic

*K*

_{3}.

V= {1,i, 0}

D= {1}

f;_{c}c∈C= {f,_{¬}f_{∧},f_{∨},f_{⊃}}(note,

A≡Bwe are defining as (A⊃B) ∧ (B⊃A) )

f_{¬}10 ii01

f_{∧}1i011 i0 iii0 00 0 0

f_{∨}1i011 1 1 i1 ii01 i0

f_{⊃}1i011 i0 i1 ii01 1 1

] Now we examine another three-valued logic called *LP*. “This is exactly the same as *K*_{3}, except that *D* = {1, *i*}” (124).

7.4.2

[In *LP*, the connective functions assign the same values for the inputs. However, we understand 1 to mean *true and true only*, 0 to mean *false and false only*, and *i* to mean *both true and false*. The output calculations correspond to our intuitions regarding these meanings. For example, suppose for *A*∧*B* that *A *is 1 and *B* is *i*. Since *B *is at least true, then *A*∧*B* is at least true. And since *B* is also at least false, *A*∧*B* is at least false. So *A*∧*B* is both true and false, or *i*, which is what the truth tables calculate it to be.]

In *LP*, *i* is thought to mean *both true and false*. With that in mind, we would understand 1 as *true and true only* and 0 as *false and false only*. The truth tables remain the same, and they also make intuitive sense as well. Priest has us consider conjunction for example: *A *∧ *B*. And suppose that *A* is 1 and *B* is *i*. Since *B* is at least true, then *A *∧ *B *is at least true. And since *B* is at least false, that means *A *∧ *B* is at least false. *A *∧ *B* thus is both true and false, or *i*. As we see in the truth tables, that is how it evaluates.

f_{¬} | |

1 | 0 |

i | i |

0 | 1 |

f_{∧} | 1 | i | 0 |

1 | 1 | i | 0 |

i | i | i | 0 |

0 | 0 | 0 | 0 |

f_{∨} | 1 | i | 0 |

1 | 1 | 1 | 1 |

i | 1 | i | i |

0 | 1 | i | 0 |

f_{⊃} | 1 | i | 0 |

1 | 1 | i | 0 |

i | 1 | i | i |

0 | 1 | 1 | 1 |

(122, section 7.3)

Consider also if *A* is 0 and *B *is *i*. So, *B* is at least false, which means *A *∧ *B* is at least false, because both conjuncts are false. And, *B* is also at least true, but still, *A *∧ *B *is false, because it only takes one conjunct to be false for the whole conjunction to be false.

7.4.3

[Unlike in *K*_{3}, the law of excluded middle holds in *LP*: ⊨* _{LP}_{ }p* ∨ ¬

*p*]

[Recall from section 7.3.7 that the law of excluded middle did not hold in *K*_{3}: ⊭_{K3 }*p* ∨ ¬*p* . In *LP*, the table will be the same:

p | ¬p | p ∨ ¬p |

1 | 0 | 1 |

i | i | i |

0 | 1 | 1 |

*LP*,

*i*is a designated value, so the law of excluded middle holds in

*LP*, because no matter the input value, the output will be 1 or

*i*, which are both designated values.]

However, the change of designated values makes a crucial difference. For example, ⊨∨ ¬_{LP}_{ }pp. (Whatever valuephas,p∨ ¬ptakes either the value 1 ori. Thus it is always designated.) This fails inK_{3}, as we saw in 7.3.7.(125)

7.4.4

[The principle of explosion, or the inference rule *Ex Falso Quodlibet*, are not valid in *LP*, unlike in *K*_{3}:

*p* ∧ ¬*p* ⊭* _{LP }q* ]

[Let us consider *p* ∧ ¬*p* ⊨ *q* for *K*_{3} and *LP*.

p q | ¬p | p ∧ ¬p | ⊨ q |

1 1 | 0 | 0 | 1 |

1 i | 0 | 0 | i |

1 0 | 0 | 0 | 0 |

i 1 | i | i | 1 |

i i | i | i | i |

i 0 | i | i | 0 |

0 1 | 1 | 0 | 1 |

0 i | 1 | 0 | i |

0 0 | 1 | 0 | 0 |

In *K*_{3}, the only designated value is 1. There is no line where the premise is 1 and the conclusion is not 1. (In fact, there is no instance where the premise is 1 anyway. In chapter 2 of Priest’s *Logic: A Short Introduction*, he says that such cases are *vacuously valid*.) So:

p∧ ¬p⊨_{K3 }q

But, in *LP*, we have *i *and 1 as our designated values. So, if we look again, we see that when *p* is *i* and *q* is 0, then the premises have a designated value while the conclusion does not.

p q | ¬p | p ∧ ¬p | ⊨ q |

1 1 | 0 | 0 | 1 |

1 i | 0 | 0 | i |

1 0 | 0 | 0 | 0 |

i 1 | i | i | 1 |

i i | i | i | i |

i 0 | i | i | 0 |

0 1 | 1 | 0 | 1 |

0 i | 1 | 0 | i |

0 0 | 1 | 0 | 0 |

So:

p∧ ¬p⊨_{K3 }q

p∧ ¬p⊭_{LP }q

In other words, the principle of explosion, or the inference rule *Ex Falso Quodlibet*, are not valid in *LP*.]

On the other hand,

p∧ ¬p⊭. Counter-model:_{LP }qv(p) =i(makingv(p∧ ¬p) =i),v(q) = 0. Butp∧ ¬pcan never take the value 1 and so be designated inK_{3}. Thus, the inference is valid inK_{3}.(125)

7.4.5

[But in *LP*, *modus ponens* is not valid: *p*, *p *⊃ *q* ⊭_{LP }*q *]

[Now let us evaluate modus ponens, *p*, *p *⊃ *q* ⊨ *q*. We will do so for *LP*.

p q | p | p ⊃ q | ⊨ q |

1 1 | 1 | 1 | 1 |

1 i | 1 | i | i |

1 0 | 1 | 0 | 0 |

i 1 | i | 1 | 1 |

i i | i | i | i |

i 0 | i | i | 0 |

0 1 | 0 | 1 | 1 |

0 i | 0 | 1 | i |

0 0 | 0 | 1 | 0 |

Were this *K*_{3}, there would be no instances where the premises are 1 and the conclusion is not. However, in *LP*, we are looking for where the premises are either 1 or *i*, and the conclusion is 0. We see that when *p* is *i* and *q* is 0. Thus, modus ponens is not valid in *LP*.]

A notable feature of LP is that modus ponens is invalid:

p,p⊃q⊭_{LP }q. (Assignpthe valuei, andqthe value 0.)(125)

7.4.6

[*Modus ponens* can be regained by changing the evaluation for the conditional connective.]

Priest notes that we can regain *modus ponens *by changing the truth function assignments for the conditional in the following way:

f_{⊃} | 1 | i | 0 |

1 | 1 | 0 | 0 |

i | 1 | i | o |

0 | 1 | 1 | 1 |

[Let us check that with a truth table:

p q | p | p ⊃ q | ⊨ q |

1 1 | 1 | 1 | 1 |

1 i | 1 | 0 | i |

1 0 | 1 | 0 | 0 |

i 1 | i | 1 | 1 |

i i | i | i | i |

i 0 | i | 0 | 0 |

0 1 | 0 | 1 | 1 |

0 i | 0 | 1 | i |

0 0 | 0 | 1 | 0 |

As we can see, there is no line where the premises are all either 1 or *i* and the conclusion 0.]

One way to rectify this is to change the truth function for ⊃ to the following:

f_{⊃} | 1 | i | 0 |

1 | 1 | 0 | 0 |

i | 1 | i | o |

0 | 1 | 1 | 1 |

(As in 7.3.8, the meaning of

A⊃BinLPcan still be expressed by ¬A∨B.) Now, ifAandA⊃Bhave designated values (1 ori), so doesB, as a moment checking the truth table verifies.(125)

[Let us check this.

A B | ¬A | ¬A ∨ B | A ⊃ B |

1 1 | 0 | 1 | 1 |

1 i | 0 | 1 | 0 |

1 0 | 0 | 1 | 0 |

i 1 | i | 1 | 1 |

i i | i | i | i |

i 0 | i | i | 0 |

0 1 | 1 | 1 | 1 |

0 i | 1 | 1 | 1 |

0 0 | 1 | 1 | 1 |

As in the previous section, I must be making a mistake, because again for me ¬*A *∨ *B *and *A* ⊃ *B *(under its new valuation) are not equivalent. When *A* is *i* and *B* is 0, then under my calculations, ¬*A *∨ *B *is *i *and *A* ⊃ *B *is 0. So I am doing something wrong. Please correct me. But we do see that if *A *and *A *⊃ *B *have designated values (1 or *i*), so does *B*.]

7.4.7

[The above change to the conditional connective’s evaluation produces *RM*_{3}.]

“This change gives the logic often called *RM*_{3}.” [Note, in section 7.12, Priest says that “*RM*_{3 }is one family of *n*-valued logics, *RM _{n}*, related to the logic

*RM*(

*R*Mingle), which we will meet in chapter 10” (139)]

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

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