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Result 5.21: Not every classical entailment is a degree-entailment in KS3, and ditto for the other three systems L3, BI3, and BE3.
Proof: For the reason just given, this follows from earlier results.
5.5 Normal Forms
For each of the four systems, we define phrases, clauses, disjunctive normal form, and conjunctive normal form the same as we did for classical propositional logic:
A literal is a phrase.
If P and Q are phrases, so is (P Q).
Every phrase is in disjunctive normal form.
If P and Q are in disjunctive normal form, so is (P Q).
A literal is a clause.
If P and Q are clauses, so is (P Q).
Every clause is in conjunctive normal form.
If P and Q are in conjunctive normal form, so is (P Q).
Recall the equivalences that we used to convert formulas to these normal forms:
P → Q |
is equivalent to |
¬P Q |
P ↔ Q |
is equivalent to |
(¬P Q) (¬Q P) |
¬(P Q) |
is equivalent to |
¬P ¬Q |
¬(P Q) |
is equivalent to |
¬P ¬Q |
P |
is equivalent to |
¬¬P |
(P Q) R |
is equivalent to |
(P R) (Q R) |
P (Q R) |
is equivalent to |
(P Q) (P R) |
(P Q) R |
is equivalent to |
(P R) (Q R) |
P (Q R) |
is equivalent to |
(P Q) (P R) |
(Implication) (Implication) (DeMorgan’s Law) (DeMorgan’s Law) (Double Negation) (Distribution) (Distribution) (Distribution) (Distribution)
All of these equivalences hold in KS3 and BI3. The implication equivalences fail in L3, and the Double Negation equivalence fails in BE3. Proof of these claims is left as an exercise.
Because all of the equivalences hold in KS3 and BI3, we can claim that every formula in these two systems is equivalent to a formula in disjunctive normal form and to a formula in conjunctive normal form. Formulas in L3 that contain the conditional or the biconditional may not be equivalent to formulas in either normal form. We showed in Section 5.2 that we can’t define either →L or ↔L in terms of ¬L,L, and L. It follows that neither P →L Q nor P ↔L Q can be equivalent to a formula in either normal form.
It turns out that each BE3 formula is equivalent to a formula in disjunctive normal form and to one in conjunctive normal form, but we can’t claim that this
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follows from the previous equivalences since Double Negation fails for BE3. However, we note that the single case where Double Negation fails occurs when the double negation appears in front of an atomic formula, such as ¬¬S. In this case we can eliminate the double negation by replacing ¬¬P with P P, since these two formulas are equivalent for any atomic formula P. (This would also work where P is a complex formula but is unnecessary since we can simply eliminate the double negation in this case.) Then the Distribution equivalences can be applied to produce a formula in either of the normal forms.
In Chapter 2 we proved that a clause of classical propositional logic is a tautology if and only if it contains a complementary pair of literals, and that a phrase of classical propositional logic is contradictory if and only if it contains a complementary pair of literals. We have shown in this chapter, in Results 5.14, 5.15, and 5.17, that the quasitautologies of KS3, BI3, and BE3 coincide with the classical tautologies. It follows that
Result 5.22: A clause of KS3, L3, BI3, or BE3 is a quasi-tautology in that system if and only if it contains a complementary pair of literals.
(The result holds of L3 because clauses contain only negation and conjunction, and in L3 these connectives are identical to the KS3 connectives.) In Chapter 2 we also proved that a phrase of classical propositional logic is contradictory if and only if it contains a complementary pair of literals. Because the quasi-contradictions of the three systems in question coincide with classical contradictions, we also have
Result 5.23: A phrase of KS3, L3, BI3, or BE3 is quasi-contradictory in that system if and only if it contains a complementary pair of literals.
And, as a consequence of these two results:
Result 5.24: A formula P of KS3, L3, BI3, or BE3 that is in conjunctive normal form is a quasi-tautology in that system if and only if each clause in P contains a complementary pair of literals.
Result 5.25: A formula P of KS3, L3, BI3, or BE3 that is in disjunctive normal form is quasi-contradictory in that system if and only if each clause in P contains a complementary pair of literals.
5.6 Questions of Interdefinability between the Systems and Functional Completeness
Although we have characterized each of the four systems KS3, L3, BI3, and BE3 independently of the others, there are obviously important connections. For example, negation is defined identically in KS3, L3 and BI3, and disjunction and conjunction
5.6 Questions of Interdefinability |
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are defined identically in the former two systems. This raises the general question, Which connectives are definable within which systems? We first establish some very general negative results:
Result 5.26: The binary connectives of KS3, L3, and BE3 are not definable in BI3.
Proof: None of the connectives in BI3 produces a formula with a classical truthvalue when any of its immediate components have the value N. But the binary connectives of the other three systems can produce such, so none of these can be defined using only the connectives of BI3.
Result 5.27: None of the connectives of KS3, L3, or BI3 are definable in BE3.
Proof: The connectives of BE3 never produce formulas with the value N. Since each of the connectives in the other systems can produce such formulas, the result follows.
As a consequence of these two results, we know that neither BI3 nor BE3 can express everything that L3 can or everything that KS3 can, nor can either of BI3 or BE3 express everything that the other system can.
We turn now to the expressive powers of KS3 and L3. Both systems can express everything that can be expressed in BI3:
Result 5.28: All of the connectives of BI3 are definable in both KS3 and L3.
Proof: Negation in BI3 is identical to negation in the other two systems. We can define BI3’s conjunction using the other two system’s conjunction, disjunction, and negation (these connectives are defined identically in those two systems) as follows:
P BI Q = def (P K/L Q) K/L ((P K/L ¬K/L P) K/L (Q K/L ¬K/L Q))
It is left as an exercise to verify this equivalence. We can then define the other BI3 connectives in terms of these two, using any of the standard classical equivalences. Alternatively, we can give direct definitions for disjunction and the conditional analogous to the preceding definition for conjunction:
P BI Q = def (P K/L Q) K/L ((P K/L ¬K/L P) K/L (Q K/L ¬K/L Q))
P →BI Q = def (¬K/LP K/L Q) K/L ((P K/L ¬K/L P) K/L (Q K/L ¬K/L Q))
On the other hand, not all of L3 is expressible within KS3:
Result 5.29: The L3 conditional is not definable in KS3.
Proof: A formula P →L Q has the value T when both P and Q have the value N. But every KS3 connective produces a formula with the value N when its immediate components (all) have the value N, so no combination of KS3 connectives can produce a formula that expresses the L3 conditional.
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Nor can any BE3 connectives be expressed within KS3:
Result 5.30: The BE3 connectives are not definable in KS3.
Proof: No KS3 connective produces a formula that has a classical truth-value when its immediate components have the value N, so no BE3 connective can be defined using KS3 connectives.
However, it turns out that every connective of the other three systems is definable in L3. We have already shown that this is true of BI3.
Result 5.31: Every KS3 connective is definable in L3.
Proof: Since KS3’s negation, conjunction, and disjunction are identical to those of L3, we need only note that the KS3 conditional and biconditional are definable using those connectives.
Result 5.32: Every BE3 connective is definable in L3.
Proof: It will suffice to show that Bochvar’s external assertion is definable in L3. The definition aP = def ¬L(P →L ¬LP) produces the table for external assertion:
P aP
T T
F F
N F
All of the other external Bochvar connectives can be defined using external assertion and Bochvar’s internal connectives, which we have already shown to be definable in L3.
Having shown that L3 is powerful enough to define all of the connectives of the other three systems, the question arises, Are all possible three-valued truthfunctions definable in L3? If they are, then L3 is a functionally complete system. We showed in Chapter 2 that the classically defined connectives ¬ and form a functionally complete system for classical logic—every possible two-valued truthfunction can be defined solely in terms of classical negation and conjunction. Turning to L3 we might want to know, for example, whether the connective # with the truth-table
P # Q
P \ Q |
|
T N F |
|
||
T |
|
T N T |
N |
|
T N N |
F |
|
F N F |
is definable in L3. The answer is yes, and we leave it as an exercise to produce a formula that has these truth-conditions—using the algorithm that we are about to present in Result 5.33. More generally, every regular three-valued truth-function can be defined in L3, where a regular truth-function is one that produces classical
5.6 Questions of Interdefinability |
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truth-values when (but not necessarily only when) applied exclusively to classical truth-values:
Result 5.33: All regular three-valued truth-functions are definable in L3.
Proof: A regular three-valued n-place truth-function can be described by the truth-table schema
P1 P2 . . . Pn |
|
|
|
|
|
||
T T . . . T |
|
v1 |
|
T |
T . . . N |
|
v2 |
|
. . . |
|
. . . |
F |
F . . . F |
|
v3n |
|
|||
where each of v1, v2, . . . , v3n is one of the values T, N, F and where vi is T or F if all of the values to the left of the vertical bar in row i are classical truth-values.
We will first provide, for each row i of the truth-function’s table that has the value vi = T, a formula Qi that has the value T in that row and F in all other rows. We’ll be using the external assertion connective, which we have already shown to be definable in L3. For each such row of the table, define the formula Qi to be P1 L P2 L . . . L Pn where
Pj = aPj if the value of Pj is T in row i,
a¬LPj if the value of Pj is F in row i, and ¬LaPj L ¬La¬LPj otherwise.
Each of these formulas Pj defined for a particular row i will have the value T when Pj has the value it has in row i and will have the value F otherwise. So the conjunction Qi will have the value T in the row i for which it is defined but will be false in each other row since it will have at least one conjunct with the value F.
Next we provide, for each row i of the truth-function’s table that has the value vi = N, a formula Qi that has the value N in that row and F in all other rows. Because the truth-function that we are considering is regular, at least one of the Pj must have the value N in such a row. For each such row i, define Qi to be P1 L P2 L . . . L Pn where
Pj = aPj if the value of Pj is T in row i,
a¬LPj if the value of Pj is F in row i, and Pj L ¬LPj otherwise.
Finally, we form a disjunction of the formulas Qi for each row i with vi = T or vi = N. This disjunction expresses the function defined in the truth-table schema: the disjunction will have the value vi for each row i such that vi = T or vi = N,andF for all other rows—thedesiredresult,sinceallotherrowshavevi = F. Except that there is one special case—if the function produces F in every row, then that function can be defined in L3 using aP1 L ¬LaP1 L P2 L . . . L Pn— this formula always has the value F.