Negating Rules

[1] p & ~ p


[2] p

[1], elimination of conjunction

[3] p ∨ q

[2], introduction of disjunction

[4] ~ p

[1], elimination of conjunction

[5] q

[3], [4], disjunctive syllogism

Rules are neither true nor false. So, it is not clear which is the logical value (if any) that is to be inverted by negation. Deontic logicians and legal theorists have often referred to the value of “validity”2. However, validity is a highly contested concept, liable to be reduced to other, more nuanced, notions3.

Here I will use a quite abstract notion of efficacy as the logical value of rules4. I will hold a rule efficacious if its propositional content is always true in so far as commands (Op and O ~ p) are concerned, and sometimes true when authorizations (~ Op and ~ O ~ p) are concerned. In other terms, a command may be said to be efficacious if it is always complied with during its normative existence (i.e. its membership in a normative system), whereas an authorization is efficacious if it is sometimes used during its normative existence.

In so far as categorical rules are concerned, negation is easily applied. If rule Op is efficacious, its contrary O ~ p and contradictory ~ Op cannot be efficacious, and vice versa. More precisely, ‘Op’ and ‘O ~ p’ are dubbed “contrary” in that they may be both inefficacious, but not both efficacious. ‘Op’ and ‘~ Op’ are contradictory in that if one is efficacious, the other one cannot be efficacious, and vice versa. As a matter of course, the same relation holds for ‘O~ p’ and ‘~ O ~ p’. ‘Op’ implies ‘~ O ~ p’ because if ‘p’ is always the case, ‘~ p’ cannot sometimes be the case. The same holds for ‘O ~ p’ and ‘~ Op’. The members of the relation of implication are usually called “subalterns.” Finally, ‘~ Op’ and ‘~ O ~ p’ are “subcontraries”: they can be both efficacious, but not both inefficacious.

When two contradictory categorical rules belong to the same system, such a system is usually regarded as trivialized, i.e. any rule whatsoever will be a consequence of it. This is easily provable, in analogy to propositional calculus:

[6] Op & ~ Op


[7] Op

[6], elimination of conjunction

[8] Op  ∨  Oq

[7], introduction of disjunction

[9] ~  Op

[6], elimination of conjunction

[10] Oq

[8], [9], disjunctive syllogism

30.2.1 Negating Conditional Rules

Things are not so easy when it comes to hypothetical (viz. conditional) rules5, i.e. those rules which connect a certain normative solution to a determinate (non-empty) set of conditional operative facts or properties: e.g. “If it rains, you ought to close the window”.

In propositional logic, denied conditionals are very simply reconstructed as “~ (p ⊃ q)”. There is a contradiction between conditionals whenever the same set allows one to derive, at the same time, “p ⊃ q” and “~(p ⊃ q)”—or equivalent sentences such as “~ (p & ~ q)” and “p & ~ q”. Any sentence will follow from a set containing such two conditionals.

By contrast, the conjunction of the two conditionals “p ⊃ q” and “p ⊃ ~ q” is consistent and equates to the negation of the common antecedent: i.e., “~ p”6.

If one “translates” such simple tenets into the terms of prescriptive discourse, surprising results are obtained.

Assuming (the controversial) tenet that a conditional rule can be reconstructed as the connection of a (generic) case with the deontic qualification of a state of affairs (in symbols: p ⊃ Oq), the negation of a conditional rule would be expressed by the formula “~ (p ⊃ Oq)”. By contrast, “p ⊃ Oq” and “p ⊃ ~ Oq” would not bring about any inconsistency.

If it were correct, some remarkable consequences would follow:


There would be a contradiction between conditional rules, whenever the same set of rules allows one to derive, at the same time, “p ⊃ Oq” and “~(p ⊃ Oq)”. From such a set any rule would follow.



Analogously to what happens in propositional logic, the conjunction of the conditional rules “p ⊃ Oq” and “p ⊃ ~ Oq” would be equivalent to the negation of the common antecedent: that is “~ p”7. A set containing such two rules would not be inconsistent or trivialized.


Both tenets are debatable indeed8. Let us examine them in this order.

30.3 Denied Conditional Rules

The (presumptive) contradiction stemming from the simultaneous presence in a normative system of two conditional rules such as “p ⊃ Oq” and “~ (p ⊃ Oq)” is not easy to construe. Understood as a relation between rules, it prima facie asserts that the second rule is incompatible with the first rule: but are we sure that the latter sentence really expresses (or is commonly taken as expressing) a rule? What does it mean to assert that “It is not the case that (if p, then it ought to be that q)”?

There are at least four possible answers to this question.

30.3.1 Equivalence of External and Internal Negations

A first reading understands the expression at hand—“~ (p ⊃ Oq)”—as a sentence according to which “p ⊃ ~ Oq”: this is a typical thesis of LS logics9, which are intended to reconstruct counterfactual sentences. In LS systems, this reading is predicated on the fact that, in ordinary language, we negate, say, the counterfactual conditional

[11] If Scott Norwood had scored the free kick against the Giants in Super-bowl XXV, then the Bills would have won four Super-bowls in a row by means of the other conditional.

[12] If Scott Norwood had scored the free kick against the Giants in Super-bowl XXV, then the Bills would have not won four Super-bowls in a row the latter being tantamount, on this reading, to the sentence “It is not the case that ‘If Scott Norwood had scored the free kick against the Giants in Super-bowl XXV, then the Bills would have won four Super-bowls in a row’”.

Analogously, it would suffice to take notice of the fact that, in prescriptive language, in order to negate the conditional rule “If it rains, then it is obligatory to close the windows”, one enacts the other conditional rule “If it rains, then it is not obligatory to close the windows”, to maintain that the negation of a conditional rule is the same rule with the denied consequent (so-called “conditional rule-denial”).

It must be observed that, if this thesis is accepted, the contradictoriness of two conditional rules exclusively depends on the configuration of an inconsistency between categorical rules, connected to the occurrence of a state of affairs which both the antecedents of the conditional rules bear upon.

Once one has established the criteria to determine in which cases two categorical rules are incompatible10, it suffices to control whether such rules are connected to the same universe of cases: if this universe is void, then the rules are categorical and an inconsistency between them occurs, as it were, in any possible world; if it is not empty, then the occurrence of the relevant case (say, p) triggers the contradiction between the two categorical rules (i.e. the consequents) only regarding the world p. We will elaborate on this point in Sect. 30.3, in dealing with conditional denial.

30.3.2 The Biconditional Reading

Another way of reconstructing “~ (p ⊃ Oq)” as equivalent to “p ⊃ ~ Oq” consists in reading the negated conditional rule as a biconditional.

Given that, for the biconditional, external negation and internal negation are equivalent11, it would be possible to maintain that the negation of a conditional rule is materially equivalent to a sentence having the same operative facts and the denied consequent: respectively, “~(p ≡ Oq)” and “(p ≡ ~ Oq)”. The second rule-formulation provides that non-q is permitted if, and only if, p occurs12. And this would be equivalent to negating the rule that provides that if, and only if, p, then q is obligatory.

This reading presents some shortcomings. If one wants to reconstruct what normally goes on in normative reasoning, it is highly problematic, for the biconditional construal of conditional rules would (conceptually) preclude the emergence of normative gaps and the subsequent application of analogy. It must also be noticed that the rational reconstruction of a prima facie conditional rule is not a genuinely interpretive act, but an act of creation of a new rule, for it introduces a biconditional clause which, ex hypothesi, was not present in the original formulation.

30.3.3 Ambivalence

The third interpretation regards the conflict between conditionals not as a relation of contradiction stricto sensu, but rather as a relation of “ambivalence”13: namely, a “pragmatic contradiction,” consisting in enacting and rejecting the same normative content. However, in this case, external negation would have a completely different meaning from the internal one. While the latter would be a quasi-semantic operator, inverting the relevant logical value, the former would be a pragmatic operator, used to represent the attitudes of the normative authority. It follows from that that one needs to introduce different symbols to represent such operators. If we symbolize the act of enactment by “!”, and the act of repealing by “¡”, we can represent an ambivalence in the following way: “! (p ⊃ Oq)” and “¡ (p ⊃ Oq).” This does not exclude the possibility of configuring a (presumed) conditional contradiction through internal negation, as the enactment of two incompatible norms, which we can represent as follows: “! (p ⊃ Oq)” and “! (p ⊃ ~ Oq)”.

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