ON THE ALLEGED ERROR OF FORMAL OBJECTIONS TO NORMATIVE ERROR THEORY

Abstract According to Streumer and Wodak, a particular type of formal objection to normative error theory fails because it rests on a questionable assumption about the logical duality of the normative concepts of permissibility and impermissibility. In this discussion, we argue that there is an error in their indictment; as such, the formal objection to normative error theory might still prevail.


Introduction
Do formal objections to normative error theory fail?According to Streumer and Wodak (2021), at least one type of them does.It fails because of the faulty assumption that certain normative concepts are mere logical duals.They argue that these concepts are not topic neutral since they "take a stand on the permissibility or impermissibility of any particular action".As such, this is not an innocuous formal assumption but a rather substantive normative claim (Streumer & Wodak 2021, p. 258).
However, contra Streumer and Wodak, we show that the duality of the target normative concepts does not only result from a substantive claim about those concepts but also from the formal machinery assumed by a given normative theory.If the theory assumes the machinery of classical logic, the normative concepts are indeed logical duals; hence, the formal objection to normative error theory holds.On the other hand, if it assumes a piece of non-classical machinery, these concepts need not be duals; hence, the formal objection to normative error theory may not hold.Thus, the assumed formal machinery has a consequence to what Streumer and Wodak have alleged as the error of a certain type of formal objection to normative error theory.Before getting into this, however, let us first rehearse what a normative error theory is and what a formal objection to it implies.

The formal objection to normative error theory
Normative error theorists hold that: (NET) Any normative judgement1 is false (since there are no normative properties).
Let us now consider the type of formal objection to NET that Streumer and Wodak (2021, pp. 254-255) envisaged.This objection assumes the classical logical law of excluded middle: (L) For every proposition p, either p or not-p t and the implied mutual exclusivity (and perhaps exhaustivity) of the normative concepts of permissibility and impermissibility: (D) Every action is either permissible or impermissible.
If we take a standard (bivalent) deontic logic (i.e., the standard logic discussed by Carr (2017) and McNamara (2019)), then "permissibility" and "impermissibility" are interdefinable in the following way (letting "Pa" and "Ia" stand for "a is permissible" and "a is impermissible", respectively): Given L, D, and ID, Ia and Pa are logical duals This means that the denial of one implies the assertion of the other.Thus, if an action is not permissible, that particular action is impermissible.On the other hand, if an action is not impermissible, then it is permissible.Now the formal objection to NET runs as follows: (i) Suppose NET is correct.
(ii) Then some arbitrary normative judgement, Ia, is false.
(iii) Given L, (ii) implies that ~Ia is true.
(iv) Given D and ID, (iii) implies that Pa is true.
(v) But Pa must be false, ex hypothesi.Contradiction! (vi) Hence, NET must be incorrect.
This reductio argument is easy to understand.Given that NET is correct, no normative judgement must be true.However, given L, D, and ID, if Ia is false, Pa is true.But this contradicts the assumption since there is at least one true normative judgement.Hence, NET cannot be correct (Tiefensee 2020;Dworkin 2011, pp. 42-44).

On the alleged error of the formal objection
Streumer and Wodak question the reasoning behind this formal objection.They argue that since an analogous reductio argument to a more plausible error theory fails, there must be something amiss with the original formal objection to NET.For them, the error lies in premise (iv). 3In particular, they claim that since the duality of normative concepts is a substantive, and not a mere logical, assumption, the falsity of Ia does not entail the truth of Pa.Their argument proceeds as follows.
Consider a purity error theorist who holds that: (PET) All propositions about the spiritual purity of objects are putatively false (since there are no spiritual properties).
To generate the formal objection to PET, let us grant L as before, but let us modify D as: (D*) Every object is either spiritually pure or impure.
Furthermore, let ID remain the same, but let "Ia" and "Pa" now represent "a is spiritually impure" and "a is spiritually pure", respectively.Given these caveats, the formal objection to PET would then run as follows: (a) Suppose PET is correct.
(c) Given L, (b) implies that ~Ia is true.
(d) Given D* and ID, (c) implies that Pa is true.
(e) But Pa must be false, ex hypothesi.Contradiction! (f) Hence, PET must be incorrect.
problematic and that a presuppositional error theorist need not reject (L) and accept truth-value gaps (ibid.).
This reductio argument is again easy to understand.Suppose that PET is correct, then no purity judgement can be true.However, given L, D*, and ID, if Ia is false, Pa must be true.But this contradicts the assumption since there is at least one true purity judgement.Hence, PET cannot be correct.Streumer and Wodak (2021, p. 258) argue that since PET is a plausible error theory, "it cannot be false in virtue of its form".But why do they think that PET is a plausible theory in the first place?For them, the purity concepts used in the argument are bogus, ex hypothesi.Since no one would think that a reductio against PET implies that some purity judgements that contain these bogus concepts are true, it must follow that the analogous formal objection to NET must not imply that some normative judgements are true.Streumer and Wodak's (2021, p. 259) indictment is further supported by the observation that "x is im-F" does not always imply "x is not F".For example, "x is im-mortal" does not imply "x is not mortal" since a table is not im-mortal simply by being not mortal.By the same line reasoning, "x is im-pure" does not imply "x is not pure" "x is im-permissible" does not imply "x is not permissible".The seeming logical duality of these latter pairs of propositions only holds given a theory's substantive assumptions.But as Streumer and Wodak argue, an error theorist could simply reject such assumptions.

Sed contra
Contra Streumer and Wodak, however, we submit that with certain qualifications, the formal objection to NET still holds.First, Streumer and Wodak's observation that "x is im-F" does not imply "x is not F" seems to be a red herring.The formal objection to NET does not hinge on the logical entailment of "x is im-F" to "x is not F".Rather, it hinges on the logical duality of the target normative concepts of permissibility and impermissibility as defined by ID.In fact, their point about this non-entailment disappears once we characterise the duality of the target concepts without using the sentential forms "x is im-F" and "x is not F".A general definition like what follows might suffice towards this end: (GD) For any proposition Cp and C*p, Cp iff ~C*p and vice-versa.
GD is obviously true given that C and C* are dual normative concepts and that we are working within a logic that abides by L.
Streumer and Wodak may respond that their observation about "x is im-F" not implying "x is not F" is only part of their response to a possible reply and is far less central to their argument.To this, we retort, fair enough, but it is nonetheless important to emphasise that nothing really hinges on this non-entailment.
Second, Streumer and Wodak claim that the reductio objection to NET only holds if we grant that permissibility and impermissibility are mere logical duals in the first place.However, they think that the duality of such concepts stem solely from D, which is not a mere formal assumption but a substantive normative claim.
But why should we think that D alone suffices for the logical duality of permissibility and impermissibility?Perhaps Streumer and Wodak are just confusing ID with D. ID does imply the logical duality of the target concepts.By fiat, if ~Ia implies Pa and ~Pa implies Ia, then permissibility and impermissibility are logical duals.Moreover, ID implies D. If permissibility and impermissibility are logical duals and they are the only two normative concepts with which actions are judged, then every action is exclusively and exhaustively judged as permissible or impermissible.
But does D alone suffice for ID?It does not seem so since it might depend on the formal machinery a normative theory assumes.If a normative theory T assumes classical logic, T countenances L. This then implies D and ID.But if T assumes a non-classical then T might not countenance L and the truth or falsity of D might not be relevant to the truth of ID since D might simply be false or is neither true nor false, or is both true and false.
Suppose that T allows action-judgement gaps, i.e., actions that may be judged as neither permissible nor impermissible.For example, think of normatively neutral actions or amoral actions that cannot be judged as either permissible or impermissible.If T allows such gaps, then it assumes a gappy logical framework.Given this framework, L is not always true since there is at least one case -viz., the actionjudgement gaps themselves -that makes it not true.Moreover, in this gappy framework, ID only trivially follows from D. The entailment from D to ID is valid since there is no case where D is true while ID is not true.But this means that the truth of D has nothing to do with the truth of ID because D, ex hypothesi, is either false or is neither true nor false.Thus, if T allows action-judgement gaps, D does not suffice for ID. 4  Now suppose that T allows action-judgement gluts, i.e., actions are judged as both permissible and impermissible. 4There are various gappy logics that could deliver such results.However, the logical framework at work here is the Strong Kleene logic (K3).K3 is a paracomplete, three-valued logic that permits truth-value gaps.In K3, L is not a logical truth, and a conclusion is a logical consequence of a set of premises just in case there is no scenario where all the premises are true but the conclusion is not true (Priest 2008, p. 122).
Think of legal systems implying contradicting norms of actions or legal dialetheia; for example, think of contradictory norms implied by (i) no person of the female sex is allowed to vote and (ii) all property holders shall have the right to vote (Priest 2006, p. 184).If T allows such gluts, then T might assume a kind of glutty logical framework.Such a framework rejects the law of non-contradiction and the classically valid principle, ex contradictione quodlibet.The glutty framework preserves the truth of L but treats D as a truthvalue glut; i.e., it is both true and false.Given this, it is easy to show that ID trivially follows from D since there is a scenario where D and ID are both gluts. 5rom these considerations, it follows that the duality of the target normative concepts cannot be grounded solely on D; it must be grounded on a theory's assumed formal machinery as well.This assumption implies the truth or nontruth of D and whether D suffices for ID.
Compare the duality of the target normative concepts with the duality of ordinary first-order predicates, F and not-F.Suppose that someone claims that everything is either F or not-F.Does this already imply the duality of F and not-F?Not immediately!Unless we are assuming classical logic, where L (and the law of noncontradiction) holds, it is not necessarily the case that because something is not-F, it must be an F. For all we know, a particular thing might be neither F nor not-F, or both F and not-F (Beall & Logan 2017, pp. 104-ff).For example, if F is a vague concept like baldness, then a person might be neither bald nor not-bald, or both bald and not-bald.
The same reasoning applies in the case of the target normative concepts.The duality of permissibility and impermissibility is not just a product of a substantive normative claim, but also of the assumed formal machinery on which the normative claim is made.To say that these concepts are logical duals means that the assertion of one implies the denial of the other.But this is a feature of a theory's assumed formal machinery and not just of the substantive theory that it endorses.Thus, contra Streumer and Wodak, if both the advocates and critics of NET employ the same formal machinery of classical logic, then the formal objection to the theory holds.
Yet Streumer and Wodak may remain unconvinced.They might still insist that if the formal objection to NET prevails, the analogous objection to PET should also prevail.For them, this means that if we accept that the formal objection to NET holds, not only should we accept that some normative judgements are true but also that some judgements about a bogus purity concept are true.But is this reasonable?It seems not.
The foregoing discussions only provided a reason to accept that the formal objections to normative and purity error theories hold if both the advocates and critics of NET and PET assume classical logic.However, whether or not one should accept that there are true normative judgements and true judgements about a bogus purity concept as a consequence of this would be an entirely separate issue altogether.After all, one's negative reason for rejecting a theory need not be a positive reason for accepting another (contrary) theory.Since we have not provided a positive reason to think that there are true normative judgements nor a reason to think that there are true judgements about a bogus purity concept, it is still up for grabs whether there are such true judgements.Thus, even if we grant that PET is a more plausible error theory than NET, this does not mean that the failure of the formal objections to both theories implies the acceptance of true judgments about a bogus purity concept.
One might interject that NET is supposed to be a second-order (descriptive) theory of actual normative discourse, according to which we reason about normative judgements in a classical way, but we mistakenly assume that some of these are true. 6As such, the whole affair about the validity of the formal objection to NET must also assume classical logic.
However, this interjection is rather too quick.While NET is indeed a second-order theory about first-order normative discourse, it does not mean it merely describes such a discourse and that such a discourse is done classically.After all, NET, like other meta-normative theories, offer a substantive metaphysical (cum semantical) claim about our normative judgements and arguments.Such claims do not merely describe how we actually conduct our normative discourse but explain the very nature of such discourse and how such discourse is possible.This explanation need not subscribe to classical logic since much of our first-order normative discourse (especially those that allow action gaps and gluts) might assume a non-classical logic.As such, assessing NET's and other second-order normative theories' plausibility need not be done classically.

Conclusion
This paper has shown that Streumer and Wodak's indictment against the formal objection to NET might be in error.The duality of permissibility and impermissibility stems from the formal machinery and substantive normative claims adopted by a given normative theory.Thus, if the advocates and critics of NET subscribe to the machinery of classical logic, the formal objection to the theory holds.
But how should advocates of NET address the formal objection, then?Streumer and Wodak (2021, p. 261) are correct in saying that error theorists could only address the formal objection by dropping either L or D. For them, while it is possible to drop L, the best option is to drop D. However, contrary to what they think, it is not enough to drop D, one must also look into the logical framework assumed by both the advocates and critics of the theory.