1In this paper I will argue that norms, of whatever kind, are functions. A function is a way of relating the elements of two sets: an initial set or domain, and a final set or codomain. For the relation to be functional, each element of the domain must be assigned a single value from the codomain, although multiple elements of the domain may receive the same value. This broad idea is compatible with different notions of function, coming from different disciplines. Thus, from mathematics I would consider a very general definitions as “[…] a function is simply a certain kind of relation between input and output: a function is any relation for which an input gives at most one output, and always the same output” (Hamkins 2021: 74) suitable enough for exploring the concept of norm as a function. But to be sure, the notion of predicative function for the characterisation of concepts that Frege (1981) presents in the text “Function and concept” is probably more adequate. The functions I am interested in go from classes of actions to one of the two values of correctness: correct (C) and incorrect (¬C).
- 1 For example, in the case of definitions, the relevant class of actions is what to call something, a (...)
2All norms are functions because they all associate correctness values with behaviour (or the result of behaviour). In other words, the codomain of a normative function is always a class of correctness values. The difference between types of norms is in the nature of the classes of actions and omissions that are under regulation.1
- 2 I owe this clarification to a suggestion made by one of the referees who evaluated this article.
- 3 The relationship between norms and the language in which they can be expressed and represented is n (...)
3If norms are functions, they are not the meaning (Sinn) of normative sentences as some theories of norms have claimed, but rather their references (Bedeutung). Normative sentences have meanings, and these meanings are an appropriate way of determining norms, i.e., functional relations going from actions to values of correctness.2 I would say that expressing a norm in natural language is an appropriate way of determining a normative function. However, since the structure that a normative sentence exhibits is very different from a structure showing how a function operates, such sentences are not a good way to represent norms.3
- 4 In the case of numerical functions, it is easy to see this. The expressions “f(x) = 2x+3” and “g(x) (...)
4Functions are neither true nor false and they can be expressed and represented in several equivalent manners.4 There are no deductive relations between functions but, if a function assigns a property to the elements of a set, then the elements of any subset of that set having elements will have that property too, which can be understood as a relation of logical consequence.
- 5 In a broad sense, this could be seen as part of an input/output logic model because the functional (...)
5Next, I will introduce a model of norms as functions.5 But before specifying its elements, I will recall what the normative use of language consists of.
- 6 The first sentence comes from the official by-law text of the Barcelona City Council. See Barcelona (...)
6Norms, whatever their ontological status, are expressible linguistically, in both natural and formal languages. But as mentioned before, norms are neither true nor false. This is one of the most widely accepted and solid theses when studying norms. This is important because norms are expressed by sentences that can be syntactically identical to sentences with a truth value, i.e., sentences that represent norms to inform about what is correct and incorrect. The sentence “The consumption of tobacco products or devices capable of releasing nicotine is prohibited on the beaches of Barcelona.” and the sentence “Smoking on the beach is prohibited from 1 July 2022.” can be both used to inform about Barcelona’s bylaws and to proscribe an action.6 This difference, as said, embodies two distinct uses of language: normative and informative. The task of sentences normatively used is not to give us information about the world, a task for sentences with truth value, but to attain its transformation. And because these two different uses can be performed by syntactically identical sentences it is often said that, despite that appearance, they have opposite directions of fit.
- 7 Although among legal philosophers it is common to use the direction of fit metaphor as presented by (...)
- 8 “Just as orders and promises achieve or fail to achieve fit between the propositional content and r (...)
7The concept of direction of fit is frequently illustrated, in varying contexts, through the example provided by Anscombe (1957)7 of a shopping list carried by a man purchasing items in town and another list written by a spy who follows him to report on the items purchased. In the popularization of the example made by Searle (1995), the focus was on distinguishing types of speech acts.8
- 9 “Anscombe’s basic insight is that just as there is a kind of mental state (belief) that is partiall (...)
8As it is well known that the shopping list example is meant to illustrate situations that can be viewed from both directions of fit. The purchased products give truth value to the information contained in a checklist made to control the fulfilment of the order to buy a collection of products. It was the original list that conferred the correctness value to the action of purchasing some products and not others. When there is a mismatch between the names on a list and a collection of products, considering a modification in the list of names as a solution indicates a descriptive matching. Conversely, if changing the products is the way to fix the mismatch, we are performing a normative matching. Anscombe’s example deals with an individual case — an order to buy some items and its fulfilment.9 In the model introduced here, the elements involved will be general norms and their compliance.
9The norm gives a normative value to the fact. The fact receives its normative value from the norm. And, the fact gives a truth value to the description, while the description receives its truth value from the fact.
10If the norm is an order to buy or not to buy one product (bread), and the description is a report of what is bought, we will have the following four cases:
(1) The order “Buy bread!” gives the normative value Correct to the fact of buying bread, and the fact of buying bread gives the truth value True to the description “Bread was bought”.
(2) The order “Buy bread!” gives the normative value Incorrect to the fact of not buying bread, and the fact of not buying bread gives the truth value False to the description “Bread was bought”.
(3) The order “Don’t buy bread!” gives the normative value Correct to the fact of not buying bread, and the fact of not buying bread gives the truth value True to the description “Bread was not bought”.
(4) The order “Don’t buy bread!” gives the normative value Incorrect to the fact of buying bread, and the fact of buying bread gives the truth value False to the description “Bread was not bought”.
11In the context of the direction of fit analysis, which is about matching words-and-world, the order is the device that assigns the normative value to the fact. Although it occupies the place of the word, this “piece of language” is what gets the job done, so to speak, of expressing the order. In a mismatch the fact needs modification because it becomes a case of disobedience.
- 10 The Argentinian mathematician María Inés Baragatti, former professor at the Universidad Nacional de (...)
12For an informative function, the domain is the word element of the pair world and word. If a function were a machine,10 well-formed sentences expressing descriptions would be introduced into the machine that gives them a truth value. For a normative function, the domain is the world element of the pair world and word. The facts would be introduced into the machine that gives them a normative value.
13But since the direction of fit is, as it is, a metaphor, what kind of formalization would be useful for “pieces of language” that are not truth-apt and play the role of the machine? The choice will be formalising norms as functions.
- 11 To be a proposition, to have a truth value, and to be true or false are internally related properti (...)
- 12 In legal theory there are a myriad definition of the term “norm”. One of the consequences of the so (...)
14The definition of “norm” that I will use is not particularly problematic or committed: every norm is a normative qualification of an action or omission. In turn, any normative qualification implies a certain distribution of correctness and/or incorrectness on individual or generic actions and/or omissions.11 But this, like any other definition of the term, only gives us a semantic substitute for the word.12 Knowing what something is for, or what it is used to achieve, permits some inferences, but does not commit us to a specific essence or nature of that something.
- 13 Although I will not deal specifically with what are structurally different types of norms, the list (...)
15An order is a norm, a command is a norm, as are a definition or a general rule of behaviour.13 The ways in which an action or an omission can be normatively qualified in natural languages are very diverse. Smoking may be forbidden in the classroom; perhaps the same system of rules permits smoking in the parking lot of the school; there may be an obligation to pay a tax when a certain income threshold is reached; perhaps there are amounts exempt from that payment. Thus, obligation, prohibition, or permission seem to be common, but not the only, ways in which actions or omissions are normatively qualified.
16Such qualifications are usually treated, when formalising normative language, as deontic operators. In doing so, an analogy is assumed to hold between the deontic operators (O) obligatory, (PH) prohibited, and permitted (P) and the alethic operators of necessity, impossibility, and possibility. But in the model of norms as functions introduced here, the deontic operators Obligatory (O), Prohibited (PH), and Permitted (P) don’t operate by analogy with alethic operators. Instead, they distribute the two normative values Correct (C) and Incorrect (¬C). Hence the model has a semantic commitment to the thesis: “Every norm distributes correctness and/or incorrectness (our normative values) over classes of actions,” defining the deontic operators as follows:
- 14 The algebraic equivalent of the logical functor & is the intersection for sets and the product for (...)
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(OA) Obligating an action A, is to qualify the performance of that action A as correct and14 its non-performance (A’) as incorrect: OA = CA ∪ ¬CA’
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(PHA) Prohibiting an action A, is to qualify the performance of that action A as incorrect and its non-performance (A’) as correct: PHA = CA’ ∪ ¬CA
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(PA) Permitting an action A, is to qualify the performance of that action A as correct and its non-performance (A’) also as correct: PA = CA ∪ ¬CA’
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17A fourth operator will be added, the Blocking operator (B), defined as:
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(BA) Blocking an action A, is to qualify the performance of that action A as incorrect and its non-performance (A’) also as incorrect: BA = ¬CA ∪ ¬CA’
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- 15 In this book (Alchourrón and Bulygin 1971: 77-78) norms are categorised as statements that correlat (...)
18These definitions of the operators O, PH, and P, which involve the normative actions of expressing obligations, prohibitions, and permits, are a reformulation of the maximal solutions as characterized in the book Normative systems by Alchourrón and Bulygin.15 There are, however, important differences. First, when norms are regarded as functions, the normative values they distribute are correctness, and its negation incorrectness, rather than the character of permitted or not permitted. In this way we use normative values (correctness and incorrectness) that are suitable for any system of norms, and we are freed from having to consider the polysemy of the term ‘permitted’ and the different forms of negation of permissibility. Of course, any use of the vocabularies of permission in natural languages of has to be accommodated when deciding which normative functions are expressed by a normative sentence.
19There is no negation of a deontic operator in a normative function. There are no negations of norms in the model. This is because when the distribution of correctness and incorrectness is carried out on classes of actions and their complements, it is not a product (conjunction) of the classes that operate, but a sum (disjunction) of them.
20So, to deny a normative function as OA would result in
¬OA = ¬(CA ∪ ¬CA’) = ¬CA ∩ CA’ = ∅
The intersection of a set with its complement, or the product of a class with its complement, will always be the empty set or empty class, whatever the qualifications, properties, or values of these classes. Of course, the introduction of a blocking operator (B), which I will discuss in section 5.3 is completely foreign to the formalisations of deontic operators.
- 16 As Alchourrón and Bulygin (1971: 42) assert: “Among norms, two subclasses are distinguished: catego (...)
21In Alchourrón and Bulygin’s model, on the other hand, norms are correlations of cases with normative solutions. This makes it essential to consider the structure of conditional norms, since they are the standard version of the regulative norm:16 each norm provides a consequence (correlates a normative solution) for certain antecedent conditions (to a generic case made up of relevant properties). A normative solution is a deontically qualified action, selected from a universe of actions, and the identity of that action is independent of the properties that integrate the generic case. But in the model presented here, each class-term (both integrating actions and generic cases) is incorporated into the configuration of the classes of actions that are normatively qualified. Consequently, conditional norms disappear from the model.
- 17 In an approach where the correction values are considered aptic in nature, a gradation could be est (...)
22The above definitions turn the O, PH, P, and B operators that distribute the value correct (C) and its negation incorrect (¬C) over classes of actions and their complements. They are therefore functions that associate classes of actions with correctness values. These values are binary and exhaustive.17 Every action or omission that is qualified by a normative function is either correct or incorrect. Antinomies are dysfunctional in the strict sense of the word. If any domain element is related to more than one codomain element, the relation is non-functional. The model is then normative, and takes the actions of producing obligations, prohibitions, and permits as governed by some conceptual principles of rationality.
23The principles of rationality shaping this model are second-level functions, which apply to all kinds of normative orders. They are:
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(P1) Every norm must offer a path of correctness to be followed, that is, it must generate a compliance space.
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(P2) Every action considered by a normative order must have assigned a normative value.
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(P3) In any normative order, correct and incorrect behaviour must be treated differently.
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(P4) In any normative order, incorrect actions and omissions (non-compliance) must be treated worse than correct actions and omissions (compliance).
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(P5) In any normative order, only incorrect behaviour (non-compliance) can be punished.
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- 18 This is in line with the principle of deontic logic according to which the conjunction of an action (...)
24With regard to P1, whatever the source of a norm, or its type, it must be conceptually possible to follow it, comply with it, obey it, fulfil its content, or to act in a way that fits the norm. If a norm does not generate a space of compliance in which the addressee has to be located to follow the norm, to comply with it, or to obey it, then we are not dealing with a norm at all. P1 tells us that blockades are incorrect:18
This can be read as ‘it is incorrect for an action and its corresponding omission to be jointly incorrect’. The iteration of the correctness value is not problematic, as it is the iteration of deontic operators. Sums and products deployed by an operator cannot be applied to themselves, but correctness and incorrectness, on the contrary, operate analogously to truth and falsehood and can be iterated. By applying the De Morgan transformation, the principle is equivalent to CA ∩ CA’. That formula represents the intersection of a set, and its complement (or the product of a class and its complement) is the space of correctness. Unfortunately, that operation equals an empty set (∅). No space of correctness, no norm.
25Regarding P2, any norm needs to assign a correctness value to the classes of actions and omissions it contemplates. The principle P2 is telling us that antinomies are irrational:
and that is equivalent to ¬CA ∪ CA. That means that if a very same action or omission is associated with the two normative values of correctness, the relationship established is no longer functional. The union of a set or the sum of a class with itself is the very same set or class, and having two values goes against being a function.
26Regarding P3, the normative order involved must treat differently what has been normatively assessed as different, given an action and the corresponding omission. Compliance and non-compliance must be treated differently. There is no need for any specific type of treatment. From severe sanctions to silence, from millionaire awards to discrete applause, whatever the treatment that a normative order offers to compliance and non-compliance they cannot be equivalent from the value perspective. No behaviour can be motivated if doing right or wrong is equally treated. This is relevant not only for reasons of instrumental rationality. If both elements in the pairs harming or not harming one’s neighbour, dancing or not dancing barefoot, studying or not studying for the final exam, are treated in the same way by the source of normativity, then it cannot be that one of the elements in the pair is right and the other wrong. It may be that both are right. Learning how to behave in order to obey social rules presupposes such a divergence:
27Regarding P4, the normative order must treat better correct behaviour, compliance, than incorrect behaviour or non-compliance. There is an internal relationship between being treated better for compliance than for non-compliance regardless of the source that gave the correctness value to the behaviour. Representing the relation “CA is treated better than ¬CA” as
- 19 This is a contingent principle. For example, the rules for manipulating substances in a pharmaceuti (...)
28According to P5 normative orders should only provide sanctions for wrong behaviour. The ways in which coercion and sanction are conceptualised are highly variable. I use a notion of sanction, loosely understood as a negative consequence foreseen for incorrect behaviour. If an action or omission is correct within a normative order, it should not be sanctioned. But what about indifference? Indifferent actions are still considered correct within the normative order as long as they should not be sanctioned. This is the case whether we are using a normative function introduced by the permitted operator (P) — which has a reading equivalent to facultative or optional — or whether the case is an instance of genuine indifference in that it has not even been considered. Again, an element of Alchourrón and Bulygin’s model, known as weak permissions, is translatable into the vocabulary of normative functions. A class of actions that is not explicitly permitted is called “weakly permitted” because it is not prohibited. In the model presented here, we do not distribute permissiveness but values of correctness, and that the values to be distributed are correctness and incorrectness means that the closure norm does not have a tautological reading. That everything that is not forbidden is allowed has an intuitive interpretation in “the default value of correctness of an action or omission is correct”.19
29The sentences used to express norms are defective — in terms of drafting — if they do not correlate all the elements of the domain with some value in the codomain (normative gap). This kind of defect, that an element of the domain has not been associated with a value of correctness, has to be remedied by the principle of rationality P5 (second-level functions), which will treat it as correct. Being indifferent for a normative order has a normative reading: that order should not punish or react against actions that are indifferent according to it, because only incorrect behaviour (non-compliance) can be punished.
30The analytical sequence would be as follows: Only cases of incorrect behaviour (non-compliance) can be punished; to be punished is to be treated worse than a correct behaviour is treated. Thus, unless the normative order has provided rewards for correct conduct, indifference and correctness are treated in the same way.
- 20 Sixteen functions can be developed with two class-terms (as a full Hasse diagram shows). See Wikime (...)
31In the following table we see some representations of normative functions when certain actions20 are obligatory or forbidden. The first column contains actions truth-functionally described and modalized by a deontic operator — in these examples, obligatory and prohibited. The second column contains the classes of actions and their complements with their corresponding value of correctness distributed. As mentioned, when noticing how this differs from the representation of maximal solutions in Alchourrón and Bulygin (1971), a union of sets or sum of classes is needed (∪), as shown in the diagram. The function assigns correctness values to certain actions and omissions. For a representation helping to visualise de distribution of correctness and incorrectness the class of correct actions is shown in red, and the class of incorrect actions is shown in white in the Venn diagrams. In the last column, I mention the names of truth-functional operators of propositional calculus because the diagrammatic representation of the normative function coincides with the representation of those operators.
- 21 Lexically it makes sense to say that “dancing” is a simple action and “dancing by jumping” is not. (...)
32Correctness tables or minterms can be used to represent the above functions, both of which are more suitable than Venn diagrams as the number of class-terms increases. Instead of using the class-term notion, the concepts of variable or property can be also used. The only thing to remember is that each class defines a property. This property may be the sum or product of others, but once its representation is generated in a model, e.g., diagrammatically or by including and excluding classes, it is only one.21 It is worth noting that the introduction of a new class-term can lead to a significant increase in both the correctness and incorrectness subclasses and the number of possible normative functions. This can render a non-scalable tool such as a Venn diagram totally ineffective.
- 22 Regarding functions, these are the number of elements of the dominium set. In legal theory these ar (...)
- 23 To understand the magnitude of the growth in the number of functions, we can refer to the well-know (...)
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Number of class-terms (n)
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Number of subclasses22 (2n = m)
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Number of possible different normative functions (2m)
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1
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2
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4
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2
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4
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16
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3
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8
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256
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4
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16
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65.356
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5
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32
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4.294.967.296
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6
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64
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“[T]he year’s harvest would not furnish the amount of corn necessary.”23
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- 24 The list of possible issues to be treated is considerably longer. Here I have selected only the thr (...)
33In the following sections I will discuss three of the consequences of applying the model to the formal representation of normative orders.24 In particular, and as already mentioned, I will add some remarks on conditional norms (5.1), denying norms (5.2), and antinomies and blockades (5.3).
34One of the central features of the model of norms as functions is that there is no need for conditional norms.
- 25 The understanding that the structure of general legal norms is conditional is a well-established po (...)
35As I said before, according to Alchourrón and Bulygin, norms are by default conditional given that they are statements that correlate generic cases, as conditions, with normative solutions, as consequences.25 Dealing with legal systems necessitates considering the conditions under which obligations, prohibitions, or permits are claimable, valid, or effective. Thus, conditions are a pervasive element in this context. However, this does not justify treating norms as a type of conditional.
- 26 The 1985 campaign by the Spanish traffic authorities was designed to discourage drunk driving and t (...)
- 27 That is usually labelled the bridge conception of conditional norms and would also include the equi (...)
36Consider, for example, a famous campaign by the traffic authorities in Spain in 1985, led by the singer Stevie Wonder.26 The campaign used the grammatically conditional sentence “If you drink, don't drive”. A frequent manner of representing the norm it expresses would be27
p → PHq
where “p” represents the case of someone who has been drinking, the class of drinking actions, and “q” the prohibited action, driving. Instead, in the model developed here this case can be represented in several manners corresponding to our normative function Fn5 where “φ” represents the class of drinking actions and “ψ” the class of driving actions:
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(Fn5)
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PH(φ & ψ) = ¬C(φ ∩ ψ) ∪ C(φ’ ∪ ψ’)
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Minterms
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Fn5
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O(φ’ ∪ ψ’) = C(φ’ ∪ ψ’) ∪ ¬C(φ ∩ ψ)
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m0 = φ’ψ’
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C
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m1 = φ’ψ
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C
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m2 = φψ’
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C
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m3 = φψ
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¬C
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- 28 Using a descriptive sentence “When I drink, I do not drive” the case that falsifies these words is (...)
37With two class-terms, as we saw, we have four subclasses to be related with correctness and incorrectness. Here, driving drunk is incorrect; and driving without being drunk, having drunk without driving, or not having drunk and not driving, are all correct. It doesn’t mean we that cannot say that what is forbidden is driving when the condition of having drunk obtains, or if one drank. But the relation between inputs and outputs in the corresponding normative function is not equivalent to that holding between the antecedent and the consequent in a conditional (material, strict, or otherwise). The representation relies — as indeed the very same song of Stevie Wonder mentioned — on the logical product of the two classes: “Don’t drive drunk”.28 The Venn diagram representation is isomorphic with that of the propositional connective “&” because an intersection of properties or a product of classes is what depicts the incorrect class of actions.
38Another usual option to accommodate conditional norms is to give to deontic operators a wide scope treating the former driving and drinking case as a prohibition represented as
PH(p→q)
But in the model of norms as functions, the scheme of the normative function Fn7 is meant to be used for cases such as: “Payment of access fees and no criminal record are required” taking “φ” for paying the access fees and “ψ” for having a clean criminal record.
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(Fn7)
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PH(φ → ψ) = C(φ - ψ) ∪ ¬C(φ - ψ)’
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Minterms
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Fn7
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m0 = φ’ψ’
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¬C
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m1 = φ’ψ
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¬C
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m2 = φψ’
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C
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m3 = φψ
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¬C
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39Now, see how the use of the conditional functor works, as it happens, in the normative function scheme Fn3 in the norm expressed by the sentence “Only those who are members of the federation can participate in the tourney”, where the representation would yield a difference between classes. Taking “φ” for participating in the tourney and “ψ” for been a member of the federation:
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(Fn3)
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O(φ → ψ) = C(φ - ψ)’ ∪ ¬C(φ - ψ)
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Minterms
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Fn3
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m0 = φ’ψ’
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C
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m1 = φ’ψ
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C
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m2 = φψ’
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¬C
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m3 = φψ
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C
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40Some further commitments are made by treating norms as functions. Since it is impossible to deny functions, it is also impossible to accept the logical negation of norms. But this does not exclude the possibility that one normative function can produce distributions of correctness and incorrectness that are the reverse to those of another normative function. Obligations and prohibitions are reverse normative functions. What one makes correct when applied to a domain, the other makes incorrect when applied to the same domain, and vice versa. That is why we can express the same normative function by an obligation and a prohibition. For example, the normative function Fn2 presented in the table above with an obligation operator (O) can also be represented with a prohibition operator (PH), thus:
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(Fn2)
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O(A v B) = C(A∪B) ∪ ¬C(A’∩B’)
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Minterms
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Fn2
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PH(A’ & B’) = ¬C(A’∩B’) ∪ C(A∪B)
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m0 = A’B’
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¬C
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m1 = A’B
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C
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m2 = AB’
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C
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m3 = AB
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C
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- 29 As in algebra, opposite numbers are related by their positive and negative signs. To find the oppos (...)
41The function proceeds from the set of actions to the set of correctness values. The identity of the function is then given by the ordered pairs that are formed by taking an element from the domain and assigning a correctness value to it from the co-domain. Therefore, we also know that the reverse29 of the function Fn2 would be the function Fn6, which can be displayed with an O operator or a PH operator as follows:
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(Fn6)
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O(A’ & B’) = C(A’∩B’) ∪¬C(A∪B)
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Minterms
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Fn6
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PH(A v B) = ¬C(A∪B) ∪ C(A’∩B’)
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m0 = A’B’
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C
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m1 = A’B
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¬C
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m2 = AB’
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¬C
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m3 = AB
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¬C
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42When we want to make right what was wrong, and vice versa, we interchange obligations and prohibitions. And there is no problem in preferring to prohibit an omission rather than to making an action obligatory, or vice versa, if, for stylistic, psychological, or pragmatic reasons, we want to express or represent a normative function in a particular way.
43But if we try to deny any of the normative functions, we obtain, strictly speaking, the empty set. Our functions are equal to the union or sum of spaces of correctness and incorrectness, one being the complement of the other. The intersection of a set with its complement, or the product of a class with its complement, will always be the empty set or empty class:
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(Denial of Fn2)
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¬O (A v B) = ¬C (A’ ∩ B’) ∩ C (A ∪ B) = ∅
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(Denial of Fn6)
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¬O (A’ & B’) = C (A∪B) ∩ ¬C (A’∩B’) = ∅
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44And what does it mean that the intersection of two sets is the empty set? φ and ψ being any couple of predicates, A ∩ B = ∅ means: ∀x, ¬(φ(x) ∧ ψ(x)) that no element has both properties. When the sets we are dealing with are complementary, we are not facing a contingency related to those predicates but a stronger case. No element can belong to a set and not belong to a set. We thus have a commitment to bivalence in building our classes of actions.
45The reverse character is also present in a permission and a blockade function, because as in obligations and prohibitions what is qualified as correct by one of them is qualified as incorrect by the other:
PA = CA ∪ CA’
BA = ¬CA ∪ ¬CA’
46To deny a normative function expressed as a permission or as a blockade is again the empty set:
¬PA = ¬CA ∩ ¬CA’= ∅
¬BA = CA ∩ CA’= ∅
47The oddity of the B operator will be treated in the next paragraph when discussing antinomies.
48By definition, in our normative model, a norm, i.e., a normative function, cannot by itself generate an antinomy because an antinomy implies that a class of actions has simultaneously been assigned with both the opposed values: correct and incorrect. This would happen if any element of the domain were related to the two elements of the codomain. In other words, it would be a non-functional relation. But if we add functions, could it not happen, as we would say from our understanding of normative orders, that a class of actions ends up being assigned both correctness values? In this model, the conflict would need to be between two or more simultaneously applicable normative functions, given their own spatial, temporal, and hierarchical scopes. Before such criteria are considered, the antinomy would be merely nominal.
49Let’s consider two normative functions:
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(Fna)
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“Keeping promises is mandatory”
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Oα = Cα ∪ ¬Cα’
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(Fnb)
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“When a promise is not kept, apologising is required”
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PH(α’&β’) = ¬C(α’∩β’) ∪ C(α∪β)
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50First, we need to sum the codomains of both functions, i.e., take into account all the class-terms used by both and consider the number of new subclasses generated by the sum. Second, we need to assign the appropriate correctness values provided by each function. Third, in the case of nominal dysfunctionality, we need to use the corresponding antinomy resolution criterion (in this case temporal). Last, the resulting normative function will be displayed:
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(Fna+b)
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O(α v β) = PH(α’&β’)
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Minterms
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Fna
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Fnb
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m0 = α’β’
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¬C
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¬C
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m1 = α’β
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¬C
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C
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m2 = αβ’
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C
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C
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m3 = αβ
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C
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¬C
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- 30 Since Chisholm (1963), there have been many different versions of the contrary-to-duty obligation p (...)
51One criticism of this solution to the so-called “contrary-to-duty obligation” problem is that it seems to assume that it is right to break promises if the breaker apologises afterwards.30 But this would be a misunderstanding of the way in which classes of actions are configured. The same happens with “unless” clauses, including justificatory self-defence, for example, which make an otherwise prohibited action correct. If killing (φ) is forbidden unless it is done in self-defence (ψ):
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Minterms
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Fnc
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Fnd
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m0 = φ’ ψ’
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C
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C
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m1 = φ’ ψ
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C
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C
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m2 = φ ψ’
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¬C
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¬C
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m3 = φ ψ
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¬C
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C
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52This shows how far natural language is from the kind of formalisation that I am proposing. It might be natural to say both that killing is prohibited and that whoever kills must be punished; and that killing in self-defence is not prohibited and that those who so act are not to be punished. But it would be a poor understanding of self-defence to insist that some killing is both correct and incorrect. With those two class-terms and, correspondingly, four logical spaces, there is no room for saying (sans phrase) that killing is incorrect. In the realm of the function, no space exists that is defined by only one characteristic. Absent characteristics are present, so to speak. So, killing not in self-defence is incorrect, and killing in self-defence correct, meaning that it cannot be punished.
53Why consider a normative operator that only distributes incorrectness as the blocking operator does?
BA = ¬CA ∪ ¬CA’
- 31 “That norms can contradict each other logically is not anything which logic, ‘by itself’, can show. (...)
54Von Wright (1963) has pointed to the scenario of an irrational authority31 that forbids us both to do and not to do an action at the same time, thus making compliance impossible. He rejects the option based on the irrationality of the will, and not for logical reasons because he recognises that norms as far as they are prescriptions cannot contradict each other. In our model, although we accept the need for amending those functions, we proceed differently.
55This option fulfils all the criteria of functionality. Even if no one — except the malign torturer — were to enact a rule that had as its content what amounts to a negation of the principle of bivalence, it could be the result of a sum of normative functions. But to avoid including such functions in normative orders, the principle (P1) of normative rationality is available.
- 32 Though I do not address this question here, the role that truth has in apophatic discourse is the r (...)
56Recall the inclusion of indifferent cases in the correct ones introduced in the first paragraphs. This depended only on the definition of incorrectness and correctness, according to which it is correct to sanction incorrect behaviour and incorrect to sanction correct behaviour.32 Now, when a kind of action and its complement are both wrong, there is no room to avoid sanctioning, since there is no possibility of complying with or satisfying the norm. When we are faced with an antinomy, we can perform a correct action even though we are at the mercy of a sanction for the incorrect, dysfunctional one. But this is not the case with blocking. It will always be necessary to create a new function (not to choose among the available ones) that selects the kind of actions that will be correct: it is a conceptual matter that, according to a norm, some action or class of actions must be correct. If the principle of bivalence is accepted, it is a conceptual matter that, according to the facts, a proposition or its negation must be true.
57Although the most common way to present the feature of general norms that prevents deducibility between them is by arguing that they are prescriptive, and prescriptions are not truth-apt, why it cannot be it is not stated.
- 33 The relation of the membership of an element to a set works in a different manner than the relation (...)
58The rule according to which the truth of a sentence implies the falsehood of its negation is invalid for ascriptions of correctness to classes of actions. From the correctness of a class of actions, the incorrectness of its complement, that is, of its omission, does not follow.33 It seems otherwise when the normative text is interpreted as expressing an obligation or prohibition, precisely because as happens with orders or commands, there is a simultaneous distribution of correctness to an action and of incorrectness to the corresponding omission, as we have seen. In those cases, orders and the general norms of behaviour are structurally equivalent and in the same way that if one has to buy bread (it is correct to buy it) not buying bread is incorrect, it is understood that if it is correct to pay taxes not doing it is incorrect.
59This is the case regardless of the assumed definitions of truth and correctness, that is, regardless of the semantic content of these predicates (Frapolli 2013). And so, in a syllogism such as “Those who do not pay taxes must be punished, and Nathan has not paid taxes, so Nathan must be punished” the normative concepts are treated as appearing in descriptive statements. What Nathan must do according to some existing set of rules is what is being reported. They are treated, therefore, as a chain of normative propositions with truth value. The modus ponens rule is safe to be used in deontic logic systems of normative propositions.
60In the model of norms as functions introduced in this paper, there are no logical relations between norms. But when a function assigns a property to the elements of a set, then the elements of any subset — having elements — of that set, have that property. This is a relation of logical consequence.
—Acknowledgment.— I am grateful to Luís Duarte d’Almeida, José Juan Moreso and Pablo Rapetti for their invaluable suggestions and comments on a previous version of this text. I discussed this paper at the meeting “Logic and computation in legal theory” (Tarello Institute, University of Genoa, Italy, June 11-14, 2024), which allowed me to draw on many of the questions raised in the debate. I would like to make special mention of the issues raised by Giovanni Battista Ratti and of the enlightening discussion on function denial with Jorge Rodríguez after the meeting. The central idea introduced here was also presented at the congress “En teoría hay mujeres (en teoría)” (September 6, 2024 in Santiago de Chile). The comments and suggestions of two anonymous referees allowed me to eliminate many inaccuracies. I am very grateful to them for sharing their ideas about the possibilities and limits of projecting the model of norms as functions.
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