Can inferences in normative reasoning be governed by some form of logic?¹

1. Introduction

My aim is to dispute so called deontic paradoxes’possible effects to moral theory. I think that an ideal moral system must be coherent, without paradoxical forms. But deontic logic, despite its logical nature, shows many formal problems, in primis that it’s possible to derive incoherent theorems, as well as paradoxes. Although, we need of an exhaustive definition of the matter. In fact, despite big literature about it, there are many logical proposals so it’s difficult to express a simple its definition. Then, we can call it a logic of normative propositions [proposizioni normative] ,² and their truth’s functions [funzioni di verità] .³ So, deontic logic is a study of logical behavior deontic sentences, as well as reasonings about deontic propositions. It’s correct to state that deontic logic is anàlogon to logic’s definition: to state the rules of a correct thinking. In fact, Berto writes:

La logica è la disciplina che studia le condizioni di correttezza del ragionamento. Il suo scopo è dunque elaborare criteri e metodi, attraverso i quali si possano distinguere i ragionamenti corretti, detti anche validi, da quelli scorretti, o invalidi.⁴

While, Frixione adds:

tra le discipline che studiano il ragionamento, la logica è la disciplina normativa per eccellenza: essa specifica a quali condizioni un ragionamento deduttivo risulta logicamente corretto.⁵

Insofar, Sartor states that:

La logica si propone di fornire strumenti per l’analisi del linguaggio e del ragionamento.⁶

So, if the reasonings are logical inferences via states’sets, then deontic logic’s objects will be normative sentences. Yet, it will study normative reasonings. So, in this paper, we will take care of deontic reasonings to explicate (1) what is their grade of rationality; and, (2) what effects they have for moral theory. In fact, a moral theory is rational because its reasonings are coherent. So, if there are some irrational types of deontic inferences, then follows the question: is moral theory irrational?

2. Deontic Paradoxes

The history of logic points out many paradoxical expressions (i. e. liar’s paradox; Russell’s paradox; and so on). In these cases, it stands out that: (i) on the hand, logic systems with paradoxes show somehow contradictions that they self put forward coherence; (ii) on the other hand, in the presence of paradoxes we must contradict logic’s normal conception: to capture correct thinking’s rules.⁷ But, logical antinomies push to study in depth more human thought. Somehow paradoxes are challenges for our intelligence. As von Wright states:

the antinomies of logic have puzzled people ever since they were discovered — and will probably continue to puzzle us forever. We should, I think, not regard them as much as problems awaiting a solution, but rather as providing a perpetual raw — material for thinking. They are important, because thinking about them challenges the ultimate questions of all logic — and therefore of all thinking.⁸

But present paradoxes have effects more heavy in deontic logic than in general logic. For few authors, deontic logic is a labyrinth. In fact, von Wright writes:

Il mio itinerario attraverso il labirinto della «logica deontica» dura ormai da più di trent’anni.⁹

Modern deontic logic’s founder is forced to acknowledge that it is very problematic branch of logic.

On the other side, same matter’s evolution consists in research of a better logical formalism. So, Artosi writes:

La storia della logica deontica è la storia di come i paradossi sono stati scoperti e rimediati con misure che hanno a loro volta dato origine a nuovi paradossi.¹⁰

These paradoxes, also called deontic paradoxes, have an important consequence for moral theory because deontic logic is wanted to formalize moral reasonings. But as in ethics there are also various antinomies, generally called moral dilemmas, so deontic formalization can suggest useful directions for moral theory.

Deontic literature presented following incoherent theorems, also called paradoxical inferences:

1. The Paradox of Ross;
2. The Paradox of Åqvist;
3. The Paradox of Good Samaritan;
4. The Paradox of Robber;
5. The Paradox of Victim;
6. The Paradox of Plato;
7. The Paradox of Sartre;
8. The Paradox of Imperatives contrary to Duty;
9. The Paradox of Epistemic Duty.

They are very hard formulations. So, we will speak about them in different sections.

2.1. The Paradox of Ross

This paradox is expressed by danish Alf Ross in 1941¹¹ first. In fact, von Wright writes:

In the debate [about the logic of imperatives] two Danes took a prominent part. One was Jørgen Jørgensen, after whom the name “Jørgensen’s Dilemma” was coined. The other was Alf Ross, inventor of the famous paradox. Both the dilemma and the paradox are still active topics of current debate.¹²

In short, by sound principle in deontic logic:

[DR1]      ⊢A→B            ___________               ⊢ OA→OB

it is possible to derive following theorem:

[PR1] ⊢ (OA∨OB) → O (A∨B) [also: Op⊃O (p∨q)]^[13]

This formula [PR1] is an deontic paradox that involves directly most important principle of Standard Deontic Logic (SDL): whatever implies what is forbidden is itself forbidden .¹³

Let’s formulate an possible semantic meaning. So, (i) the letter A stands for “Nicola helps Giorgio”; and, (ii) the letter B stands for “Nicola kills Giorgio”. Obtained formula [PR1], we come from following reasoning:

(1) OA [read: it is obligatory that: Nicola helps Giorgio];

∴ (2) OA∨OB (propositional logic by (1)) [read: it is obligatory that: Nicola helps Giorgio, or it is obligatory that: Nicola kills Giorgio];

∴ (3) O (A∨B) (by (2) and [PR1]) [read: it is obligatory that: Nicola helps or kills Giorgio] .

Obviously, latter theorem (3) is paradoxical: can one obligation be immoral? As Al — Hibri Cox states:

clearly, the obligation expressed by (3) is satisfiable by killing Jones [’Giorgio’], which is absurd.¹⁴

For Mazzarese, the paradox of Ross is founded on particular nature of standard systems of deontic logic. In fact, they admit following formula: [F] Op ⊃ O (p∨q) [it is obligatory p implies: it is obligatory p-or-q] . In effect, the formula [F] is paradoxical because states that if an action is obligatory, then it’s obligatory that action or whatever other action. In Italian: «se un’azione è obbligatoria, allora è obbligatoria quell’azione o qualsiasi altra».¹⁵

So, it’s a conclusion very surprising and unpalatable.

2.2. The Paradox of Åqvist

Like last case, present paradox collides with the principle [DR1] and the principle [P] .

So, it follows a set of propositions: (1) it is the case that Nicola is refrain to robe Giorgio; (2) it is the case that Samaritan helps Giorgio who is robbed by Nicola. The sentences (1) and (2) can be formalized in the following manner: the letter (A) stands for “Nicola robes Giorgio”; and, the letter (B) stands for “Samaritan helps Giorgio”. Then, we derive following reasoning:

(1) O ∼A [read: it is obbligatory that: Nicola doesn’t robe Giorgio];

(2) O (A∧B) [read: it is obbligatory that: Nicola robes Giorgio and Samaritan helps Giorgio];

∴ (3) OA (by [DR1] and (2)) [read: it is obligatory that: Nicola robes Giorgio] .

Obviously, the inference (3) is counter-intuitive because there’s a contradiction between ban expressed by state (1) and deontic conclusion expressed by state (3). So, this paradox is the result of contradiction between propositions (1) and (3). In fact, it is surprising to ascertain that the conclusion (3) can be derived by sentence (1).

Besides, the state (3)

is not a logician’s paradox, like Russell’s class paradox; it reveals no logical antinomy or contradiction within calculus.¹⁶

In fact, the sentence (3) only, even if logically and soundly derived, involves

when interpreted, a result which is not only surprising, but unpalatable.¹⁷

For many authors, this result is based on a formal limit of SDL. In fact, real meaning of the state (2) in not a disjunction, but an entailment. Unfortunately, formal language of SDL can’t represent the implications.

2.3. The Paradox of Good Samaritan

The paradox concerns a modified version of principle [P] .¹⁸ Al — Hibri Cox, in fact, writes:

whatever implies what is forbidden is itself forbidden.¹⁹

So, there have following cases (1) if Good Samaritan helps Giorgio who is robbed, then Giorgio is robbed; (2) it is forbidden that Giorgio is robbed. Now, the letter (A) stands for “Good Samaritan helps Giorgio”; and, letter (B) stands for “Giorgio is robbed”. Then, we will have following reasoning:

(11) (A∧B) → B; [read: or Good Samaritan helps Giorgio or Giorgio is robbed implies Giorgio is robbed];

(21) FB [read: it’s forbidden: Giorgio is robbed];

∴ (31) F (A∧B) (by (1) — (2) and FR) [read: it’s forbidden that: Good Samaritan helps Giorgio who is robbed] .

Obviously, the sentence (31) is absurd. Perhaps, it is a partial consequence of formal limit of propositional logic, on which SDL is founded.

2.4. The Paradox of Robber

This paradox involves the principle [P] . There exist following propositions:

(1) if a robber is disappointed for his rob, then there was a rob;

(2) it’s forbidden to rob.

Like Good Samaritan, we will derive following inference: (3) it’s forbidden that the robber disappoint his rob. Obviously, deontic consequence (3) is paradoxical.

2.5. The Paradox of Victim

Also this paradox involves the principle [P] . We have the following sentences:

(1) if one victim by a robber deplores his fate, then there was a rob;

(2) it’s forbidden to rob.

So, it follows that: (3) it’s forbidden that a victim deplores his fate. The conclusion (3) has dubious meaning.

2.6. The Paradox of Plato

First version is in Plato:

Ti faccio un caso: se uno ha ricevuto armi da un amico sano di mente e se le sente richiedere da quell’amico impazzito, chiunque dovrebbe dire, a mio avviso, che non bisogna ridargliele e che non sarebbe giusto chi gliele ridesse.²⁰

This paradox derives by sound principle of SDL that the obligations doesn’t conflict.²¹

Well, let’s assume that one person left us his gun with promising to return him when he will ask for again it. Some days after, our person asks for again his gun because he wants to kill his wife who is unfaithful to. In this situation, we have an unpleasant state of two opposite duties: (a) (duty) return the gun; and, (b) (duty) don’t return the gun. In fact, we promised to return it if asked again for. So, we have a duty to return the gun. But, it’s easy to hypothesize its possible use. So, we have another duty too: don’t return the gun.²²

Therefore, we must face a paradoxical situation, although it isn’t as last paradox at all. In fact, in this case, the authors speak about moral dilemmas, or deontic dilemmas.

2.7. The Paradox of Sartre

Its original expression is in Sartre:

citerò il caso di un mio allievo, venuto a chiedermi consiglio nelle circostanze seguenti. Nella sua famiglia i rapporti tra il padre e la madre si erano guastati e d’altra parte il padre tendeva a collaborare con i tedeschi; il figlio maggiore era caduto durante l’offensiva germanica del ’40, mentre il figlio minore, i mio allievo, giovane dotato di sentimenti un po’primitivi ma generosi, lo voleva vendicare. La madre viveva sola con l’unico figlio rimastole, affranta per il mezzo tradimento del marito e per la fine dell’altro figlio, e vedeva in lui la sola consolazione. Quel giovane in quel momento poteva scegliere tra partire per l’Inghilterra e arruolarsi nelle Forze Francesi di Liberazione — e quindi abbandonare la madre — o restare presso la madre e consolarne l’esistenza. Si rendeva ben conto che la donna viveva solo per lui e che il suo andarsene via — e forse la sua morte — l’avrebbero gettata nella disperazione.²³

Its’a very famous paradox. It also shows an interesting deontic profile.

Well, let’s assume that one guy, who lose his brother during the War, wants to enlist in the forces of Free France. But, he have another duty too: to take care of his ill mother. So, the guy is in a dilemma: to choose one between two opposite duties, to enlist in army; and, to take care of ill mother. Still, on person can’t choose because opposite duties are equal, have same importance.²⁴

In short, this situation is like the latter. In fact, we have a moral dilemma.

2.8. The Paradox of Imperatives contrary to Duty

This is a important paradox for checking the coherence of SDL.

Let’s assume that one deontic logic accepts the principle [A1]^[26] and there exist following facts: (i) generally, we can neglet to do our duties; (ii) we ought to do the better in a possible case made by fact (i). So, we must to face following situation:

(1) Nicola robs Giorgio;

(2) Nicola ought don’t rob Giorgio;

(3) If Nicola doesn’t robe Giorgio, then he won’t punish;

(4) If Nicola robs Giorgio, then he ought to be punished.

Chisholm calls (4) an imperative contrary to duty.²⁵ In fact, (4) is an imperative contrary to duty expressed by (2).

This paradox shows deontic logic’s formal limits. Let’s adopt following formalism: the letter (A) stands for “Nicola robs Giorgio” and the letter (B) stands for “Nicola is punished”. So, to express (1) — (4), Al — Hibri believes that it’s necessary to write (3) as ‘O (∼A→∼B) ‘and (4) as ‘ (A→OB) ‘.²⁶ But by (1) — (4) we derive the conclusion: (5) it’s obligatory that: Nicola is punished. Instead, by (2) and (3) (via (1) and the theorem [T0]),²⁷ we derive following conclusion: (6) it’s obligatory that: Nicola isn’t punished. But, the conjunction of (4) and (6) contrasts with the principle [A1] .

So Al — Hibri states:

we need to find an adequate way for representing (3) and (4) in SDL_²⁸

Why does it happen? Probably,

deontic logic tells us what would be the case if all obligation were fulfilled, but fails to tell us would we ought to do in the real world, where obligations can be violated.²⁹

In other words, deontic logic fails in concrete cases because it expresses an ideal rationality of moral discourse, not concrete nature of singular moral cases. Besides, it seems to us that it can’t capture causes’mechanism between related conditions.

2.9. The Paradox of Epistemic Duty

Well, let’s add to SDL following epistemic constants: K = “I know that… ”; and, epistemic theorem ⊢ KA→A. So, it is possible to obtain following propositions: (1) Nicola ought to refrain robbing Giorgio; (2) I ought to know that Nicola robs Giorgio. Now, the letter (A) stands for “Nicola robs Giorgio”. So, we have following reasoning:

(11) O∼A [read: It is obligatory that: Nicola doesn’t rob Giorgio];

(21) OKA [read: it is obligatory that: I known that Nicola robs Giorgio] .

But by preceding theorem and constant, via substitution of (21) by epistemic theorem, we can to derive following deontic conclusion:

(31) OA [read: it is obligatory that: Nicola robs Giorgio] .

Still, (31), jointly with (11), is inconsistent with the principle [A1] . Here is the paradox.

Therefore, this paradoxical conclusion is based on

the premise that John ought to know that Mary is stealing surprisingly entails, in standard deontic logic, that Mary ought to steal.³⁰

3. What effects for Moral Theory?

The paradoxes invalidate host theoretical system’s rationality. Since its origins, deontic logic was an attempt to formalize moral reasoning, moral theory’s language. Now, if it seems irrational, or incoherent in many achievements, is moral theory irrational too? As Cremaschi states:

la logica deontica è rilevante per il ragionamento pratico.³¹

While Artosi deems that:

Filosofi morali e giuristi sembrano essere i clienti naturali del logico deontico. E, tuttavia, non sembra che si siano stabiliti buoni rapporti né con gli uni, né con gli altri. Colpa del venditore, probabilmente, che non sa presentare bene la merce. In realtà, molte cose possono aver contribuito a creare e a diffondere l’impressione che la logica deontica è irrilevante per l’etica e per il diritto.³²

Then, if deontic logic is incoherent, so inconsistent, then entire moral theory is irrational. This is a result unpalatable. In fact, it implies to expel practical reason from rational domain. Instead, Poli states:

il pensiero pratico è pensiero sul mondo in relazione a specifici concetti essenzialmente pratici. Pensiamo praticamente quando emettiamo ordini e comandi e quando prendiamo decisioni.³³

Then, we must consider practical reasoning as thinking’s portion. In fact, von Wright writes:

il pensiero pratico è pur sempre pensiero e, come tale, deve soddisfare i requisiti e le leggi della logica.³⁴

Then, how can we face deontic paradoxes? For many authors, deontic paradoxes are made by a particular tension between common knowledge of norms and specific formal job. Makinson states:

There is a singular tension between the philosophy of norms and the formal work of deontic logicians.³⁵

One tension between common normative intuitions and the formalism of deontic logic.³⁶

So, we must to add following remarks on the matter:

(a) SDL is a logic that formalize the behavior of deontic conceptions (as well as their normative propositions), while settling what is content of normative proposition is ethics’task;³⁷

(b) SDL is possible but it hasn’t functions exorbitant from its formal structure.

Still, nothing prevent us to believing that in future logic’s development can to extend deontic logic’s functions and its suitable for ethical propositions. In fact, Grana writes:

la logica nella sua evoluzione, come abbiamo già accennato precedentemente, ha esteso il suo campo d’indagine dalle proposizioni conoscitive ed assertive, vere o false, a proposizioni normative, allorché è riuscita a catturare in modo rigoroso quel dominio dell’instabile, del vago, del lacunoso, che era appunto escluso dai parametri angusti della logica standard. Questo nuovo dominio è stato esteso alle norme, settore in cui più che in ogni altro era necessaria una buona razionalizzazione […] oggi sappiamo che questo dominio è più vasto ed investe settori come l’etica, il diritto e più in generale la filosofia.³⁸

In last years, many model of a new deontic logic are proposed, defeasible deontic logic³⁹ and a paraconsistent deontic logic.⁴⁰ These proposal are based on modifications of SDL’s axioms. So, it seems to us to face principal problems: (1) relations of cause between deontic moods; (2) relations of conditional (primary and secondary) between deontic propositions; (3) deontic moods iterated; (4) iterated moods; (5) defeasibility and introduction of times and agency in deontic propositions; (6) coherence’s bonds. Feldman states:

it seems to me that the solutions to the paradoxes require a system that has at least the following features: (a) it must be able to express some sort of conditional obligations for which factual detachment fails; (b) it must be able to express the idea that something may be obligatory as of one time, but not-obligatory at some other time; (c) it must be able to express the idea that something may be obligatory for one person, but not for others. Insofar as other systems lacks these features, I cannot see how they can provide adequate solutions to the paradoxes.⁴¹

The future will say to us if it is possible this advanced deontic logic.

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