PERMITTED EVENTS = LEGAL EVENTS

AEP does not seem interesting from the point of view of logic.

3.2.2. Theory 2: All Legal Events are Either Permitted or Forbidden (AEPF)

By adding two specific axioms to non-specific axioms,A1. ∀ x (LEV (x) Ɂ (PER (x) ∨ FOR (x))), A2. ∃ x (PER (x) ∧ FOR (x)), We will ge<sup>t</sup> a deontic theory: AEPF. This corresponds to the following Venn diagram:

3.2.3. Theory 3: All Legal Events are Either Permitted or Ordered or Forbidden (AEPOF)

By adding three specific axioms to non-specific axioms, A1. ∀ x (LEV (x) Ɂ (OBL (x) ∨ PER (x) ∨ FOR (x))), A2. ∃ x (PER (x) ∧ FOR (x)), A3. ∀ x (OBL (x) → PER (x)), We will ge<sup>t</sup> a deontic theory: AEPOF. This corresponds to the following Venn diagram:

3.2.4. Theory 4: All Legal Events are Either Permitted or Ordered or Forbidden or Irrelevant (AEPOFI)

By adding five specific axioms to non-specific axioms, A1. ∀ x (LEV (x) Ɂ (OBL (x) ∨ PER (x) ∨ FOR (x) ∨ IRR (x))), A2. ∃ x (PER (x) ∧ FOR (x)), A3. ∃ x (IRR (x) ∧ FOR (x)),

A4. ∃ x (PER (x) ∧ IRR (x)), A5. ∀ x (OBL (x) → PER (x)), We will ge<sup>t</sup> a deontic theory: AEPOFI. This corresponds to the following Venn diagram:

3.2.5. Existence of Legal Events

In the deontic theories set out above, we do not prejudge whether there are legal events. To determine this, a specific axiom should be added to each of these systems:

A0. ∃ x LEV (x).

#### 3.2.6. Selected Theorems of Legal Event Theories

Selected theorems of the theories of legal events are presented below. We omit proofs, because they are quite simple and intuitive.

AEPF, AEPOF, AEPOFI include, in particular, the following theorems:

T1. ∀ x (PER (x) ∧ FOR (x)); T2. ∀ x ( PER (x) ∨ FOR (x)); T3. ∀ x (PER (x) → FOR (x)); T4. ∀ x (FOR (x) → PER (x)). Of course, we also have in AEPOF and AEPOFI the following theorems: T5. ∀ x (OBL (x) → FOR (x)); T6. ∀ x (FOR (x) → OBL (x)); T7. ∀ x ( PER (x) → OBL (x)). Theorems T1–T7 have close equivalents in deontic propositional logics. On the other hand, in AEPF and AEPOF, we have T8. ∀ x (LEV (x) → (PER (x) ∨ FOR (x))); T9. ∀ x (LEV (x) → ( PER (x) → FOR (x))); T10. ∀ x (LEV (x) → ( FOR (x) → PER (x))); And consequently, we also have T11. ∀ x (LEV (x) → (PER (x) Ɂ FOR (x))) which follows from T3, T10; T12. ∀ x (LEV (x) → (FOR (x) Ɂ PER (x))) which follows from T4, T9. Theorems T8–T12 have equivalents in deontic propositional logics. The

predecessor of these theorems indicates, however, that the relations described by the successor occur only for legal events and not just for any events.

*3.3. Theories of Simple Acts*

The domain of the theories of acts is the set of situations as understood in accordance with Section 2 above. Thus, all propositions of these theories are propositions about situations.

We distinguish four binary predicates: ACT (x, y)—read "replacement x by y is an act";

PER (x, y)—read "replacement x by y is permitted";

FOR (x, y)—read "replacement x by y is forbidden";

OBL (x, y)—read "replacement x by y is ordered".

The specific axioms of these theories are selected in such a way that they determine the relations between sets of ordered, forbidden and permitted acts.

We consider only one such theory below, which is an extension of AEPOF.

3.3.1. Theory: All Acts are Either Permitted or Obligatory or Forbidden (AAPOF)

Every act is a legal event. Thus, the first three AAPOF-specific axioms are the exact counterparts of the AEPOF-specific axioms:

A1. ∀ x y (ACT (x, y) Ɂ (OBL (x, y) ∨ PER (x, y) ∨ FOR (x, y)));

A2. ∃ x y (PER (x, y) ∧ FOR (x, y));

A3. ∀ x y (OBL (x, y) → PER (x, y)).

These three axioms determine the relations between any situations x and y, forming one legal event (i.e., forming a sequence of situations < x, y >).

The next three AAPOF-specific axioms define relations involving three situations, x, y, z, forming two legal events (i.e., forming two sequences of situations: < x, y > and < x, z >).

Axiom A4 states that every act is a choice:

A4. ∀ x y (ACT (x, y) → ∃ z (ACT (x, z) ∧ y - z))

(In each choice situation, there are at least two options).

Axiom A5 confirms that the orders are consistent:

A5. ∀ x y z (OBL (x, y) → (y - z → FOR (x, z))

(If in a choice situation x, an option y is ordered, then all other options are prohibited in x).

On the other hand, the axiom A6 states that not everything is forbidden:

A6. ∀ x y (FOR (x, y) → ∃ z (ACT (x, z) ∧ y - z ∧ FOR (x, z)))

(If in a choice situation x, an option y is forbidden, then some other option is not forbidden in x). As in the case of the theories of legal events, we do not prejudge whether acts exist. To determine

> AEPOF:

this, it would be necessary to add the specific axiom A0 to AAPOF:

A0. ∃ x y ACT (x, y).

(There are choice situations).

3.3.2. Selected Theorems of AAPOF that are Equivalent to Theorems of AEPOF

```
In AAPOF, we have exact equivalents of theorems T1–T12 ofT1. ∀ x y  (PER (x, y) ∧ FOR (x, y));
T2. ∀ xy( PER (x, y) ∨  FOR (x, y));
T3. ∀ x y (PER (x, y) →  FOR (x, y));
T4. ∀ x y (FOR (x, y) →  PER (x, y));
T5. ∀ x y (OBL (x, y) →  FOR (x, y));
T6. ∀ x y (FOR (x, y) →  OBL (x, y));
T7. ∀ xy( PER (x, y) →  OBL (x, y));
T8. ∀ x y (ACT (x, y) → (PER (x, y) ∨ FOR (x, y)));
T9. ∀ x y (ACT (x, y) → ( PER (x, y) → FOR (x, y)));
T10. ∀ x y (ACT (x, y) → ( FOR (x, y) → PER (x, y)));
T11. ∀ x y (ACT (x, y) → (PER (x, y) Ɂ  FOR (x, y)));
T12. ∀ x y (ACT (x, y) → (FOR (x, y) Ɂ  PER (x, y))).
```
3.3.3. Selected AAPOF Theorems Specific to Acts

In AAPOF, we also have theorems that do not have their exact counterparts in AEPOF, which are the consequences of adding specific axioms A4–A6 to the system:

∧

T13. ∀ x y z (OBL (x, y) → (y - z → PER (x, z))

(If an option y is ordered in a choice situation x, then no other option is permitted in x); T14. ∀ x y z (OBL (x, y) → (y - z → OBL (x, z))

(If an option y is ordered in a choice situation x, then no other option is ordered in x);

T15. ∀ x y z (OBL (x, y) ∧ OBL (x, z) → y = z)

(If, in a choice situation, two options are ordered, they are identical);

T16. ∀ x y z (y - z → (OBL (x, y) ∧ OBL (x, z)))

(In any choice situation, different options cannot be ordered together);

T17. ∀ x y (FOR (x, y) → ∃ z (y - z ∧ PER (x, z)))

(If an option y is forbidden in a choice situation x, then some other option z is permitted in x); T18. ∀ x y (OBL (x, y) → ∃ z (y - z ∧ FOR (x, z)))

(If an option y is ordered in a choice situation x, then some other option z is forbidden in x); T19. ∀ x y z (y - z → (OBL (x, y) → PER (x, z)))

(If an option y is ordered in a choice situation x, then no other option is permitted in x);

T20. ∀ x y z (ACT (x, y) ∧ ACT (x, z) ∧ y - z ∧ ∀ w (ACT (x, w) → (w = y ∨ w = z)) → (FOR (x, y) FOR (x, z)))

(If there are exactly two options in a choice situation, both cannot be forbidden);

T21. ∀ x y z (ACT (x, y) ∧ ACT (x, z) ∧ y - z ∧ ∀ w (ACT (x, w) → (w = y ∨ w = z)) → (FOR (x, y) →PER (x, z)))

(If, in a choice situation, there are exactly two options, then if one of them is forbidden, the other is permitted);

T22. ∀ x y z (ACT (x, y) ∧ ACT (x, z) ∧ y - z ∧ ∀ w (ACT (x, w) → (w = y ∨ w = z)) → (PER (x, y) ∨ PER (x, z)))

(If, in a choice situation, there are exactly two options, then at least one of them is permitted);

T23. ∀ x y z (ACT (x, y) ∧ ACT (x, z) ∧ y - z ∧ ∀ w (ACT (x, w) → (w = y ∨ w = z)) → ( PER (x, y) → PER (x, z)))

(If, in a choice situation, there are exactly two options, then if one of them is not permitted, the other is permitted);

T24. ∀ xyzw (FOR (x, y) ∧ ∀ z (FOR (x, z) → y = z) → (ACT (x, w) ∧ w - y → PER (x, w)))

(If, in a choice situation, exactly one option is prohibited, then any other option is permitted).

## *3.4. Theories of Compound Acts*

In deontic propositional logics, deontic operators apply to conjunction or alternative of propositions; for example,

$$\mathcal{O} \left( \mathbf{p} \wedge \mathbf{q} \right) \to \mathcal{O} \,\mathbf{p} \wedge \mathcal{O} \,\mathbf{q},$$

$$\mathcal{O} \,\mathbf{p} \wedge \mathcal{O} \,\mathbf{q} \to \mathcal{O} \,(\mathbf{p} \wedge \mathbf{q}),$$

$$\mathcal{O} \,\mathbf{p} \to \mathcal{O} \,(\mathbf{p} \vee \mathbf{q}).$$

Such sentences are intended to formalize the intuition that an obligation, prohibition or permission may relate to situations where one is part of the other.

This intuition can be expressed more precisely by developing AAPOF into the theory of compound acts. We do this by adding axioms defining relations between situations, some of which are parts of the others.

To do so, we need to distinguish further one unary predicate "AT (x)", one binary predicate "ε (x, y)" and one ternary predicate "= + (x, y, z)":

AT (x)—read "x is an atomic situation";

ε (x, y)—read "x is a part of y";

= + (x, y, z)—read "x is the sum (composition) of y and z".

Below, we will write "x ε y" instead of "ε (x, y)" and "x = y + z" instead of "= + (x, y, z)".

## 3.4.1. AAPOF for Compound Acts

First, we will list axioms that will determine when a situation is a part of another situation, when a situation is the sum (composition) of other situations, and when a situation is an atomic situation.

We use Wolniewicz's approach to define the relation of "being a part of":

A7. ∀ x x ε x; A8. ∀ x y z (x ε y ∧ y ε z → x ε z); A9. ∀ x y (x ε y ∧ y ε x → x = y). We also add the A10 axiom for atomic situations: A10. ∀ x (AT (x) Ɂ ∀ y (y ε x → y = x)) (Every atom is a situation that has no proper parts). Then, we introduce the sum (composition) of situations:

A11. x = y + z Ɂ y ε x ∧ z ε x ∧ ∀ w (AT (w) → (w ε x → (w ε y ∨ w ε z)))

(A situation x is the sum (composition) of situations y and z, when they are parts of it, and each atom of the situation x is a part of the situation y or a part of the situation z).

Using the concept of a part of situation, we can express the intuition that a part of a situation has the same deontic modality as this situation:

A12. ∀ x x1 y y1 (x1 ε x ∧ y1 ε y → (OBL (x, y) → (ACT (x1, y1) → OBL (x1, y1))));

A13. ∀ x x1 y y1 (x1 ε x ∧ y1 ε y → (PER (x, y) → (ACT (x1, y1) → PER (x1, y1))));

A14. ∀ x x1y y1(x1ε x ∧ y1ε y → (FOR (x, y) → (ACT (x1, y1) → FOR (x1, y1)))).

In turn, using the concept of the sum (composition) of situations, we can express intuition, according to which any situation has the same deontic modality as its parts:

A15. ∀ x x1 x2 y y1 y2 (x = x1 + x2 ∧ y = y1 + y2 → (OBL (x1, y1) ∧ OBL (x2, y2) → OBL (x, y))); A16. ∀ x x1 x2 y y1 y2 (x = x1 + x2 ∧ y = y1 + y2 → (PER (x1, y1) ∧ PER (x2, y2) → PER (x, y))); A17. ∀ x x1 x2 y y1 y2 (x = x1 + x2 ∧ y = y1 + y2 → (FOR (x1, y1) ∧ FOR (x2, y2) → FOR (x, y))).

3.4.2. Selected AAPOF Theorems Specific to Compound Acts

The consequences of adopting additional specific axioms A7–A17 include, but are not limited to, the following examples:

T25. ∀ x x1 x2 y y1 y2 (x = x1 + x2 ∧ y = y1 + y2 → (OBL (x, y) → (ACT (x1, y1) ∧ ACT (x2, y2) → (OBL (x1, y1) ∧ FOR (x2, y2))))

(If an act is ordered, it is not that one part of it is ordered and the other is forbidden);

T26. ∀ x x1 x2 y y1 y2 (x = x1 + x2 ∧ y = y1 + y2 → (OBL (x, y) → (ACT (x1, y1) ∧ ACT (x2, y2) → (PER (x1, y1) ∧ FOR (x2, y2))))

(If an act is ordered, it is not that one part of it is permitted and the other is forbidden);

T27. ∀ x x1 x2 y y1 y2 (x = x1 + x2 ∧ y = y1 + y2 → (OBL (x, y) → (ACT (x1, y1) ∧ ACT (x2, y2) → (FOR (x1, y1) ∨ FOR (x2, y2))))

(If an act is ordered, it is not that any part of it is forbidden); T28. ∀ x x1 x2 y y1 y2 (x = x1 + x2 ∧ y = y1 + y2 → (OBL (x1, y1) ∧ OBL (x2, y2) → PER (x, y))) (If acts are ordered, their composition is permitted); T29. ∀ x x1 x2 y y1 y2 (x = x1 + x2 ∧ y = y1 + y2 → (PER (x1, y1) ∧ PER (x2, y2) → FOR (x, y)))

(If acts are permitted, their composition is not forbidden);

T30. ∀ x x1 x2 y y1 y2 (x = x1 + x2 ∧ y = y1 + y2 → (FOR (x1, y1) ∧ FOR (x2, y2) → PER (x, y))) (If acts are forbidden, their composition is not permitted).

The above relations are useful for reconstructing legal reasoning *a maiori ad minus* and *a minori ad maius*, as well as for reconstructing other similar reasonings.

## **4. Discussion**

A comparison of axioms and theorems of considered deontic theories with axioms and theorems of deontic propositional logics indicates that a number of properties of obligation, prohibition and permission are similarly defined in both approaches.

In particular, the basic theorems of legal event theories, i.e., T1–T7, have close equivalents in deontic propositional logics.

In turn, although theorems T8–T12 have equivalents in deontic propositional logics, their predecessor indicates that successive relations occur only for legal events, not for any events.

For example, T12

$$\forall \mathbf{x} \text{ (LEV (x)} \rightarrow \text{(FOR (x)} \star \text{ "PER (x)))}$$

is a counterpart to the definition of prohibition in propositional logics:

$$\mathbf{^F p} =\_{\mathbf{def}} \mathbf{\top} \mathbf{P} \mathbf{p} \mathbf{-}$$

Interestingly, in none of the four theories of legal events under consideration have we a counterpart of the definition of obligation in propositional logics,

$$\mathcal{O} \text{ } \mathcal{P} \equiv\_{\text{def}} \mathcal{T} \mathcal{P} \text{ } \mathcal{T}\_{\mathcal{P}'} $$

which is based on a definition from modal (aletic) logics:

$$
\Box \mathbf{P} \equiv\_{\text{def}} \Box \diamond \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P} \cdot \Box \mathbf{P}
$$

This is because the expression " P p" has no equivalent in any of these theories. However, this is not the case in theory of acts, where T19

$$\forall \mathbf{y} \,\,\mathbf{z} \,\,\mathbf{(y} \neq \mathbf{z} \to (\text{OBL} \ (\mathbf{x}, \mathbf{y}) \to \ \top \text{PER } (\mathbf{x}, \mathbf{z})))$$

is a counterpart of the aforementioned definition of obligation in propositional logics:

$$\mathcal{O} \text{ } \mathcal{P} =\_{\text{def}} \mathcal{T} \mathcal{P} \text{ } \mathcal{T} \mathcal{P} \text{ } \mathcal{P} \text{ }$$

Although, of course, the following proposition is not an AAPOF's theorem:

> ∀ x y z (y - z → (OBL (x, y) Ɂ PER (x, z))).

Further, T16

$$\forall \mathbf{x} \,\,\mathbf{y} \,\,\mathbf{z} \,\,(\mathbf{y} \neq \mathbf{z} \to \mathbf{J} \,\,(\mathbf{OBL} \,(\mathbf{x}, \mathbf{y}) \wedge \mathbf{OBL} \,(\mathbf{x}, \mathbf{z})))$$

is a counterpart of the theorem

$$\Box(\mathcal{O}\_P \land \mathcal{O}\_P \mathsf{I}\_P),$$

while T20

$$\begin{array}{c} \mathsf{V} \text{ x y z (ACT (x, y) \land ACT (x, z) \land y \neq z \land \forall w (ACT (x, w) \rightarrow (w = y \lor w = z)) \rightarrow \\ \mathsf{T} \text{ (FOR (x, y) \land FOR (x, z)))} \end{array}$$

is a counterpart of the theorem

 (F p ∧ F p).

In turn, T22

$$\begin{array}{c} \mathsf{V} \text{ x y z (ACT (x, y) \land ACT (x, z) \land y \neq z \land \forall w (ACT (x, w) \rightarrow (w = y \lor w = z)) \rightarrow } \\ \qquad \qquad \qquad \qquad \qquad \text{(PER (x, y) \lor PER (x, z)))} \end{array}$$

is a counterpart of the theorem

$$\mathsf{P}\_{\mathsf{P}} \mathsf{P} \lor \mathsf{P} \top \mathsf{P} \cdot$$

On the other hand, the A12 axiom

$$\forall \mathbf{x} \,\mathbf{x}\_1 \,\mathbf{y} \,\mathbf{y}\_1 \,(\mathbf{x}\_1 \,\varepsilon \ge \mathbf{x} \land \mathbf{y}\_1 \,\varepsilon \,\mathbf{y} \to (\text{OBL} \,(\mathbf{x}, \mathbf{y}) \to (\text{ACT} \,(\mathbf{x}\_1, \mathbf{y}\_1) \to \text{OBL} \,(\mathbf{x}\_1, \mathbf{y}\_1))))$$

is a distant counterpart of the theorem

O (p ∧ q) → O p.

In turn, axiom A15

$$\forall \mathbf{x} \,\, \mathbf{x}\_1 \,\, \mathbf{x}\_2 \,\, \mathbf{y} \,\, \mathbf{y}\_1 \,\, \mathbf{y}\_2 \,\, (\mathbf{x} = \mathbf{x}\_1 + \mathbf{x}\_2 \wedge \mathbf{y} = \mathbf{y}\_1 + \mathbf{y}\_2 \rightarrow \text{(OBL (x}\_1, \mathbf{y}\_1) \wedge \text{OBL (x}\_2, \mathbf{y}\_2) \rightarrow \text{OBL (x, y))})$$

is a distant counterpart of the theorem

$$\mathcal{O} \,\mathsf{p} \wedge \mathsf{O} \,\mathsf{q} \to \mathsf{O} \,\mathsf{(p} \wedge \mathsf{q}) \cdot$$

Similarly, axiom A13

$$\forall \mathbf{x} \,\,\mathbf{x}\_1 \,\,\mathbf{y} \,\,\mathbf{y}\_1 \,\,(\mathbf{x}\_1 \,\,\varepsilon \,\mathbf{x} \wedge \mathbf{y}\_1 \,\,\varepsilon \,\mathbf{y} \to (\text{PERR} \,(\mathbf{x}, \mathbf{y}) \to (\text{ACT } (\mathbf{x}\_1, \mathbf{y}\_1) \to \text{PERR } (\mathbf{x}\_1, \mathbf{y}\_1))))$$

is a distant counterpart of the theorem

$$\mathbb{P}\left(\mathbf{p}\wedge\mathbf{q}\right)\to\mathbb{P}\left.\mathbf{p}\cdot\mathbf{q}\right|$$

While the axiom A16

$$\forall \mathbf{x} \,\,\mathbf{x}\_1 \,\,\mathbf{x}\_2 \,\,\mathbf{y} \,\,\mathbf{y}\_1 \,\,\mathbf{y}\_2 \,\,(\mathbf{x} = \mathbf{x}\_1 + \mathbf{x}\_2 \wedge \mathbf{y} = \mathbf{y}\_1 + \mathbf{y}\_2 \to \text{(PER (x}\_1, \text{y}\_1) \wedge \text{PER (x}\_2, \text{y}\_2) \to \text{PER (x, y))}$$

is a distant counterpart of the theorem

$$\mathbf{P} \cdot \mathbf{P} \land \mathbf{P} \cdot \mathbf{q} \to \mathbf{P} \text{ (}\mathbf{p} \land \mathbf{q}\text{)}\text{.}$$

Similarly, the A14 axiom

$$\forall \mathbf{x} \,\,\mathbf{x}\_1 \,\,\mathbf{y} \,\,\mathbf{y}\_1 \,\, (\mathbf{x}\_1 \,\,\varepsilon \,\mathbf{x} \wedge \mathbf{y}\_1 \,\,\varepsilon \,\mathbf{y} \to (\text{FOR } (\mathbf{x}, \mathbf{y}) \to (\text{ACT } (\mathbf{x}\_1, \mathbf{y}\_1) \to \text{FOR } (\mathbf{x}\_1, \mathbf{y}\_1))))$$

is a distant counterpart of the proposition

$$\vdash (\mathbf{p} \land \mathbf{q}) \to \mathbf{F} \,\mathbf{p} \cdot \mathbf{p} \,\mathrm{d}\mathbf{q}$$

While the axiom A17

$$\forall \mathbf{x} \,\,\mathbf{x}\_1 \,\,\mathbf{x}\_2 \,\,\mathbf{y} \,\,\mathbf{y}\_1 \,\,\mathbf{y}\_2 \,\,(\mathbf{x} = \mathbf{x}\_1 + \mathbf{x}\_2 \wedge \mathbf{y} = \mathbf{y}\_1 + \mathbf{y}\_2 \to \left(\text{FOR}\left(\mathbf{x}\_1, \mathbf{y}\_1\right) \wedge \text{FOR}\left(\mathbf{x}\_2, \mathbf{y}\_2\right) \to \text{FOR}\left(\mathbf{x}, \mathbf{y}\right)\right)\,,$$

is a distant counterpart of the proposition

$$\mathcal{F}\mathbf{p} \wedge \mathcal{F}\mathbf{q} \to \mathcal{F}(\mathbf{p} \wedge \mathbf{q}) \cdot \mathbf{p}$$

As can be seen, the axioms and theorems of the deontic theories constructed above are usually not the exact equivalents of theorems of deontic propositional logics. They reflect additional restrictions that are necessary for expressing obligation, prohibition and permission in accordance with intuition, but which are inexpressible in propositional logics.

## **5. Conclusions**

Due to the discussed restrictions, the presented systems avoid the non-intuitive properties of propositional deontic logics.

Firstly, deontic sentences do not apply to all domains. They are sentences about legal events, and in particular about acts.

Secondly, in the presented systems we have no equivalents of many non-intuitive sentences of propositional deontic logics, such as those considered in the introduction:

$$\begin{array}{c} \bullet \ (\mathsf{P} \ (\mathsf{p} \land \mathsf{q}) \to \mathsf{q}) \to (\mathsf{O} \ (\mathsf{p} \land \mathsf{q}) \to \mathsf{O} \ \mathsf{q}), \\\\ \bullet \ (\mathsf{p} \to (\mathsf{p} \lor \mathsf{q})) \to (\mathsf{O} \ \mathsf{p} \to \mathsf{O} \ (\mathsf{p} \lor \mathsf{q})). \end{array}$$

It is a consequence of the accepted limitation that, in the presented systems, deontic sentences are sentences about legal events, and not sentences about any states of a ffairs.

Thirdly, it is also noteworthy that—for obvious reasons—in the deontic theories presented above, not even far counterparts of propositions that would include iterations of deontic operators exist.

Fourthly, the presented systems have no equivalents to the paradoxical statements of propositional deontic logics such as those considered in the introduction:

$$
\mathcal{O} \not\supset \mathcal{O} \not\supset \mathcal{O} \not\supset \mathcal{q} \mathcal{N}
$$

$$
\mathcal{F} \not\supset \mathcal{p} \to \mathcal{O} \not\supset \mathcal{q} \mathcal{N}
$$

Once again, it is a consequence of the accepted limitation that, in the presented systems, deontic sentences are sentences about legal events, and not sentences about any states of a ffairs.

In addition, some axioms and theorems of the deontic theories presented above do not have counterparts in propositional logics at all, and at the same time reflect important intuitions related to deontic modalities. Examples include the A4, A5 and A6 axioms and some theorems obtained with the help of these axioms.

Furthermore, thanks to Wolniewicz's situation ontology, the presented systems are based on a clear concept of deontic modalities: orders, bans and permits are simply sets of legal events.

In the presented approach, a distinction is also made between the deontic properties of any legal events and the deontic properties of acts. The former are described in AEPF, AEPOF, AEPOFI. The latter are expressed, e.g., by axioms A4 to A6 and A7 to A17 of the AAPOF system. Axioms such as A7 to A17 of the AAPOF system also show that it is possible to formally consider the relations between an act and its parts, which is important for the legal applications of deontic logics.

All this leads to the conclusion that deontic theories built on the first-order predicate logic and inspired by Wolniewicz's situation ontology are worthy of attention and development.

**Funding:** This research received no external funding.

**Acknowledgments:** I would like to thank Kazimierz Trz ˛esicki for encouraging me to write this article. I would also thank all the appointed reviewers of the article for their valuable remarks and suggestions, and my son Jakub Malec for the first review of the article and his suggestions.

**Conflicts of Interest:** The author declares no conflict of interest.
