**Proof.**


$$\begin{aligned} \text{15. } \begin{aligned} \text{15. } \begin{cases} \text{G. } \begin{aligned} \text{G. } \begin{pmatrix} \text{G. } \begin{pmatrix} \varphi \end{pmatrix} \rightarrow \begin{pmatrix} F \ \begin{pmatrix} \varphi \end{pmatrix} \rightarrow \begin{pmatrix} \text{G. } \begin{pmatrix} \varphi \end{pmatrix} \end{pmatrix} \end{aligned} \\ \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \begin{aligned} \text{16. } \end{aligned} \end{aligned} \\ \end{aligned} \\ \end{aligned} \end{aligned} \end{aligned} \end{aligned} \end{cases} \end{aligned} \end{cases} \end{aligned} \right) \end{aligned} \rightlefth{\text{A1} }$$

**Lemma 11. IKt***P* (*ϕ* → *ψ*) → (*<sup>H</sup>ϕ* → *Pψ*)

Proof analogous to the proof of the previous lemma. We will show that the " new " *A*5 and *A*5 axioms are we can derive from the' 'old' '8 and 8 axioms.

**Lemma 12.** *G*¬*ϕ* → ¬*Fϕ IKt Fϕ* → ¬*G*¬*ϕ*

**Proof.**

> 1. **IKt** *G*¬*ϕ* → ¬*Fϕ* assumption 2. **IKt** (*<sup>G</sup>*¬*ϕ* → ¬*Fϕ*) → (*<sup>F</sup>ϕ* → ¬*G*¬*ϕ*) axiom 1 3. **IKt** (*<sup>F</sup>ϕ* → ¬*G*¬*ϕ*) 1,2,MP

It is likewise proved that:

#### **Lemma 13.** *H*¬*ϕ* → ¬*Pϕ* **IKt***Pϕ* → ¬*H*¬*ϕ*

Thus, we have shown that the given axioms are equivalent. In further considerations we will use "new" axiomatics of **IKt**.

#### **8. The Adequacy of IKt Relative to Modified Semantics**

The natural question is the question about the relationship between modified semantics and the assumed set of axioms for **IKt**.

**Theorem 14.** *The* **IKt** *axioms are true in any model, and the* **IKt** *inference rules are infallible.*

**Proof.** We will prove only *A*2, *A*4 axioms and *RH* rule. Proofs for the other rules and axioms is carried out in analogous manner.

**A2'** For any M, *mi*(<sup>∈</sup> <sup>M</sup>), and *<sup>t</sup>*(∈ *Ti*): M, *mi*, *t* |= *H* (*ϕ* → *ψ*) → (*<sup>H</sup>ϕ* → *<sup>H</sup>ψ*).

Suppose for some M, *mi* (∈ M) and *<sup>t</sup>*(∈ *Ti*) : M, *mi*, *t H* (*ϕ* → *ψ*) → (*<sup>H</sup>ϕ* → *<sup>H</sup>ψ*).

Therefore, from the condition of the truth for the implications, there is a state of knowledge *mj*, *mi* ≤ *mj*, such that:

$$\left|\mathfrak{M},m\_{\mathfrak{j}},\mathfrak{t}\right| = H\left(\mathfrak{q} \to \mathfrak{\psi}\right),\tag{23}$$

$$\text{\textquotedblleft \textquotedblright} \mathcal{M}, m\_{\mathfrak{j}}, \mathfrak{t} \not\models H\!\!\!/ \mathfrak{q} \longrightarrow H\!\!\!/ \mathfrak{q} \tag{24}$$

From (24) and the condition of the truth for the implications, in a certain state of knowledge *mk*, with a level of knowledge not lesser than the level of knowledge of the state *mj*, i.e., such that *mj* ≤ *mk*:

$$\mathfrak{M}, m\_k, t \rightleftharpoons \varPi \mathfrak{q},\tag{25}$$

$$\text{\textquotedblleft } \mathfrak{M}, m\_k, t \not\vdash \!H\psi . \tag{26}$$

> From (25) and the condition of the truth for the *H* operator we get:

for any state of knowledge *ml* such that *mk* ≤ *ml* and forany *t*1 ∈ *Tl*suchthat*t*1*Rlt*holds:M,*ml*,*t*1|= *ϕ*.(27)

From (26) and the condition of the truth for the *H* operator, there is a state *mp* sucht that *mk* ≤ *mp*) and there is a moment *t*2 ∈ *Tp* such that *<sup>t</sup>*2*Rpt*, in which:

$$2\mathfrak{N}\_{\prime}m\_{p\prime}t\_2 \nleq\psi. \tag{28}$$

Because *mk* ≤ *mp* and *<sup>t</sup>*2*Rpt* therefore from (27) we have that at the moment *t*2 holds M, *mp*, *t*2 |= *ϕ*. Hence, from (28) and the condition of the truth of the implications we get:

$$2\mathfrak{N}, m\_{\mathfrak{p}^\vee} t\_2 \upharpoonright \mathfrak{q} \to \mathfrak{q} . \tag{29}$$

From (23) and the condition of the truth of the operator *H* we have:

for any *mr* such that *mj* ≤ *mr* and

> for any *t*3 ∈ *Tr* such that *t*3*Rrt* holds : M, *mr*, *t*3 |= *ϕ* → *ψ*. (30)

Because: *mj* ≤ *mk*, *mk* ≤ *mp*, so from the transitivity of the relationship ≤ we ge<sup>t</sup> *mj* ≤ *mp*. The moment *t*2 is such that *<sup>t</sup>*2*Rpt*. Therefore, from (30) we have:

$$
\mathfrak{M}, m\_{p\prime} t\_2 = \varphi \to \psi.
$$

This is contrary to 29.

**A4'** For any M, *mi* (∈ M) and *<sup>t</sup>*(∈ *Ti*): M, *mi*, *t* |= *H* (*ϕ* → *ψ*) → (*<sup>P</sup>ϕ* → *<sup>P</sup>ψ*).

Suppose for some M, *mi*(<sup>∈</sup> M) and *<sup>t</sup>*(∈ *Ti*) M, *mi*, *t H* (*ϕ* → *ψ*) → (*<sup>P</sup>ϕ* → *<sup>P</sup>ψ*).

Thus, from the condition of the truth of the implications, in a certain state of knowledge *mj*, such that *mi* ≤ *mj* we have:

$$\left| \mathfrak{M}, m\_j, t \right| = H \left( \varphi \to \psi \right), \tag{31}$$

$$P\mathfrak{M}, m\_{\mathfrak{j}}, t \not\models P\mathfrak{q} \to P\mathfrak{q}.\tag{32}$$

From (32) and the condition of the truth of the implications, in some state of knowledge *mk*, such that *mj* ≤ *mk* :

$$\mathcal{D}\mathfrak{A}, m\_k, t \rightleftharpoons P\,\mathfrak{q},\tag{33}$$

and

$$2\mathfrak{N}, m\_k, t \not\equiv P\psi. \tag{34}$$

From (33) and the condition of the truth of the *P* operator we have:

$$\text{If there exists } t\_1 \left( \in T\_k \right), t\_1 R\_k t \text{ such that } \mathfrak{M}, m\_k, t\_1 \left| = \wp. \tag{35}$$

From (34) and the condition of the truth of the *P* operator we obtain:

$$\text{does not exist moment of time } t\_2 \left( \in T\_k \right), t\_2 \mathbb{R}\_k t\_\prime \text{ such that } \mathfrak{M}, m\_k, t\_2 \succeq \psi. \tag{36}$$

> Let us consider the moment *t*1 satisfying (35). Because *t*1*Rkt*, so from (36) we have:

$$2\mathfrak{N}, m\_k, t\_1 \nleq \psi. \tag{37}$$

If M, *mk*, *t*1 |= *ψ*, it would be against (36).

From (35), (37) and the condition of the truth of the implications, we ge<sup>t</sup> that M, *mk*, *t*1 *ϕ* → *ψ*. From (31) and condition the truth of the operator *H* we have:

for any *ml* such that *mj* ≤ *ml* and for any *t*3 (∈ *Tl*) such that *t*3*Rlt* : M, *ml*, *t*3 |= *ϕ* → *ψ*. (38)

Because *mj* ≤ *mk*, *t*1*Rkt* and M, *mk*, *t*1 (*ϕ* → *ψ*), so we ge<sup>t</sup> a contradiction with (38). **RH** If M |= *ϕ*, then M |= *Hϕ*.

Let us assume that M |= *ϕ*. So for any *mi* and for any *<sup>t</sup>*(∈ *Ti*) holds M, *mi*, *t* |= *ϕ*. So especially for any *<sup>t</sup>*1(<sup>∈</sup> *Ti*) such that *t*1*Rit* : M, *mi*, *t*1 |= *ϕ*. So for any *<sup>t</sup>*(∈ *Ti*) holds M, *mi*, *t* |= *Hϕ*. Because we were considering any *mi*, therefore M |= *Hϕ*.

Adequacy **IKt** with respect to modified semantics was demonstrated by Surowik [**?** ].

**Theorem 15.** Σ **IKt** *ϕ iff* Σ |=**IKt** *ϕ*.

The proof of this theorem is similar to the proof of the adequacy theorem demonstrated by Ewald in [**?** ].

#### **9. Mutual Undefinability in IKt Operators** *H***,** *P* **and** *G***,** *F*

We will now prove theorems that show some special properties of the **IKt** system, essentially distinguishing this system from systems built on the basis of classical logic. For the formula to be the tautology of the **IKt** system, it needs to be true at any time, in any state of knowledge. To show that a formula is not true, it is enough to indicate the state of knowledge and the moment in which this formula is not true.

We will show that some relationships between the operators *H* and *P* and *G* and *F* holds in the system **Kt** but do not occur between the equivalents of these operators in the system **IKt**.

## **Theorem 16.**

