**Proof.**


$$\begin{array}{llll} 1. & \vdash\_{\mathsf{IK}\_{\mathsf{t}}} (H\varphi \to \neg P\neg\varphi) \to (P\neg\varphi \to \neg H\varphi) & & \text{axiom 1, }\\ 2. & \vdash\_{\mathsf{IK}\_{\mathsf{t}}} P\neg\varphi \to \neg H\varphi & & 1, \text{case (b).MP.} \end{array}$$

The proofs of the cases (d), (e) and (f) are similar, so we skip them.

## **10. Summary**

Temporal logic systems can be built in a variety of ways. They can be based on classical logic, but also, as we presented in this article, based on intuitionistic logic. The discussed systems are minimal systems, which means that no properties have been imposed on the time structure. One can, however, enrich these systems with additional specific axioms, build a temporal logic systems adequate to various time structures, e.g., reflexive, symmetrical, transitive, linear or branched. However, while in tense logic systems based on classical logic, the thesis of logical determinism can be rejected by modifying the structure of time and assuming, as a semantic time, a branching time into the future, in tense logics based on intuitionistic logic, modification of the time structure is not necessary. Formulas expressing the thesis of logical determinism are not theses of the minimal system because of its basic properties, no matter what time structure is adopted as a semantic time.

There is a relationship between the systems being discussed. Each thesis of the **IKt** system is also the thesis of **Kt**, so:

## **IKt** ⊂ **Kt**.

In addition, as we have shown in this article, intuitionistic temporal logic can be used to represent knowledge that changes over time. Intuitionistic logic and knowledge are closely related. This epistemic approach is the epicenter of Brouwer's intuitionistic explanation of truth as provability by an ideal mathematician, or more generally by an ideal cognitive subject. Kripke's intuitionistic models are good tools for modelling the evolutionary learning process of the cognitive subject.

The intuitionistic temporal logic **IKt** has many advantages when we understand it as a formal tool for the logical representation of knowledge changing over time. Knowledge is implemented in this system on a semantic level in a natural way. In a natural way, by means of a set of partially ordered states of knowledge, the way of acquiring knowledge is also modeled. However, this system has some imperfections and limitations. The first is the limited applicability of this system. Due to the adopted monotonicity of knowledge, i.e., a fact recognized in a given state of knowledge is known in all states of knowledge with a not lower level of knowledge, this system is a good tool for a modelling of mathematical or logical knowledge that changes over time.

**Conflicts of Interest:** The authors declare no conflict of interest.
