*3.16. Theorem C I I* |→ *C I*

**Proof.** Based on (25) and (15), (17) we can see that {*CII*}⊂{*C I*}. Strictly speaking, the first-principle model is equal to the special result-principle model {*CII*} = {*C I*}0<*ηm*<<sup>1</sup> for *η<sup>m</sup>* = *ηC*.

However, we will show directly that the principle *CII* applied to the processes of heat flow (*q*, 0, *q*) allows heat to flow downwards ↓ 0 ↓, but does not allow spontaneous flows upwards ↑ 0 ↑:

$$\frac{-q}{T\_1} + \frac{q}{T\_2} \ge 0 \quad \Rightarrow \quad \frac{T\_1 - T\_2}{T\_1 T\_2} q \ge 0 \quad \Rightarrow \quad q \ge 0. \tag{36}$$

This condition expresses the essence of the *C I* principle (assuming *T*<sup>1</sup> > *T*2), so the proof need not include other processes. On the other hand, the generality of the formal implication is secured by the inclusion of the *CII* principle model in the *C I* principle model.

Thus, six formal implications were proved and six other formal implications were rebutted. The proof was made double: the first method was based on the inclusion relationship of models, and the second method was direct proof (not by contradiction). The rebuttals, on the other hand, were threefold: the first method showed the absence of inclusion relationship of models, the second method was based on one or two counterexamples, and the third method indicated errors in the alleged pseudo-proof by contradiction.

#### **4. Results**

The proved and disproved implications of the four principles (*C*0, *C I*, *K*, *CII*) are summarized in Table 3. Thus, only the two principles *C*0 and *CII* turned out to be equivalent. These principles are also stronger than the *C I* and *K* principles that follow from the previous ones. Thus, what Smoluchowski postulated in 1914 is being implemented: *"We call the Carnot principle as the second law of thermodynamics since Clausius time"* [41,42]. It is worth noting that the equivalence of the principles *C*0 and *CII* occurs in the conceptual system of thermodynamics assuming convexity and completeness of models. These additional rules were needed to give general meaning to the principles *C*0, *C I*, and *K*. The convexity assumption in the sense of internality of addition is implicitly commonly used, and the completeness rule only extends this assumption. The *CII* principle, on the other hand, does not need these additional rules.

**Table 3.** The resulting matrix of mutual formal implications (or the lack of them) of the four principles: Carnot, Clausius I, Kelvin, Clausius II. Each implication was proved or disproved, and was additionally analyzed on the basis of the structure of the models (sets of processes).


Another way to show relationships between principles is to represent relationships between sets of all their processes belonging to all models. It has been marked, in a simplified manner, in the diagram of Figure 5. We see that the largest set of possible processes is allowed by the Kelvin principle. The only process that is not covered by the Kelvin principle, and which the Clausius I principle does not exclude, is a *perpetuum mobile* type II. Both the Clausius I principle and the Kelvin principle allow for a *perpetuum mobile* type III and, thus, differ significantly from the principles of Carnot and Clausius II. The Kelvin principle additionally includes processes of spontaneous flow of heat upwards (towards a higher temperature).

**Figure 5.** Resulting diagram showing the relationship between the fundamental principles crucial to the II law of thermodynamics. Additionally, the location of the fictitious *perpetuum mobile* processes of three types was marked, along with the heat flow process "upwards" *π*↑. *Perpetuum mobile π<sup>I</sup>* goes beyond a dozen considerations, and it would be even more difficult to locate *π*<sup>0</sup> here. On the other hand, the physical process (diagram), consistent with the II law of thermodynamics (with principles *C*0 or *CII*), was denoted by *d*.

The relationships between the principles are also shown in Figure 6 in a "logical square" design. One diagonal of the square indicates equivalence, and the other diagonal is not marked, as it is neither equivalence nor implication.

**Figure 6.** The resulting "logical square" of the implications of the four principles related to the II law of thermodynamics. This law is determined equivalently by the principles on the main diagonal. On the other hand, non-diagonal principles result from diagonal ones, but they are weaker and do not result from each other.

#### **5. Conclusions**

Of the four principles (Carnot, Clausius I, Kelvin, Clausius II) pretending to formulate the II law of thermodynamics, only the Carnot and Clausius II principles turned out to be equivalent and strong principles, that forbid the decrease of the entropy of a heat-insulated system. The principles of Clausius I and Kelvin turned out to be less restrictive principles that say little about the reversibility of processes, and even allow impossible processes. These principles are true, but they are also so obvious that their predictions contribute very little. Moreover, the principles of Clausius I and Kelvin turned out to be, strictly speaking, independent (they do not imply each other). However, if we omit the *perpetuum mobile* of the II kind, it can be said, that the Clausius I principle is stronger than the Kelvin principle.

The most elegant formulation of the II law of thermodynamics is the entropic principle of Clausius II. The only competing formulation of this law, as it turns out, is the equivalent Carnot principle. Carnot principle is often *implicite* treated as the equivalent of the II law of thermodynamics, but *explicite* was not included in the formulation of this principle. Instead, the formulations of the II law mistakenly included the Clausius I principle and the Kelvin principle (as well as the Caratheodory principle). It can be said that it has not been noticed that the Carnot condition *η* ≤ *η<sup>C</sup>* is stronger than the Kelvin condition in terms of Ostwald *η* < 1. The II law of thermodynamics excludes not only the *perpetuum mobile* type II, but also the *perpetuum mobile* type III, that is a heat engine with an efficiency greater, than that of the Carnot engine. In the context of Carnot, who was a historical pioneer of the II law of thermodynamics, the distractor was the lack of knowledge of the I law of thermodynamics in his time. Carnot used the concept of indestructible heat (*caloric*), but in the sense of heat, not energy. Nevertheless, it did not prevent him from formulating correct conclusions regarding the II law of thermodynamics.

So how to explain the incorrect proof of the equivalence of Clausius I and Kelvin principles with the II law of thermodynamics? The first cause and distractor may be the frequent consideration of only Clausius I and Kelvin principles, regardless of the entropy formulation (Clausius II), or even more so the Carnot principle. In this approach, we lose the stronger context of the II law of thermodynamics. This, however, does not explain the alleged proof of a substantially incorrect implication of *K* |→ *C I*. Here a second cause appears, based on the formal implication |→ distractor, which is a material implication ⇒. The formal implication requires more subtle methods of proving, than the material implication. The main difference is the necessity to use the universal quantifier in formal implication, i.e., the necessity to check all the processes belonging to the model of the principle, which is the antecedent of the implication.

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

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Many thanks to Ludomir Newelski for his consultation on advanced logic in the context of formal implication and model theory. I also thank Martin Bier for verifying the thermodynamic aspects of the work.

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