On a Proof of the Inconsistency of The Classical Propositional Calculus and The Intuitionistic Propositional Calculus

1. Introduction

The issue of consistency or inconsistency of a given formal system, is the most fundamental issue concerning such system.

The literature devoted to different aspects of consistency and/or inconsistency and/or contradictions in and/or of formal logical systems or in and/or of mathematics or other sciences, is very rich. See for e.g., Refs. (Alegre 2025)—(Beirlaen etal. 2013), (van Bendegem 2002)—(Bueno 2002), (Carbone and Semmes 1997)—(Carnieli and Malinowski 2018), (Coddington 2025)—(Curry 1942), (Dubois and Prade 2015)—(Engesser etal. 2009), (Estrada-González and del Rosario Martínez-Ordaz 2018), (Fazio etal. 2025)—(Goddard 2000), (Grant 2020)—(Grygiel 2010), (Heller 2018), (Hernández and Hernández-Quiros 2024)—(Hohol 2010), (Jaśkowski 1948), (Kahle 2015)—(Krajewski 2007), (Ligęza and Nalepa 2009)—(Malinowski G. 2009), (Marcelino etal. 2018)—(Omori and Wansing 2025), (Paleo 2018)—(Picollo 2023), (Pietryga 2004), (Priest 2002)—(Raftery 2013), (Raspa 1999), (Robles 2012), (Rodríguez-Consuegra 2025)—(Smith etal. 2023), (Stępień and Stępień 2017)—(Tanaka and Girard 2023), (Tennant 2024)—(Walicki 2024), (Weber and DeClercq 2002)—(Wiśniewski 2021), (Woods 2014), (Wybraniec-Skardowska 2016), (Yu 1992)—(Żandarowska 1966).

Probably, the most known example of an inconsistent system, is Frege’s system presented in II volume of his “Grundgesetze der Arithmetik”. Its inconsistency was proved by Russell in 1903 (see Ref. (Marciszewski 1981), cf. (Besler 2016)). On the other hand, Russell did not understand Gödel’s results, which in his interpretation, implied inconsistency of Arithmetic (Priest 2022). Another well-known case of the system, which inconsistency had been proven, is the original Church’s system of $\lambda$-calculus—its inconsistency was proved by Kleene and Rosser in 1935, (Zuloaga 2025).

One needs the Classical Propositional Calculus CPC in order to construct the Classical Calculus of Quantifiers (Classical Calculus of Predicates, first-order logic). This last one is necessary to construct the Classical Functional Calculus. Classical Functional Calculus is needed to formalize the Peano Arithmetic System. So, the significance of the issue of consistency or inconsistency of the classical propositional calculus, is obvious: in the case of inconsistency of CPC, we have spreading of “the infection of inconsistency” to Peano Arithmetic System—this is inconsistent, too.

As one can read in (McFarlane 2021): “There is no good reason to assume that mathematics must be consistent. If mathematics concerns a supersensible realm of objects, why should we assume they’re like ordinary empirical objects with respect to consistency? If it is a free human creation, why can’t it be inconsistent? However, classical logic forces a mathematical theory to be either consistent or trivial (meaning that it entails everything).”

One of the non-classical propositional calculus is the Intuitionistic Propositional Calculus. The main difference between this system and Classical Propositional Calculus CPC, is such that in this last one, the principle of indirect proof (proof by contradiction), has been rejected. In accordance with intuitionistic point of view, existence of a given object can be accepted, provided that there exists a proof, which establishes this object (Polak 2014).

We can also tell on an impact of inconsistency (in a broad sense), not only on logical systems or on the branches of mathematics, but also on philosophy or semantics (Patterson 2007), on some aspects concerning belief (Brożek 2001), artificial intelligence (Gabbay and Hunter 1991), functionality of mind (Grygiel 2010, Hohol 2010), or psychology (Rudnicki 2020). There in Ref. (Engesser etal. 2009) some inconsistency of Birkhoff-von Neumann quantum logic was discussed.

In 2010 Voevodsky delivered a talk entitled “What if current foundations of mathematics are inconsistent?” (Voevodsky 2010), where he considered the issue of probably inconsistency of first-order Arithmetic System. In 2011 Nelson claimed he had proved inconsistency of the Arithmetic System (Nelson 2011), (Chow 2018). However, soon Tao and Tausk found independently an error in Nelson’s proof mentioned above (Chow 2011).

In (Durante 2019) some aspect of the (hypotetic) inconsistency of intuitionistic logic has been analysed.

In (Walicki 2024) among others, some situations, when certain paradoxes cause inconsistencies in formal logical systems, were considered.

In 2009 one of the authors of this paper (TJS) found a first sign telling on inconsistency of the classical logic, when he was checking with ŁTS their proof of the consistency of Peano Arithmetic System PA (done by using only resources of PA), and later it turned out that these both topics: proving consistency of Peano Arithmetic System PA and inconsistency of the Classical Propositional Calculus CPC, are strictly connected (a short discussion of this issue is presented in the Section 5 “Conclusions” of this paper). Within next years certain versions of proof of the inconsistency of the Classical Propositional Calculus were constructed by TJS and ŁTS.

The aim of this paper is to present:

• the proof of inconsistency of metatheory (metalogic) of the Classical Propositional Calculus (the zero-order logic, the classical propositional logic)
• the proof of (meta)inconsistency of this last one
• the proof of inconsistency of the Classical Propositional Calculus in the absolute sense (Post’s sense)
• the proof of inconsistency of the Intuitionistic Propositional Calculus in the absolute sense (Post’s sense)

This paper has the following structure. There in the section 2, a notation is introduced and certain well-known notions and some well-known theorems are repeated. The next section includes a proof of a Lemma (Lemma 3.1). Proofs of The Main Results are placed in the section 4. The section 5 is devoted to some conclusions.

The inconsistency of the Classical Propositional Calculus was announced in Refs. (Stępień and Stępień 2017)—(Stępień and Stępień 2020). The section 5 is devoted to some discussion and conclusions. There in a preprint (Stępień 2023) was published another proof of inconsistency of the Classical Propositional Calculus. However, it turned out that version of this proof needed to be changed.

In November 2025 such changed version of this proof was submitted to Journal of Mathematics and System Science. The proof of inconsistency of the Classical Propositional Calculus included in this paper (consisting on the proof of Lemma 3.1 and the proofs of The Main Results), differs from those both versions and includes some two new results (mentioned in the points C and D above).

2. Preliminaries

The symbols: $\to , ~, \vee , \wedge , \equiv$ denote the connectives of: implication, negation, disjunction, conjunction and equivalence, respectively. $\mathcal{N}=\{\mathrm{1,2},\dots \}$ denotes the set of all natural numbers.

Next,${ At}_0=\left\{p,q,r,\dots ,p_1,p_2,\dots ,q_1,q_2\dots ,r_1,r_2,\dots \right\}$ denotes the set of all propositional variables. The symbol $S_0$ denotes the set of all well-formed formulas, which are built in the usual manner from propositional variables by means of logical connectives (and parentheses, if they are necessary). We denote the well-formed formulas by Greek letters. So, $S_0=\{\alpha ,\beta ,\mathsf{\gamma },\delta ,\dots ,\lambda ,\dots ,\mathsf{\Phi },\dots ,\mathsf{\Psi },\mathsf{\Omega }\}$. $R_{S_0}$ denotes the set of all rules over $S_0$. The symbol ${\mathfrak{M}}_2$ denotes the classical two-valued matrix and $Z_2$ is the set of all formulas valid in the matrix ${\mathfrak{M}}_2$. The symbols $\Rightarrow ,\neg ,\mathbb{V},\&,\iff$ denote metasystem connectives: metaimplication, metanegation, metadisjunction, metaconjunction and metaequivalence, correspondingly. The symbols $\forall ,\exists$ denote the metalogical quantifiers: general and existential one, correspondingly. This is important to stress here that we assume for these quantifiers all laws of the Classical Calculus of Quantifiers (Pogorzelski 1975), and this last one is based on the Classical Propositional Calculus (CPC).

Next, $r_0$ is the symbol of Modus Ponens (detachment rule), in the classical propositional calculus. We can define this rule in the following way (Pogorzelski 1975): Definition 2.1. (of Modus Ponens) $\left(X,\beta \right)\in r_0\iff (\exists \alpha \in S_0)(X=\left\{\alpha ,\alpha \to \beta \right\})$ for any $X\subseteq S_0$ and $\alpha ,\beta \in S_0$. The simplest way of noting this rule is the scheme: $r_0: {\displaystyle \frac{\Psi \to \Omega ,\Psi }{\Omega }}$. Hence, $R_0=\left\{r_0\right\}$. We remind also the definition of substitution rule $r_{*}$. One can define this rule in two steps: at first, one describes operation of substitution, next one defines the rule of substitution (Pogorzelski 1994). Definition 2.2. (of operation of substitution) Let $e$ be the following mapping $e:{At}_0⟶S_0$, then we define the operation of substitution in the following way:

• $h^e\left(\alpha \right)=e\left(\alpha \right)$ for $\alpha \in {At}_0$
• $h^e\left(\alpha \to \beta \right)=h^e\left(\alpha \right)\to h^e\left(\beta \right)$
• $h^e\left(\alpha \wedge \beta \right)=h^e\left(\alpha \right)\wedge h^e\left(\beta \right)$
• $h^e\left(\alpha \vee \beta \right)=h^e\left(\alpha \right)\vee h^e\left(\beta \right)$
• $h^e\left(\alpha \equiv \beta \right)=h^e\left(\alpha \right)\equiv h^e\left(\beta \right)$
• $h^e\left(\sim \alpha \right)=\sim h^e\left(\alpha \right)$, for any $\alpha ,\beta \in S_0$.

The operation $h^e:S_0⟶S_0$ defined in such way, is determined uniquely by $e:{At}_0⟶S_0$ i.e., if $e_1=e_2$, then $h^{e_1}=h^{e_2}$ for arbitrary $e_1,e_2:{At}_0⟶S_0$.

Definition 2.3. (of substitution rule)

$$\left(X,\beta \right)\in r_{*}\iff \left[\right(\exists \alpha \in S_0\left)\right(\exists e:{At}_0⟶S_0)$$

$\left(X=\left\{\alpha \right\}\wedge \beta =h^e\left(\alpha \right))\right]$, for any $X\subseteq S_0$

and $\beta \in S_0$. A simpler version of this Definition (Pogorzelski 1994): $(\alpha ,\beta )\in r_{*}\iff \left(\exists e:{At}_0⟶S_0\right)\left(\beta =h^e\left(\alpha \right)\right)$. The scheme of the substitution rule is following: $\alpha /h^e\left(\alpha \right)$ for any $e:{At}_0⟶S_0$.

We recall also a very well-known definition and a very well-known theorem from set algebra (Ben-Ari 2018), cf. (Grzegorczyk 1974), (Rasiowa 1973):

Definition 2.4. Let $A$ and $B$ be sets. A is a subset of B (we denote this, as $A\subseteq B$), iff every element of $A$ is an element of $B$ i.e., $x\in A \to x\in B$.

Theorem 2.1. $A=B$ iff $A\subseteq B$ and $B\subseteq A$.

The formula $A\subset B$ denotes that $A\subseteq B$ and $A\ne B$. Then, of course,${ At}_0\subset S_0$. The couple $〈R,X〉$ is called as a system, whenever $R\subseteq R_{S_0}$, and $X\subseteq S_0$.

Thus, $〈R_0,Z_2〉$ denotes the system of the classical propositional calculus (Pogorzelski 1975), (Pogorzelski and Wojtylak 2008), where $Z_2$ is the set of all tautologies of the Classical Propositional Calculus. Next, $S_0^0=\{\phi \in S_0:\phi \notin Z_2 \& ~\phi \notin Z_2\}$. Hence, $S_0^0$ is the set of only such well-formed formulas that neither they nor their negations are the tautologies of the Classical Propositional Calculus (CPC). Obviously, this set is infinite. Next, we define $S_0^I$ as certain finite subset of $S_0^0$ such that there to $S_0^I$ belong certain conjunctions and certain disjunctions and negations of these conjunctions and these disjunctions, such that each propositional variable occurs in all these formulas at most once. For example, this set can be, as follows:

$S_0^I=\{p_1, ~p_1,p_2\wedge p_3,~\left(p_2\wedge p_3\right),q_1\vee q_2,$

$\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right), ~\left[\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right)\right],$

$\sim \left(p_6\vee p_7\right)\wedge q_5, ~\left[~\left(p_6\vee p_7\right)\wedge q_5\right],$

$\sim \left(q_1\vee q_2\right),q_6,~q_6,\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right),$

$~\left[\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right)\right]\},$

where $p_i,q_j\in {At}_0$ ($i,j\in \{1,\dots ,11\})$.

Let $Sb\left(X\right)$ denote the set of all substitutions of the formulas belonging to the set $X$ by expressions belonging to the set $S_0$, see (Pogorzelski 1975), (Pogorzelski 1994). In particular, ${Sb(A}_2)$ is the set of all subsitutions of the axioms of Classical Propositional Calculus i.e., this is a substitutional closure of the set $A_2$ by expressions belonging to the set $S_0$. There is also another version of the Classical Propositional Calculus i.e., $〈R_{0*},A_2〉$, where $R_{0*}=\{r_0,r_{*}\}$. The set $A_2$ is the set of axioms of Classical Propositional Calculus, and we write here its elements are given for e.g., in (Pogorzelski & Wojtylak 2008), cf. (Pogorzelski 1975, Pogorzelski 1994):

[1.]

$p\to (q\to p)$

[2.]

$\left(p\to \left(p\to q\right)\right)\to (p\to q)$

[3.]

$\left(p\to q\right)\to \left[\left(q\to s\right)\to \left(p\to s\right)\right]$

[4.]

$p\to p\vee q$

[5.]

$q\to p\vee q$

[6.]

$\left(p\to s\right)\to \left[\left(q\to s\right)\to \left(p\vee q\to s\right)\right]$

[7.]

$p\wedge q\to p$

[8.]

$p\wedge q\to q$

[9.]

$\left(p\to q\right)\to \left[\left(p\to r\right)\to \left(p\to q\wedge r\right)\right]$

[10.]

$p\to (~p\to q)$

[11.]

$(p\to ~p)\to ~p$

[12.]

$~~p\to p,$

where $p,q,r,s\in {At}_0$. The set of axioms of the Intuitionistic Propositional Calculus IPC is: $A_{int}=A_2\setminus \{~~p\to p\}$. Hence, there in IPC the equivalence: $~~p\equiv p$ is invalid (this is, of course, valid in CPC). Several examples of other unprovable laws in IPC (but, of course, provable in CPC), are (obviously, under assumption of consistency of IPC), as follows (Pogorzelski 1994):

[A.]

$(~p\to ~q)\to (q\to p)$

[B.]

$(~p\to p)\to p$

[C.]

$p\vee \sim p$

[D.]

$(p\to q)\to (~p\vee q),$

where $p,q\in {At}_0$. Analogically, ${Sb(A}_{int})$ is the set of all substitutions of the axioms of Intuitionistic Propositional Calculus i.e., this is a substitutional closure of the set $A_{int}$ by expressions belonging to the set $S_0$. Thus, $〈R_0,{Sb(A}_{int})〉$ denotes the system of the Intuitionistic Propositional Calculus (Pogorzelski & Wojtylak 2008).

Now we repeat some well-known properties of Tarski’s consequence operation (axioms of Tarski’s theory T of consequence operation), and some well-known definitions of: consistency in the traditional sense ($〈R,X〉\in {Cns}^T)$ and consistency in the Post’s sense ($〈R,X〉\in {Cns}^A)$. For any $X\subseteq S_0$ and $R\subseteq R_{S_0}$, $Cn(R,X)$ is the smallest subset of $S_0$, containing $X$, and closed under the rules belonging to $R$, where $R\subseteq R_{S_0}$ see Ref. (Pogorzelski 1975), cf. (Andrews 2002), (Marciszewski 1981), (Mirek 2021), (Öner etal. 2018), (Pogorzelski and Wojtylak 2008), (Rautenberg 2010), (Sher 2022), (Tarski 1956), (Wasilewska 2019), (Woleński 2002), (Wójcicki 1970), (Wójcicki 1984), (Wybraniec-Skardowska and Waldmajer 2011), (Żandarowska 1966). Then:

$a_1)X\subseteq Cn\left(R,X\right),$

$a_2)X\subseteq Y\Rightarrow Cn\left(R,X\right)\subseteq Cn\left(R,Y\right),$

$a_3)R\subseteq R^{\prime }\Rightarrow Cn\left(R,X\right)\subseteq Cn\left(R^{\prime },X\right),$

$a_4)Cn\left(R,Cn\left(R,X\right)\right)=Cn\left(R,X\right),$

$a_5)Cn\left(R,X\right)=\bigcup \{Cn\left(R,Y\right):Y\subseteq X\&\stackrel{̿}{Y}<{\mathrm{\aleph }}_0\}.$

Obviously (cf. Pogorzelski 1975, Pogorzelski & Wojtylak 2008): $Cn\left(R_0,Z_2\right)=Cn\left(R_{0*},A_2\right)=Cn\left(R_0,Sb\left(A_2\right)\right)$ and $Z_2=Cn\left(R_0,Z_2\right)$. Next, the Intuitionistic Propositional Calculus IPC is a subsystem of the Classical Propositional Calculus CPC i.e., (Pogorzelski & Wojtylak 2008): $Cn\left(R_0,Sb\left(A_{int}\right)\right)\subseteq Cn\left(R_0,Sb\left(A_2\right)\right)$. Of course, (see (Pogorzelski & Wojtylak 2008)): $Cn\left(R_0,Sb\left(A_{int}\right)\right)=Cn\left(R_{0*},A_{int}\right)$. In this paper we prove inconsistency of CPC and IPC.

We demonstrate now, how the rules: substitution and Modus Ponens work. Namely, they and consequence operation allow to get new theses (laws) of the given system (for e.g., Classical Propositional Calculus CPC), from other laws of such system. Let us take into account two well-known laws of CPC: $p\to (q\to p)$—simplification law and $~~p\to p$—strong double negation law.

The simplification law tells that each true proposition (sentence) is implicated by arbitrary proposition (sentence). They are both the theses of CPC, i.e., $p\to \left(q\to p\right), ~~p\to p\in Cn(R_{0*},A_2)$, where where $p,q\in {At}_0$. Since $~~p\to p$ as the law of CPC, is obviously also true proposition, then we can apply for this simplification rule. Namely, we make the following substitution in a): $r_{*} p/~~p\to p$ i.e., we substitute the double negation law for $p$ and we get (the double negation law appears now instead the “old” propositional variable $p$ in simplification law):

$\left(~~p\to p\right)\to \left[q\to \left(~~p\to p\right)\right]\in Cn(R_{0*},A_2)$, where $p,q\in {At}_0$. Hence, we can apply also Modus Ponens and to detach the consequent of this above implication from its antecedent, then we have $\left[q\to \left(~~p\to p\right)\right]\in Cn(R_{0*},A_2)$, where $p,q\in {At}_0$.

Now we repeat the well-known definition of the consistency in the traditional sense see Ref. (Pogorzelski 1975), cf. (Grzegorczyk 1974), (Pogorzelski and Wojtylak 2008), (Robles 2012), (Sher 2022), (Tarski 1956), (Wasilewska 2019), (Wójcicki 1984): Definition 2.5. $〈R,X〉\in {Cns}^T\iff \left(\neg \exists \alpha \in S_0\right)$ $[\alpha \in Cn\left(R,X\right)\&~\alpha \in Cn\left(R,X\right)]$.

So, the fact that the given system is consistent in the traditional sense, means that there does not exist such formula that both: this formula and its negation are the laws of this system. However, if we consider systems, where the connective of negation is absent, then the notion of consistency in the traditional sense is useless in the case of such systems. One of them is the classical implicational propositional Tarski-Bernays’ calculus (another example of such system: the positive implicational Hilbert’s calculus). So, we need another notion of consistency for such system, i.e., the consistency in the absolute sense (Post’s sense), see (Pogorzelski 1975), (Pogorzelski and Wojtylak 2008), (Robles 2012):

Definition 2.6. $〈R,X〉\in {Cns}^A\iff Cn\left(R,X\right)\ne S_0$.

Hence, one can derive any sentence in the system inconsistent in the traditional sense, of course, if Duns-Scottus law (Ex Falso Quodlibet, EFQ, principle of explosion): $\left(\alpha \wedge ~\alpha \right)\to \beta$, is valid there, because such system is inconsistent also in absolute sense cf. (Andrews 2002), (Tennant 2024). Such systems are called also as “trivial systems” cf. (Friend & del Rosario Martínez-Ordaz 2018). Some logical systems reject EFQ—these are the so-called paraconsistent logics cf. (Barrio, Pailos & Szmuc 2018). Certain comparison of Intuitionistic Logic with Paraconsistent Logic was done for e.g., in (Stopa 2022).

Next, we recall the well-known definitions concerning completeness (Pogorzelski 1975), (Pogorzelski and Wojtylak 2008).

Definition 2.7.  $〈R,X〉\in {Cpl}^T\iff$

$$(\forall \alpha \in \mathrm{S})[\alpha \in Cn\left(R,X\right)\mathbb{V}~\alpha \in Cn\left(R,X\right)].$$

Definition 2.8.  $〈R,X〉\in {Cpl}^A\iff$

$$(\forall \alpha \in \mathrm{S}\setminus Cn\left(R,X\right))[Cn\left(R,X\cup \left\{\alpha \right\}\right)=S].$$

$〈R,X〉\in {Cpl}^T$ denotes that the system $〈R,X〉$ is complete in the traditional sense (or in syntactical sense). $〈R,X〉\in {Cpl}^A$ denotes that the system $〈R,X〉$ is complete in the absolute sense or in Post’s sense, see (Pogorzelski 1975), (Pogorzelski and Wojtylak 2008), cf. (Marciszewski 1981). The fact that the set $S_0^0$ is non-empty, follows from the fact that the Classical Propositional Calculus is incomplete in the traditional sense.

Now we repeat some well-known basic Theorems, the so-called meta-theorems. The first one is the so-called Deduction Theorem, sometimes called also as Tarski-Herbrand Theorem (Pogorzelski 1975), (Pogorzelski and Wojtylak 2008), cf. (Bolzano 1837), (Buss 2023), (Ershov and Palyutin 1984), (Herbrand 1930), (Srivastava 2008), (Tarski 1930), (Tarski 1956):

Theorem 2.2. $\Psi \in Cn\left(R_0,Z_2\cup X\cup \left\{\Phi \right\}\right)⟺\left(\Phi \to \Psi \right)\in Cn\left(R_0,Z_2\cup X\right)$, for each $X\subseteq S_0$ and for any $\Phi ,\Psi \in S_0$. The two next meta-theorems are called, as Tarski’s Theorems on: Consistency and Inconsistency, correspondingly. One version of Tarski’s Theorem on Consistency is given in (Pogorzelski 1975) and (Pogorzelski & Wojtylak 2008): Theorem 2.3. $Cn\left(R_0,Z_2\cup X\cup \left\{~\alpha \right\}\right)\ne S_0 \iff \alpha \notin Cn\left(R_0,Z_2\cup X\right).$ An immediate result of this Theorem is the following Tarski’s Theorem on Inconsistency cf. (Pogorzelski & Wojtylak 2008):

Theorem 2.4. $Cn\left(R_0,Z_2\cup X\cup \left\{~\alpha \right\}\right)=S_0 \iff$$\alpha \in Cn\left(R_0,Z_2\cup X\right)$. Let us repeat also that the Deduction Theorem, as well as Tarski’s Theorem on Consistency and Tarski’s Theorem on Inconsistency hold for the system $〈R_0,Z_2〉$, but not for the system $〈R_{0*},A_2〉$. We repeat now the well-known theorems concerning consistency and incompleteness of the Classical Propositional Calculus (Pogorzelski 1975), (cf. (Pogorzelski & Wojtylak 2008):

Theorem 2.5. $〈R_0,Z_2〉\in {Cns}^T$.

Theorem 2.6. $〈R_0,Z_2〉\in {Cns}^A$.

Theorem 2.7. $〈R_0,Z_2〉\in {Cns}^A\Rightarrow 〈R_0,Z_2〉\notin {Cpl}^T$.

Theorem 2.8. $〈R_0,Z_2〉\in {Cns}^A\Rightarrow 〈R_0,Z_2〉\notin {Cpl}^A$.

We finish this section, giving the very important theorem, namely Glivenko Theorem, which establishes a connection between CPC and IPC (Polak 2014), (Moschovakis 2024):

Theorem 2.9. $\left(\forall \alpha \in S_0\right)[\alpha \in Cn\left(R_{0*},A_2\right)\iff ~~\alpha \in Cn\left(R_{0*},A_{int}\right)]$. This Theorem suffices to prove other very important fact (proven by Gödel in 1933), namely equiconsistency of the Classical Propositional Calculus and the Intuitionistic Propositional Calculus (Polak 2014), cf. (Gödel 1986), (Moschovakis 2024):

Theorem 2.10.

$$〈R_{0*},A_2〉\notin {Cns}^A\iff 〈R_{0*},A_{int}〉\notin {Cns}^A.$$

3. A Lemma

Lemma 3.1.

$$\left(\forall \delta \in S_0^I\right)\left(\exists {\omega }^{\prime }\in S_0^I\right)\left[\delta \to ~\omega \prime \in Cn\left(R_0,Z_2\right)\right],$$

where

$$S_0^0=\{\phi \in S_0:\phi \notin Z_2 \& ~\phi \notin Z_2\},$$

$$S_0^I\subset S_0^0,$$

$S_0^I=\{p_1, ~p_1,p_2\wedge p_3,\sim \left(p_2\wedge p_3\right),q_1\vee q_2,$

$\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right), ~\left[\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right)\right],$

$\sim \left(p_6\vee p_7\right)\wedge q_5, ~\left[~\left(p_6\vee p_7\right)\wedge q_5\right],$

$\sim \left(q_1\vee q_2\right),q_6,~q_6,\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right),$

$~\left[\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right)\right]\},$

where $p_i,q_j\in {At}_0$ ($i,j\in \{1,\dots ,11\})$.

Proof. Let

1) $\neg \left(\forall \delta \in S_0^I\right)\left(\exists {\omega }^{\prime }\in S_0^I\right)\left[\delta \to ~\omega \prime \in Cn\left(R_0,Z_2\right)\right]$,

where

2) $S_0^0=\{\phi \in S_0:\phi \notin Z_2 \& ~\phi \notin Z_2\}$,

3) $S_0^I\subset S_0^0,$4)

$S_0^I=\{p_1, ~p_1,p_2\wedge p_3,\sim \left(p_2\wedge p_3\right),q_1\vee q_2,$

$\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right), ~\left[\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right)\right],$

$\sim \left(p_6\vee p_7\right)\wedge q_5, ~\left[~\left(p_6\vee p_7\right)\wedge q_5\right],$

$\sim \left(q_1\vee q_2\right),q_6,~q_6,\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right),$

$~\left[\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right)\right]\},$

5) $p_i,q_j\in {At}_0$ ($i,j\in \{1,\dots ,11\})$.

Hence, we obtain

6) $\left(\exists \delta \prime \in S_0^I\right)\left(\forall \omega \in S_0^I\right)[\delta \prime \to ~\omega \notin Cn\left(R_0,Z_2\right)]$,

where

7) $S_0^0=\{\phi \in S_0:\phi \notin Z_2 \& ~\phi \notin Z_2\}$,

8) $S_0^I\subset S_0^0$,

9) $S_0^I=\{p_1, ~p_1,p_2\wedge p_3,\sim \left(p_2\wedge p_3\right),q_1\vee q_2,$

$\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right), ~\left[\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right)\right],$

$\sim \left(p_6\vee p_7\right)\wedge q_5, ~\left[~\left(p_6\vee p_7\right)\wedge q_5\right],$

$\sim \left(q_1\vee q_2\right),q_6,~q_6,\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right),$

$~\left[\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right)\right]\},$

10) $p_i,q_j\in {At}_0$ ($i,j\in \{1,\dots ,11\})$.

Next, from 6)—10), by the Theorem 2.3, we have

11) $\left(\exists {\delta }^{\prime }\in S_0^I\right)\left(\forall \omega \in S_0^I\right)$

$$\left[Cn\right(R_0,Z_2\cup \{\delta \prime \}\cup \left\{\omega \right\})\ne S_0],$$

where

12) $S_0^0=\{\phi \in S_0:\phi \notin Z_2 \& ~\phi \notin Z_2\}$,

13) $S_0^I\subset S_0^0$,

14) $S_0^I=\{p_1, ~p_1,p_2\wedge p_3,\sim \left(p_2\wedge p_3\right),q_1\vee q_2,$

$\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right), ~\left[\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right)\right],$

$\sim \left(p_6\vee p_7\right)\wedge q_5, ~\left[~\left(p_6\vee p_7\right)\wedge q_5\right],$

$\sim \left(q_1\vee q_2\right),q_6,~q_6,\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right),$

$~\left[\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right)\right]\},$

15) $p_i,q_j\in {At}_0$ ($i,j\in \{1,\dots ,11\})$.

On the other hand, we have

16) $\left(\exists \delta \prime \in S_0^I\right)\left[Cn\right(R_0,Z_2\cup \{\delta \prime \}\cup \left\{\omega \right\})=S_0]$,

where

17)$S_0^0=\{\phi \in S_0:\phi \notin Z_2 \& ~\phi \notin Z_2\}$,

18) $S_0^I\subset S_0^0$,

19) $S_0^I=\{p_1, ~p_1,p_2\wedge p_3,\sim \left(p_2\wedge p_3\right),q_1\vee q_2,$

$\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right), ~\left[\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right)\right],$

$\sim \left(p_6\vee p_7\right)\wedge q_5, ~\left[~\left(p_6\vee p_7\right)\wedge q_5\right],$

$\sim \left(q_1\vee q_2\right),q_6,~q_6,\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right),$

$~\left[\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right)\right]\},$

20) $p_i,q_j\in {At}_0$ ($i,j\in \{1,\dots ,11\})$,

21) $\omega \in A^{*}\cap S_0^I$,

22) $A^{*}=\{\alpha \in S_0:\left(\forall \psi \in A\right)\left[\alpha =~\psi \right]\}$,

23) $A=Cn\left(R_0,Z_2\cup \{\delta \prime \}\right)$.

A contradiction. $∎$

4. The Main Results

Theorem 4.1. $〈R_0,Z_2〉\notin {Cns}^A$.

Proof.

I) Let $〈R_0,Z_2〉\in {Cns}^A$.

Then, by Lemma 3.1, we have

II) $\left(\forall \delta \in S_0^I\right)\left(\exists {\omega }^{\prime }\in S_0^I\right)\left[\delta \to ~\omega \prime \in Cn\left(R_0,Z_2\right)\right]$,

where

III) $S_0^0=\{\phi \in S_0:\phi \notin Z_2 \& ~\phi \notin Z_2\}$,

IV) $S_0^I\subset S_0^0$,

V) $S_0^I=\{p_1, ~p_1,p_2\wedge p_3, \sim \left(p_2\wedge p_3\right),q_1\vee q_2,$

$\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right), ~\left[\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right)\right],$

$\sim \left(p_6\vee p_7\right)\wedge q_5, ~\left[~\left(p_6\vee p_7\right)\wedge q_5\right],$

$\sim \left(q_1\vee q_2\right),q_6,~q_6,\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right),$

$~\left[\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right)\right]\},$

VI) $p_i,q_j\in {At}_0$ ($i,j\in \{1,\dots ,11\})$.

Hence, we get

VII) $\left(\exists {\omega }^{\prime }\in S_0^I\right)\left[~\omega \prime \in Cn\left(R_0,Z_2\right)\right]$,

where

VIII) $\delta =\omega \prime$,

IX) $S_0^0=\{\phi \in S_0:\phi \notin Z_2 \& ~\phi \notin Z_2\}$,

X) $S_0^I\subset S_0^0$,

XI) $S_0^I=\{p_1, ~p_1,p_2\wedge p_3, \sim \left(p_2\wedge p_3\right),q_1\vee q_2,$

$\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right), ~\left[\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right)\right],$

$\sim \left(p_6\vee p_7\right)\wedge q_5, ~\left[~\left(p_6\vee p_7\right)\wedge q_5\right],$

$\sim \left(q_1\vee q_2\right),q_6,~q_6,\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right),$

$~\left[\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right)\right]\},$

XII) $p_i,q_j\in {At}_0$ ($i,j\in \{1,\dots ,11\})$.

We can say already now that we have contradiction between steps VII), IX), X), XI) and XII). However, we can go further. Namely, because $Cn\left(R_0,Z_2\right)=Cn(R_{0*},A_2)$, we get

XIII) $\left(\exists \omega \prime \in S_0^I\right)[~\omega \prime \in Cn\left(R_{0*},A_2\right)]$,

where

XIV) $S_0^0=\{\phi \in S_0:\phi \notin Z_2 \& ~\phi \notin Z_2\}$,

XV) $S_0^I\subset S_0^0$,

XVI) $S_0^I=\{p_1, ~p_1,p_2\wedge p_3, \sim \left(p_2\wedge p_3\right),q_1\vee q_2,$

$\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right), ~\left[\left(p_4\vee q_3\right)\wedge \left(p_5\vee q_4\right)\right],$

$\sim \left(p_6\vee p_7\right)\wedge q_5, ~\left[~\left(p_6\vee p_7\right)\wedge q_5\right],$

$\sim \left(q_1\vee q_2\right),q_6,~q_6,\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right),$

$~\left[\left(~p_8\wedge p_9\right)\vee \left(p_{10}\wedge p_{11}\right)\right]\},$

XVII) $p_i,q_j\in {At}_0$ ($i,j\in \{1,\dots ,11\})$.

Let us remember that by the definition of the set $S_0^I$, ${\omega }^{\prime }$ is certain generalized conjunction or certain generalized disjunction or negation of one of them such that each propositional variable occurs in $\omega \prime$ at most once.

Then, we can make such substitution(s) inside $\omega \prime$ that we obtain certain formula, which is homogeneous with respect to the negated one propositional variable or unnegated one propositional variable. In other words, there after such substitution, appears only the negation of the same propositional variable or there appears only this variable without negation, for e.g.,:

$p_1,~p_1,(p_1\vee p_1)\wedge \left(p_1\vee p_1\right)$, $(p_1\wedge p_1)\vee \left(p_1\wedge p_1\right)$, $(~p_1\wedge {\sim p}_1)\vee \left({\sim p}_1\wedge {\sim p}_1\right)$ etc. etc.

Let us illustrate this by taking into account for e.g., the first formula belonging to the set $S_0^I$ (one can check easily that the analogical reasoning with the same final result, can be repeated for each other formula from the set $S_0^I$) and from XIII)—XVII), we get

XVIII) ${~p}_1\in Cn\left(R_{0*},A_2\right)$,

where

XIX) $\omega =p_1$,

XX) $p_1\in {At}_0$.

Now, we make the following substitution:

$$r_{*}: p_1/p_1\to p_1.$$

Hence, we get:

XXI) $~(p_1\to p_1)\in Cn\left(R_{0*},A_2\right)$,

where

XXII) $p_1\in {At}_0$.

Next, from XXI)—XXII), we obtain

XXIII) $p_1\wedge \sim p_1 \in Cn\left(R_{0*},A_2\right)$,

where

XXIV) $p_1\in {At}_0$.

Then, from XXIII)—XXIV), and from Duns-Scottus law (Ex Falso Quodlibet), we have

XXV) $Cn\left(R_{0*},A_2\right)=Cn\left(R_0,Z_2\right)=S_0$.

In other words

XXVI) $〈R_{0*},A_2〉\notin {Cns}^A \& 〈R_0,Z_2〉\notin {Cns}^A$.

This contradicts the step I). $∎$

Theorem 4.2. $〈R_0,{Sb(A}_{int})〉\notin {Cns}^A$.

Proof. This Theorem follows immediately from Theorem 4.1 and Theorem 2.10. $∎$

5. Conclusions

Firstly, as one can see in the section 4 “The Main Results”, according to the principle of proving by contradiction, the thesis, which is to be proven, has been negated. In other words, the first step of the proof of The Main Result is: $〈R_0,Z_2〉\in {Cns}^A$ i.e., we have assumed consistency of the Classical Propositional Calculus CPC, and at the end we have received certain contradiction between the steps VII), IX), X), XI) and XII). So, we have obtained the contradiction between two metalogical statements: $~\omega \prime \in Cn\left(R_0,Z_2\right)$ and $~\omega \prime \notin Cn\left(R_0,Z_2\right)$, under the assumption that $〈R_0,Z_2〉\in {Cns}^A$ (i.e., the Classical Propositional Calculus CPC is consistent in the absolute sense), remembering that $Z_2=Cn\left(R_0,Z_2\right)$.

Obviously, we have immediately the following conclusion, cf. (Stępień & Stępień 2025), (da Costa & French 2022):

Theorem 5.1.:The metatheory of the Classical

Propositional Calculus is

inconsistent.

We repeat, what we have written in the Section 2, namely that we assume for the metalogical quantifiers: $\forall$ and $\exists$, all laws of the Classical Calculus of Quantifiers, and this last one is formulated basing on the Classical Propositional Calculus.

So, the following Definition can be formulated (Stępień & Stępień 2025):

Definition 5.1.: We tell that the Classical Propositional Calculus is metainconsistent in such sense that the inconsistency of the Classical Propositional Calculus is coupled to the inconsistency of its metatheory.

The notion of metainconsistency of the Classical Logic, appears in (McAllister 2022), however, there the notion of meta-inconsistency is considered in somewhat another context. The Author of (McAllister 2022) has generalized the results included in (Barrio etal. 2020), where a non-uniqueness of the Classical Logic at each finite inferential level had been shown.

However, in this paper we have gone further, in comparison to our previous paper (Stępień & Stępień 2025). Namely, owing to the structure of the formula $\omega \prime$, i.e., each propositional variable occurs there at most once, by making certain proper substitution(s) for the propositional variable(s), we can get that

[a.]

$p_i\in Cn\left(R_{0*},A_2\right)$

or

[b.]

${~p}_i\in Cn\left(R_{0*},A_2\right),$

where $i\in \mathbb{N}$ and $p_i\in {At}_0$. In this paper, we have considered the case [b.]—the step XXII) in the section 4 “The Main Results” (in this case: $i=1$).

Now, we make the proper substitution: in the case [a.] the substitution is $r_{*}:p_i/~(p_i\to p_i)$, and in the case [b.] the substitution is $r_{*}:p_i/p_i\to p_i$. Of course, we can substitute for $p$ any contrtautology of CPC (in the case [a.]) or any tautology of CPC (in the case [b.]). Anyway, as the result, in each case we obtain (after doing proper substitution i.e., $r_{*}:p_i/~(p_i\to p_i)$ in the case [a.] or $r_{*}:p_i/(p_i\to p_i)$ in the case [b.]): $~(p_i\to p_i)\in Cn\left(R_{0*},A_2\right)$, where $p_i\in {At}_0$. Hence, this leads to: $Cn\left(R_{0*},A_2\right)=S_0$ i.e., $〈R_{0*},A_2〉\notin {Cns}^A \& 〈R_0,Z_2〉\notin {Cns}^A$.

Next, from this and from Theorem 2.10, we have obtained right away: $〈R_0,{Sb(A}_{int})〉\notin {Cns}^A$.

So, both: the Classical Propositional Calculus and the Intuitionistic Propositional Calculus are inconsistent in the absolute sense (Post’s sense).

Obviously, the results presented in this paper, seem to be very strange. However, there in the Ref. (Stępień & Stępień 2017) the consistency of the Arithmetic System had been proven, using only the resources of this System. On the face of it, this seems to contradict the famous Gödel’s Second Incompleteness Theorem for e.g., (de Fatima Batistela 2022), (Krajewski 2007), (Müller-Stach 2024), (Murawski 1999). However, let’s recall that according to the Gödel’s Second Incompleteness Theorem, one cannot prove consistency of the Arithmetic System, using only the resources of this System, but if this System is consistent. If we take into account inconsistency of Classical Propositional Calculus, this causes inconsistency of Classical Calculus of Quantifiers, and of course, this causes that Classical Functional Calculus is inconsistent, too. This last one is necessary to formalize the Arithmetic System. So, an impact of inconsistency of Classical Propositional Calculus on the issue of consistency of Arithmetic System, is obvious. As we have written this in the Section 1, we have spreading of “the infection of inconsistency” to Peano Arithmetic System, i.e., since the Classical Propositional Calculus is inconsistent, then Peano Arithmetic System is inconsistent, too. Obviously, this has further consequences i.e., the set theory (ZF theory and ZFC theory) is also inconsistent, and this causes the inconsistency of other branches of mathematics: algebra, mathematical analysis, functional analysis, geometry, topology etc. As this is well-known fact, mathematics is certain language and a tool of physics. So, in the context of the results presented in this paper, this language or tool has a serious problem.

Next, let’s notice that if one uses only the truth tables, and checks, whether a given formula is a (contr)tautology, then the classical propositional calculus seems to work properly i.e., there is not any contradiction, at least at a glance. The same situation is, when we obtain new laws of the classical propositional calculus, using only the inference rules: Modus Ponens and the subtitution rule, and the set of axioms. The problem seems to begin, when we use the Deduction Theorem i.e., Theorem 2.2, and we prove by contradiction. This technique is possible owing to among others, the Deduction Theorem (for e.g., (Buss 2023)). So, we have used Deduction Theorem indirectly, when we were proving Lemma 3.1 and later the Theorems included in the Section 4 “The Main Results”. This is so, because the Deduction Theorem is necessary to prove other well-known meta-theorems: Theorem 2.3 and Theorem 2.4. The Theorems: Theorem 2.2 and Theorem 2.4 were applied in the proof of the Theorems included in the Section 4 “The Main Results”, and Theorem 2.3 was used in the proof of Lemma 3.1.

One should add here that proving by contradiction causes some doubts, for e.g., (Goodstein 1948), (Piekarczyk 2019). So, the Deduction Theorem and technique of proving by contradiction seem to be a source of problems with the consistency of the Classical Propositional Calculus, but this issue requires to be investigated.

One should mention also here that this is a very interesting issue (for e.g., in the context of issue of inconsistency), that there is no logic (i.e., consequence relation), which is simultaneously determined by any class of orthomodular lattices and which admits the Deduction Theorem (Malinowski 1990), (Fazio and Mascella 2026).

There in Ref. (Weber etal. 2016) a question on necessity of assumption of truth tables consistency had been asked, and appearing of an inconsistency, in the context of the truth tables, was demonstrated there (as the Authors of Ref. (Weber etal. 2016) have established there), by using the case of liar paradox. In this paper mentioned above, a construction of truth tables in a consistency-independent paraconsistent setting was presented. The Authors of Ref. (Weber etal. 2016) had been working there just using paraconsistent metatheory.

On the other hand, there in Ref. (Tanaka and Girard 2023) were presented some arguments against classical paraconsistent metatheory.

Anyway, we would like to stress here that we have used any truth tables neither in the proof of Lemma 3.1 nor in the proofs of The Main Results in this current paper, and any liar paradox has not been involved here. In this paper we have applied: some laws of the classical propositional calculus (i.e., $~\left(\alpha \to \beta \right)\equiv \alpha \wedge ~\beta$, $\left(\alpha \wedge ~\alpha \right)\to \beta$, $\left(\alpha \to ~\alpha \right)\to ~\alpha$, where $\alpha$, $\beta \in S_0$), Modus Ponens rule $r_0$, substitution rule $r_{*}$, certain properties of consequence operation, Definition 2.1—Definition 2.6, Theorem 2.3 and Theorem 2.4. We have concluded the inconsistency of Intuitionistic Propositional Calculus from the inconsistency of the Classical Propositional Calculus and Theorem 2.10. So, we have done the proof of inconsistency of CPC using its metalogic, and in order to prove iconsistency of IPC, we have not used its metalogic (we simply have applied Theorem 2.10). Metalogic of IPC had been presented for e.g., in (Citkin 2010).

We would like to stress also that the proofs of inconsistency of the Classical Propositional Calculus (CPC) presented in this paper and in our previous paper (Stępień & Stępień 2025), are the first proofs of this result. Analogical remark concerns also the proof of the inconsistency of the Intuitionistic Propositional Calculus, included in this paper.

There in the literature is among others, well-known the proof done by Russell, who proved the inconsistency of Frege’s system (Besler 2016), (Marciszewski 1981), and by Kochen and Specker (Kochen and Specker 1967). Namely, as one can read in (Engesser etal. 2009), “Let us recall here a phenomenon first observed by Kochen and Specker, namely that Birkhoff-von Neumann quantum logic is in a sense ‘classically inconsistent’.” As this was explained in (Engesser etal. 2009), Kochen and Specker proved in (Kochen and Specker 1967) that there exists a classical tautology, which negation is provable in all consequence relations of a Hilbert space logic presented by a three dimensional Hilbert space. Hence, we can say about the inconsistency in the traditional sense of Birkhoff-von Neumann quantum logic.

The author of some other proof of inconsistency of logic system is Northrop, who claimed he had proved some inconsistency of Aristotelian logic (Northrop 1928), and Goddard claimed this same in (Goddard 2000) (let us notice that he had not cited there Northrop’s paper), cf. a comment in (Başoğlu 2019). There in (Goddard 1998) the issue of inconsistency of the traditional logic was considered, too. However, as Goddard had written there, in order to prove inconsistency of the traditional logic, he had considered there an extension of Aristotelian logic, by using negative terms, complex terms, quantified predicates, a theory of obversion etc. In contrary to him, we have proved here the inconsistency of pure Classical Propositional Calculus and of the Intuitionistic Propositional Calculus. By the way, we would like to add that neither Northrop nor Goddard used operation of consequence in their proofs—we have used in our proof, this operation.

This is worth to mention here that there in literature is known problem called as “scandal of deduction”, referred by in (Hintikka 1973), cf. (Hernández and Hernández-Quiroz 2024). As he has written. “scandal of deduction” is caused by the “tautological” or “analytical” nature of deduction. Namely, a conclusion deduced in a valid way from a set of premises, is said to be contained in that set. The “tautological” nature of deduction follows just from the Deduction Theorem (Hernández and Hernández-Quiroz 2024).

Some remarks on the case of inconsistent metatheory, are included in Refs. (Hertrich-Woleński 2015), (Visser 1991), (Woods 2014) (inconsistency of the so-called Nudel’man’s metatheory, was proven in Ref. (Ganov 1995)). Certain aspects of paraconsistent metatheory are discussed for e.g., in (Nelson 2015).

One can point out some possible directions of further research connected with the results presented in this current paper. Namely, since Deduction Theorem and the technique of proving by contradiction (indirect proof), seem to be the source of the inconsistency of the Classical Propositional Calculus CPC (metalogic is based on the Classical Calculus of Quantifiers and this last one is formulated on the ground of CPC), then, this issue needs to be analysed and fixed. We stress also here that there in the Intuitionistic Propositional Calculus IPC the Deduction Theorem holds, but the technique of proving by contradiction is unavailable—the inconsistency of IPC proven in this paper, follows from the inconsistency of CPC and Gödel’s Theorem of equiconsistency of CPC and IPC (Theorem 2.10).

We would add also that there in (Wójcicki 1999) an interesting analysis of certain issue connected to Tarski’s consequence operation was done. So, this can be another probable direction of research in the context of the results presented in this current paper, too.

As far as some possible applications of the results presented in this paper, are concerned, one should investigate consequences of the inconsistency of the Classical Propositional Calculus for the issue of Artificial Intelligence (AI). The problem with AI becomes more and more significant, due to among others, increasing number of AI chatbots, which began to cheat humans (The Guardian 2026). So, this is the important issue to investigate this problem, in the context of the inconsistency of the Classical Propositional Calculus (which causes the inconsistency of Classical Logic).