On the Inconsistency of: The Classical Propositional Calculus and Its Metatheory

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, ${At}_0=\left\{p,q,r,\dots p_1^1,p_2^1,\dots ,p_1^2,p_2^2,\dots ,p_1^k,p_2^k,\dots \right\}$, (where $k\in \mathcal{N}$), 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. We denote the well-formed formulas by small Greek letters (the subscripts, superscripts and/or accents can also be used).

So, $S_0=\left\{\alpha ,\beta ,\gamma ,\delta ,\dots ,\phi \dots ,{\alpha }_0,\stackrel{00}{\delta },\dots \right\}$.

$R_{S_0}$ denotes the set of all rules over $S_0$. $E\left(\mathfrak{M}\right)$ is the set of all formulas valid in the matrix $\mathfrak{M}$. The symbol ${\mathfrak{M}}_2$ denotes the classical two-valued matrix and $Z_2$ is the set of all formulas valid in the matrix ${\mathfrak{M}}_2$ (see 84, cf. 1,10,35,39,53,64,68,85,92,114).

The symbols $\Rightarrow ,\neg ,\mathbb{}\mathbb{V},\mathbb{}\&,\iff ,\mathbb{}\forall ,\exists$ are metalogical symbols (they denote correspondingly: metaimplication, metanegation, metadisjunction, metaconjunction, metaequivalence and the metalogical quantifiers: general and existential one).

Next, $r_0$ is the symbol of Modus Ponens in the classical propositional calculus. Hence, $R_0=\left\{r_0\right\}$. The formula $X\subset Y$ denotes that $X\subseteq Y$ and $X\ne Y$. 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 $R\subseteq R_{S_0}$.

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 (see 84,85).

Now we repeat some well-known properties of operation of consequence and some well-known definitions (see [84], cf. [1,85,114]). Let $R\subseteq R_{S_0}$ and $X\subseteq S_0$. Then:

• $a_1)\mathrm{}\mathrm{}X\subseteq Cn\left(R,X\right),$
• $a_2)\mathrm{}\mathrm{}X\subseteq Y\mathrm{}\Rightarrow Cn\left(R,X\right)\subseteq Cn\left(R,Y\right),$
• $a_3)\mathrm{}\mathrm{}R\subseteq R^{\mathrm{\text{'}}}\Rightarrow Cn\left(R,X\right)\subseteq Cn\left(R^{\mathrm{\text{'}}},X\right),$
• $a_4)\mathrm{}\mathrm{}Cn\left(R,Cn\left(R,X\right)\right)\subseteq Cn\left(R,X\right),$
• $a_5)\mathrm{}\mathrm{}Cn\left(R,X\right)=\bigcup \{Cn\left(R,Y\right):Y\subseteq X\mathrm{}\mathrm{}\&\mathrm{}\mathrm{}\stackrel{̿}{Y}<{\mathrm{\aleph }}_0\}$.

Definition 1.1. $〈R,X〉\in {Cns}^T\mathrm{}\iff \left(\neg \exists \alpha \in S_0\right)[\alpha \in Cn\left(R,X\right)\mathrm{}\mathrm{}\&\mathrm{}\mathrm{}~\alpha \in Cn\left(R,X\right)]$.

Definition 1.2. $〈R,X〉\in {Cns}^A\mathrm{}\iff Cn\left(R,X\right)\ne S_0$.

$〈R,X〉\in {Cns}^T$ denotes that the system $〈R,X〉$ is consistent in the traditional sense. $〈R,X〉\in {Cns}^A$ denotes that the system $〈R,X〉$ is consistent in the absolute sense or in Post’s sense (see 84, cf. 85,86,114).

Now we repeat some well-known basic Theorems, the so-called metatheorems. The first one is the so-called Deduction Theorem, sometimes called also as Tarski-Herbrand Theorem (see 84, cf. 16,54,85,104):

Theorem 1.1. $\left(\forall \alpha \in S_0\right)\left(\forall \beta \in S_0\right)\left(\forall X\subseteq S_0\right)$

$$\begin{gathered} [\beta \in Cn(R_0,Z_2\cup X\cup \left\{\alpha \right\})\Rightarrow \\ (\alpha \to \beta )\in Cn(R_0,Z_2\cup X\left)\right]. \end{gathered}$$

The two next metatheorems are correspondingly, the so-called Theorem on Consistency and Theorem on Inconsistency (see 84, cf. 16,54,85,89,104):

Theorem 1.2.$\left(\forall \alpha \in S_0\right)\left(\forall X\subseteq S_0\right)[Cn\left(R_0,Z_2\cup X\cup \left\{~\alpha \right\}\right)\ne S_0\iff \alpha \notin Cn\left(R_0,Z_2\cup X\right)]$.

Theorem 1.3.$\left(\forall \alpha \in S_0\right)\left(\forall X\subseteq S_0\right)[Cn\left(R_0,Z_2\cup X\cup \left\{\alpha \right\}\right)=S_0\iff ~\alpha \in Cn\left(R_0,Z_2\cup X\right)]$.

At the end of this section we repeat the well-known theorems on consistency of the classical propositional calculus (see 84, cf. 85):

Theorem 1.4. $〈R_0,Z_2〉\in {Cns}^T$.

Theorem 1.5. $〈R_0,Z_2〉\in {Cns}^A$.

3. A Lemma

Lemma 3.1.

$\left(\forall {\alpha }_0\in A_1^{\mathrm{\text{'}}}\right)\left(\forall \stackrel{00}{\delta }\in S_0\right)\left(\forall \delta \in S_0\right)$

$\left(\forall \phi \in S_0\right)\left[Cn\right(R_0,Z_2\cup A^{\mathrm{*}}\cup A^{\mathrm{*}\mathrm{*}}\cup$

$\left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup \{~\stackrel{00}{\delta }\to ~\phi \})=S_0]$,

where

$A^{\mathrm{*}}=\left\{{\alpha }_0\to \left(~\stackrel{00}{\delta }\to \phi \right)\right\}$,

$A^{\mathrm{*}\mathrm{*}}=\left\{\stackrel{00}{\delta }\to \delta \right\}$,

$A_1=Cn(R_0,Z_2\cup A^{\mathrm{*}}\cup A^{\mathrm{*}\mathrm{*}}\cup$$\left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup \left\{~\stackrel{00}{\delta }\to ~\phi \right\})$,

$A_1^{\mathrm{\text{'}}}=Cn(R_0,Z_2\cup A^{\mathrm{*}\mathrm{*}}\cup \left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup$$\left\{~\stackrel{00}{\delta }\to ~\phi \right\})$.

Proof. Let

1) $\neg \left(\forall {\alpha }_0\in A_1^{\mathrm{\text{'}}}\right)\left(\forall \stackrel{00}{\delta }\in S_0\right)\left(\forall \delta \in S_0\right)$$\left(\forall \phi \in S_0\right)\left[Cn\right(R_0,Z_2\cup A^{\mathrm{*}}\cup A^{\mathrm{*}\mathrm{*}}\cup$$\left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup \{~\stackrel{00}{\delta }\to ~\phi \})=S_0]$,

where

2) $A^{*}=\left\{{\alpha }_0\to \left(~\stackrel{00}{\delta }\to \phi \right)\right\}$,

3) $A^{**}=\left\{\stackrel{00}{\delta }\to \delta \right\}$,

4) $A_1=Cn(R_0,Z_2\cup A^{\mathrm{*}}\cup A^{\mathrm{*}\mathrm{*}}\cup$$\left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup \left\{~\stackrel{00}{\delta }\to ~\phi \right\})$,

5) $A_1^{\mathrm{\text{'}}}=Cn(R_0,Z_2\cup A^{\mathrm{*}\mathrm{*}}\cup \left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup$$\left\{~\stackrel{00}{\delta }\to ~\phi \right\})$.

From 1) – 5), we get

6) $\left(\exists {\alpha }_0^{\mathrm{\text{'}}}\in A_1^{\mathrm{\text{'}}}\right)\left(\exists \stackrel{00}{\delta }\mathrm{\text{'}}\in S_0\right)\left(\exists \delta \mathrm{\text{'}}\in S_0\right)$$\left(\exists \phi \mathrm{\text{'}}\in S_0\right)\left[Cn\right(R_0,Z_2\cup A^{\mathrm{*}}\cup A^{\mathrm{*}\mathrm{*}}\cup$$\left\{{\alpha }_0^{\mathrm{\text{'}}}\to \left(\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\delta \mathrm{\text{'}}\right)\right\}\cup \{~\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\phi \mathrm{\text{'}}\})\ne S_0]$,

7) $A^{\mathrm{*}}=\left\{{\alpha }_0^{\mathrm{\text{'}}}\to \left(~\stackrel{00}{\delta }\mathrm{\text{'}}\to \phi \mathrm{\text{'}}\right)\right\}$,

8) $A^{\mathrm{*}\mathrm{*}}=\left\{\stackrel{00}{\delta }\mathrm{\text{'}}\to \delta \mathrm{\text{'}}\right\}$,

9) $A_1=Cn(R_0,Z_2\cup A^{\mathrm{*}}\cup A^{\mathrm{*}\mathrm{*}}\cup$$\left\{{\alpha }_0^{\mathrm{\text{'}}}\to \left(\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\delta \mathrm{\text{'}}\right)\right\}\cup \left\{~\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\phi \mathrm{\text{'}}\right\})$,

10) $A_1^{\mathrm{\text{'}}}=Cn(R_0,Z_2\cup A^{\mathrm{*}\mathrm{*}}\cup \left\{{\alpha }_0^{\mathrm{\text{'}}}\to \left(\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\delta \mathrm{\text{'}}\right)\right\}\cup$$\left\{~\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\phi \mathrm{\text{'}}\right\})$.

From 6) – 10), we obtain

11) $\left(\exists {\alpha }_0^{\mathrm{\text{'}}}\in A_1^{\mathrm{\text{'}}}\right)\left(\exists \stackrel{00}{\delta }\mathrm{\text{'}}\in S_0\right)\left(\exists \delta \mathrm{\text{'}}\in S_0\right)$$\left(\exists {\phi }^{\mathrm{\text{'}}}\in S_0\right)[~{\stackrel{00}{\delta }}^{\mathrm{\text{'}}}\to {\phi }^{\mathrm{\text{'}}},{\stackrel{00}{\delta }}^{\mathrm{\text{'}}}\to {\delta }^{\mathrm{\text{'}}},{\stackrel{00}{\delta }}^{\mathrm{\text{'}}}\to ~{\delta }^{\mathrm{\text{'}}},$${\phi }^{\mathrm{\text{'}}}\to {\stackrel{00}{\delta }}^{\mathrm{\text{'}}},{\stackrel{00}{\delta }}^{\mathrm{\text{'}}},{\delta }^{\mathrm{\text{'}}},~{\delta }^{\mathrm{\text{'}}}\in A_1\&A_1=S_0]$,

where

12) $A^{*}=\left\{{\alpha }_0^{\text{'}}\to \left(~\stackrel{00}{\delta }\text{'}\to \phi \text{'}\right)\right\}$,

13) $A^{\mathrm{*}\mathrm{*}}=\left\{\stackrel{00}{\delta }\mathrm{\text{'}}\to \delta \mathrm{\text{'}}\right\}$,

14) $A_1=Cn(R_0,Z_2\cup A^{\mathrm{*}}\cup A^{\mathrm{*}\mathrm{*}}\cup$$\left\{{\alpha }_0^{\mathrm{\text{'}}}\to \left(\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\delta \mathrm{\text{'}}\right)\right\}\cup \left\{~\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\phi \mathrm{\text{'}}\right\})$,

15) $A_1^{\mathrm{\text{'}}}=Cn(R_0,Z_2\cup A^{\mathrm{*}\mathrm{*}}\cup \left\{{\alpha }_0^{\mathrm{\text{'}}}\to \left(\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\delta \mathrm{\text{'}}\right)\right\}\cup$$\left\{~\stackrel{00}{\delta }\mathrm{\text{'}}\to ~\phi \mathrm{\text{'}}\right\})$,

what contradicts the steps 6) – 10).

□

4. The Main Result

Theorem 4.1.: $〈{\mathit{R}}_0,{\mathit{Z}}_2〉\notin {\mathit{C}\mathit{n}\mathit{s}}^{\mathit{A}}.$Proof.

Let

I) $〈R_0,Z_2〉\in {Cns}^A$.

By Lemma 3.1, we have

II) $\left(\forall {\alpha }_0\in A_1^{\text{'}}\right)\left(\forall \stackrel{00}{\delta }\in S_0\right)\left(\forall \delta \in S_0\right)$

$\left(\forall \phi \in S_0\right)\left[Cn\right(R_0,Z_2\cup A^{\mathrm{*}}\cup A^{\mathrm{*}\mathrm{*}}\cup$

$\left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup \{~\stackrel{00}{\delta }\to ~\phi \})=S_0],$where

III) $A^{*}=\left\{{\alpha }_0\to \left(~\stackrel{00}{\delta }\to \phi \right)\right\}$

IV) $A^{**}=\left\{\stackrel{00}{\delta }\to \delta \right\}$

V) $A_1=Cn(R_0,Z_2\cup A^{\mathrm{*}}\cup A^{\mathrm{*}\mathrm{*}}\cup$

$\left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup \{~\stackrel{00}{\delta }\to ~\phi \})$

VI) $A_1^{\mathrm{\text{'}}}=Cn(R_0,Z_2\cup A^{\mathrm{*}\mathrm{*}}\cup$

$\left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup \{~\stackrel{00}{\delta }\to ~\phi \})$.

Hence, by Theorem 1.3, we obtain

VII) $\left(\forall {\alpha }_0\in A_1^{\text{'}}\right)\left(\forall \stackrel{00}{\delta }\in S_0\right)\left(\forall \delta \in S_0\right)$

$\left(\forall \phi \in S_0\right)\left[{\alpha }_0,\stackrel{00}{~\delta },~\phi \in Cn\left(R_0,Z_2\cup A^{\mathrm{*}\mathrm{*}}\cup \left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup \left\{~\stackrel{00}{\delta }\to ~\phi \right\}\right)\right]$,

where

VIII) $A^{**}=\left\{\stackrel{00}{\delta }\to \delta \right\}$

IX) $A_1^{\mathrm{\text{'}}}=Cn(R_0,Z_2\cup A^{\mathrm{*}\mathrm{*}}\cup$

$\left\{{\alpha }_0\to \left(\stackrel{00}{\delta }\to ~\delta \right)\right\}\cup \{~\stackrel{00}{\delta }\to ~\phi \})$.

From VII) – IX), by Duns-Scottus law (Ex Falso Quodlibet), we get

X) $\left(\forall {\alpha }_0\in A_1^{\text{'}}\right)\left[A_1^{\text{'}}=S_0\right]$,

where

XI) $A^{**}=\left\{{\alpha }_0\to {\alpha }_0\right\}$

XII) $A_1^{\mathrm{\text{'}}}=Cn(R_0,Z_2\cup A^{\mathrm{*}\mathrm{*}}\cup$

$\left\{{\alpha }_0\to \left({\alpha }_0\to ~{\alpha }_0\right)\right\}\cup \{~{\alpha }_0\to ~{\alpha }_0\})$,

and where

XIII) $\stackrel{00}{\delta }\in \left\{{\alpha }_0\right\}$

XIV) $\delta \in \left\{{\alpha }_0\right\}$

XV) $\phi \in \left\{{\alpha }_0\right\}$.

Hence, we get

XVI) $\left(\forall {\alpha }_0\in S_0\right)$

$\left[Cn\left(R_0,Z_2\cup \left\{{\alpha }_0\to \left({\alpha }_0\to ~{\alpha }_0\right)\right\}\right)=S_0\right]$.

Then, from XVI), by Theorem 1.3. we obtain

XVII) $\left(\forall {\alpha }_0\in S_0\right)\left[{\alpha }_0\in Cn\left(R_0,Z_2\right)\right]$.

Hence, by Definition 1.1, we have

XVIII) $〈R_0,Z_2〉\notin {Cns}^T,$when

XIX) ${\alpha }_0\in \left\{p\wedge \sim p\right\}$.

Then, by Duns-Scottus law (Ex Falso Quodlibet), and from the fact that $Cn\left(R_0,Z_2\right)\subseteq S_0$, we have

XX) $Cn\left(R_0,Z_2\right)=S_0$.

Hence, from Definition 1.2, we get

XXI) $〈R_0,Z_2〉\notin {Cns}^A$,

what contradicts the step I). □

5. Conclusions

If we formulate certain analogon of Definition 1.1. for the case of metatheory of the classical propositional calculus, then from Theorem 4.1. and from Theorem 1.5., we get right away the following conclusion (cf. [32]):

Theorem 5.1: The metatheory of the classical propositional calculus is inconsistent.

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 first sight. The same situation is, when we obtain new laws of the classical propositional calculus, using only the inference rules and the set of axioms.

There in [110] the question on necessity of assumption of truth tables consistency had been asked, and there appearing of the inconsistency, in the context of the truth tables, was demonstrated (as the Authors of [110] 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 [110] had been working there just using paraconsistent metatheory. On the other hand, there in [102] were presented some arguments against classical paraconsistent metatheory.

Anyway, we would like to stress here that we have not used any truth tables in this current paper. In the steps XVIII) – XIX) of the proof of the Main Result, we have obtained that the classical propositional calculus is inconsistent in the traditional sense, when ${\alpha }_0\in \left\{p\wedge \sim p\right\}$ (so, ${\alpha }_0$ is some contrtautology), however any liar paradox has not been involved here. We have applied: some laws of the classical propositional calculus, Modus Ponens rule $r_0$, Theorem 1.3, Definition 1.1. and Definition 1.2.

Some remarks on the case of inconsistent metatheory, are included in [58,87] and [109] (inconsistency of the so-called Nudelman’s metatheory, was proved in [45]).