Prerpints.org logo
Preprint
Article

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

Altmetrics

Downloads

259

Views

380

Comments

0

This version is not peer-reviewed

Submitted:

07 December 2023

Posted:

08 December 2023

You are already at the latest version

Alerts
Abstract
The classical propositional calculus (zero-order logic, classical propositional logic), is the most fundamental two-valued logical system. In this paper we present a proof of inconsistency of the classical propositional calculus. Then, we get right away the conclusion that the metatheory of the classical propositional calculus is inconsistent.
Keywords: 
Subject: Computer Science and Mathematics  -   Logic
The issue, whether a given formal system is consistent, is the most fundamental issue for such system.
Many people have been dealing with different aspects of consistency and/or inconsistency in and/or of formal logical systems or in and/or of mathematics or other sciences for e.g. 2-34,36,37,40- 51,55,57-63,65-67,70-83,87,88,91,93,94,98,102,105-107,109-113,115.
Probably, the most known example of a system, which inconsistency was (correctly) proved, is Frege’s system presented in II volume of his “Grundgesetze der Arithmetik”. This inconsistency was proved by Russell in 1903 68 (cf. 13).
The classical propositional calculus is necessary to construct the classical calculus of quantifiers (classical calculus of predicates, first-order logic), and this last one is necessary to construct the classical functional calculus. Classical functional calculus is needed to formalize the Arithmetic System.
So, the significance of the issue of consistency or inconsistency of the classical propositional calculus, is obvious.
One can also consider 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 81, and on some applications of logic in computer science 44, functionality of mind 52,56 or psychology 95.
In 38 classical inconsistency of the best-known quantum logic (Birkhoff-von Neumann quantum logic), was discussed.
In 49 Goddard claimed he had proved inconsistency of traditional logic (cf. 50). However, as he wrote this in his paper, in order to prove inconsistency, he 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 prove here inconsistency of pure classical propositional calculus (the details are given beneath).
In 2010 Voevodsky delivered a talk entitled "What if current foundations of mathematics are inconsistent?" 108, where he focused on the issue of probably inconsistency of first-order Arithmetic System. In 2011 Nelson claimed he had proved inconsistency of the Arithmetic System, 26,76. However, soon Tao and Tausk found independently an error in Nelson’s proof mentioned above, 26.
Our paper concerns a more fundamental issue, namely inconsistency of the classical propositional calculus. The aim of this paper is to present a proof of the theorem that the classical propositional calculus (the zero-order logic, the classical propositional logic), is inconsistent in the traditional sense and in the absolute sense.
This paper is organized as follows. In the section 2, we introduce a notation and we repeat certain well-known notions (among others, the notions: operation of consequence, a system, consistency in the traditional sense, consistency in the absolute sense) and some well-known theorems. In the next section, we prove some Lemma (Lemma 3.1). The section 4 includes a proof of inconsistency of classical propositional calculus (this result was announced in 99,100,101). The section 5 is devoted to some conclusions.

2. Preliminaries

The symbols: → ,     ~ ,     ∨ ,     ∧ ,     ≡  denote the connectives of: implication, negation, disjunction, conjunction and equivalence, respectively. N = { 1,2 , … }  denotes the set of all natural numbers.
Next, A t 0 = p , q , r , … p 1 1 , p 2 1 , … , p 1 2 , p 2 2 , … , p 1 k , p 2 k , … , (where k ∈ 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 = α , β , γ , δ , … , φ … , α 0 , δ 00 , … .
R S 0  denotes the set of all rules over S 0 . E ( M )  is the set of all formulas valid in the matrix M . The symbol M 2  denotes the classical two-valued matrix and Z 2  is the set of all formulas valid in the matrix M 2  (see 84, cf. 1,10,35,39,53,64,68,85,92,114).
The symbols ⇒ , ¬ , V , & , ⇔ , ∀ , ∃  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 = { r 0 } . The formula X ⊂ Y  denotes that X ⊆ Y  and X ≠ Y . For any X ⊆ S 0  and R ⊆ R S 0 , C n ( R , X )  is the smallest subset of S 0 , containing X , and closed under the rules belonging to R , where R ⊆ R S 0 .
The couple R , X  is called as a system, whenever R ⊆ R S 0 , and X ⊆ 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 ⊆ R S 0  and X ⊆ S 0 . Then:
  • a 1 ) X ⊆ C n R , X ,
  • a 2 ) X ⊆ Y ⇒ C n R , X ⊆ C n R , Y ,
  • a 3 ) R ⊆ R ' ⇒ C n R , X ⊆ C n R ' , X ,
  • a 4 ) C n R , C n R , X ⊆ C n R , X ,
  • a 5 ) C n R , X = ⋃ { C n R , Y : Y ⊆ X & Y ̿ < ℵ 0 } .
Definition 1.1. R , X ∈ C n s T ⇔ ¬ ∃ α ∈ S 0 [ α ∈ C n R , X & ~ α ∈ C n R , X ] .
Definition 1.2. R , X ∈ C n s A ⇔ C n R , X ≠ S 0 .
R , X ∈ C n s T  denotes that the system R , X  is consistent in the traditional sense. R , X ∈ C n s 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. ∀ α ∈ S 0 ∀ β ∈ S 0 ∀ X ⊆ S 0
[ β ∈ C n ( R 0 , Z 2 ∪ X ∪ { α } ) ⇒ ( α → β ) ∈ C n ( R 0 , Z 2 ∪ X ) ] .
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. ∀ α ∈ S 0 ∀ X ⊆ S 0 [ C n R 0 , Z 2 ∪ X ∪ ~ α ≠ S 0 ⇔ α ∉ C n R 0 , Z 2 ∪ X ] .
Theorem 1.3. ∀ α ∈ S 0 ∀ X ⊆ S 0 [ C n R 0 , Z 2 ∪ X ∪ α = S 0 ⇔ ~ α ∈ C n R 0 , Z 2 ∪ X ] .
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 ∈ C n s T .
Theorem 1.5. R 0 , Z 2 ∈ C n s A .

3. A Lemma

Lemma 3.1.
∀ α 0 ∈ A 1 ' ∀ δ 00 ∈ S 0 ∀ δ ∈ S 0
∀ φ ∈ S 0 [ C n ( R 0 , Z 2 ∪ A * ∪ A * * ∪
α 0 → δ 00 → ~ δ ∪ { ~ δ 00 → ~ φ } ) = S 0 ] ,
where
A * = α 0 → ~ δ 00 → φ ,
A * * = δ 00 → δ ,
A 1 = C n ( R 0 , Z 2 ∪ A * ∪ A * * ∪ α 0 → δ 00 → ~ δ ∪ ~ δ 00 → ~ φ ) ,
A 1 ' = C n ( R 0 , Z 2 ∪ A * * ∪ α 0 → δ 00 → ~ δ ∪ ~ δ 00 → ~ φ ) .
Proof. Let
1) ¬ ∀ α 0 ∈ A 1 ' ∀ δ 00 ∈ S 0 ∀ δ ∈ S 0 ∀ φ ∈ S 0 [ C n ( R 0 , Z 2 ∪ A * ∪ A * * ∪ α 0 → δ 00 → ~ δ ∪ { ~ δ 00 → ~ φ } ) = S 0 ] ,
where
2) A * = α 0 → ~ δ 00 → φ ,
3) A * * = δ 00 → δ ,
4) A 1 = C n ( R 0 , Z 2 ∪ A * ∪ A * * ∪ α 0 → δ 00 → ~ δ ∪ ~ δ 00 → ~ φ ) ,
5) A 1 ' = C n ( R 0 , Z 2 ∪ A * * ∪ α 0 → δ 00 → ~ δ ∪ ~ δ 00 → ~ φ ) .
From 1) – 5), we get
6) ∃ α 0 ' ∈ A 1 ' ∃ δ 00 ' ∈ S 0 ∃ δ ' ∈ S 0 ∃ φ ' ∈ S 0 [ C n ( R 0 , Z 2 ∪ A * ∪ A * * ∪ α 0 ' → δ 00 ' → ~ δ ' ∪ { ~ δ 00 ' → ~ φ ' } ) ≠ S 0 ] ,
7) A * = α 0 ' → ~ δ 00 ' → φ ' ,
8) A * * = δ 00 ' → δ ' ,
9) A 1 = C n ( R 0 , Z 2 ∪ A * ∪ A * * ∪ α 0 ' → δ 00 ' → ~ δ ' ∪ ~ δ 00 ' → ~ φ ' ) ,
10) A 1 ' = C n ( R 0 , Z 2 ∪ A * * ∪ α 0 ' → δ 00 ' → ~ δ ' ∪ ~ δ 00 ' → ~ φ ' ) .
From 6) – 10), we obtain
11) ∃ α 0 ' ∈ A 1 ' ∃ δ 00 ' ∈ S 0 ∃ δ ' ∈ S 0 ∃ φ ' ∈ S 0 [ ~ δ 00 ' → φ ' , δ 00 ' → δ ' , δ 00 ' → ~ δ ' , φ ' → δ 00 ' , δ 00 ' , δ ' , ~ δ ' ∈ A 1 & A 1 = S 0 ] ,
where
12) A * = α 0 ' → ~ δ 00 ' → φ ' ,
13) A * * = δ 00 ' → δ ' ,
14) A 1 = C n ( R 0 , Z 2 ∪ A * ∪ A * * ∪ α 0 ' → δ 00 ' → ~ δ ' ∪ ~ δ 00 ' → ~ φ ' ) ,
15) A 1 ' = C n ( R 0 , Z 2 ∪ A * * ∪ α 0 ' → δ 00 ' → ~ δ ' ∪ ~ δ 00 ' → ~ φ ' ) ,
what contradicts the steps 6) – 10).
□

4. The Main Result

Theorem 4.1.: R 0 , Z 2 ∉ C n s A . Proof.
Let
I) R 0 , Z 2 ∈ C n s A .
By Lemma 3.1, we have
II) ∀ α 0 ∈ A 1 ' ∀   δ 00 ∈ S 0 ∀ δ ∈ S 0
∀ φ ∈ S 0 [ C n ( R 0 , Z 2 ∪ A * ∪ A * * ∪
α 0 → δ 00 → ~ δ ∪ { ~ δ 00 → ~ φ } ) = S 0 ] , where
III) A * = α 0 → ~ δ 00 → φ
IV) A * * = δ 00 → δ
V) A 1 = C n ( R 0 , Z 2 ∪ A * ∪ A * * ∪
α 0 → δ 00 → ~ δ ∪ { ~ δ 00 → ~ φ } )
VI) A 1 ' = C n ( R 0 , Z 2 ∪ A * * ∪
α 0 → δ 00 → ~ δ ∪ { ~ δ 00 → ~ φ } ) .
Hence, by Theorem 1.3, we obtain
VII) ∀ α 0 ∈ A 1 ' ∀   δ 00 ∈ S 0 ∀ δ ∈ S 0
∀ φ ∈ S 0 α 0 , ~ δ 00 , ~ φ ∈ C n R 0 , Z 2 ∪ A * * ∪ α 0 → δ 00 → ~ δ ∪ ~ δ 00 → ~ φ ,
where
VIII) A * * = δ 00 → δ
IX) A 1 ' = C n ( R 0 , Z 2 ∪ A * * ∪
α 0 → δ 00 → ~ δ ∪ { ~ δ 00 → ~ φ } ) .
From VII) – IX), by Duns-Scottus law (Ex Falso Quodlibet), we get
X) ∀ α 0 ∈ A 1 ' A 1 ' = S 0 ,
where
XI) A * * = α 0 → α 0
XII) A 1 ' = C n ( R 0 , Z 2 ∪ A * * ∪
α 0 → α 0 → ~ α 0 ∪ { ~ α 0 → ~ α 0 } ) ,
and where
XIII) δ 00 ∈ { α 0 }
XIV) δ ∈ { α 0 }
XV) φ ∈ { α 0 } .
Hence, we get
XVI) ∀ α 0 ∈ S 0
C n R 0 , Z 2 ∪ α 0 → α 0 → ~ α 0 = S 0 .
Then, from XVI), by Theorem 1.3. we obtain
XVII) ∀ α 0 ∈ S 0 α 0 ∈ C n R 0 , Z 2 .
Hence, by Definition 1.1, we have
XVIII) R 0 , Z 2 ∉ C n s T , when
XIX) α 0 ∈ p ∧   ∼ p .
Then, by Duns-Scottus law (Ex Falso Quodlibet), and from the fact that C n R 0 , Z 2 ⊆ S 0 , we have
XX) C n R 0 , Z 2 = S 0 .
Hence, from Definition 1.2, we get
XXI) R 0 , Z 2 ∉ C n s 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 α 0 ∈ p ∧   ∼ p (so, α 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]).

6. Acknowledgments

The author would like to thank to Dr Teodor J. Stępień (author’s Father), for their common scientific cooperation on the issues of mathematical logic and foundations of mathematics. They were working among others, on the issue of inconsistency of the classical propositional calculus.
Unfortunately, because of some reasons independent on them, they were not able to finish their common work on this issue.
The author would like also to thank to the Editorial Board of the Journal of Mathematics and System Science for the permission to publish this paper in a preprint repository, before publishing this paper in this Journal.
The author declares no conflict of interest.

Note

1
Stępień, T. University of the National Education Commission, Kraków, Poland. E-mail: sfstepie@cyf-kr.edu.pl, lukasz.stepien@up.krakow.pl, URL: ltstepien.up.krakow.pl

References

  1. Andrews, P.B. An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Applied Logic Series; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 27. [Google Scholar]
  2. Armour-Garb, B. Consistent Inconsistency Theories. Inquiry 2007, 50, 639–654. [Google Scholar] [CrossRef]
  3. Bair, J.; Błaszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K.U.; Katz, M.G.; Kutateladze, S.S.; McGaffey, T.; Schaps, D.M.; Sherry, D. ; Shnider, SIs Mathematical History Written by the Victors? Notes Am. Math. Soc. 2013, 60, 886–904. [Google Scholar]
  4. Barrio, E. and Pailos, F. and Szmuc, D. What is a Paraconsistent Logic. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Cham, Switzerland, 2018. [Google Scholar]
  5. Başoğlu, Y.R. , “What is the O-Corner Interpretation and Does It Save the Traditional Square of Opposition ? Felsefe Ark. –Arch. Philos. 2019, 51, 37–59. [Google Scholar]
  6. Batens, D. From Copernicus to Ptolemy: Inconsistency and Method. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 1–33. [Google Scholar]
  7. Batens, D. Tutorial on Inconsistency-Adaptive Logics. In New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics; Beziau, J.Y., Chakraborty, M., Dutta, S., Eds, *!!! REPLACE !!!*, Eds.; Volume 152. Springer: New Delhi, India. 2015. [Google Scholar]
  8. Becker Arenhart, J.R. The Price of True Contradictions About the World. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Switzerland, 2018; pp. 11–31. [Google Scholar]
  9. Beirlaen, M. , Straßer, C. & Meheus, J. An Inconsistency-Adaptive Deontic Logic for Normative Conflicts. J. Philos. Log. 2013, 42, 285–315. [Google Scholar]
  10. Ben-Ari, M. Mathematical Logic for Computer Science; Springer: London, UK, 2012. [Google Scholar]
  11. van Bendegem, J.P. Inconsistencies in the History of Mathematics. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 43–58. [Google Scholar]
  12. Bertossi, L.; Hunter, A.; Schaub, T. Introduction to Inconsistency Tolerance; In Inconsistency, Tolerance, Bertossi, L., Hunter, A., Schaub, T., Eds.; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
  13. Besler, G. Philosophical and Mathematical Correspondence between Gottlob Frege and Bertrand Russell in the years 1902–1904. Some Uninvestigated Topics. Folia Philos. 2016, 35, 85–100. [Google Scholar]
  14. Besnard, P.; Schaub, T.; Tompits, H.; Woltran, S. Representing Paraconsistent Reasoning via Quanitfied Propositional Logic. In Inconsistency Tolerance; Bertossi, L., Hunter, A., Schaub, T., Eds.; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
  15. Błaszczyk, P.; Fila, M. Cantor on Infinitesimals Historical and Modern Perspectives. Bull. Sec. Log. 2020, 49, 149–179. [Google Scholar] [CrossRef]
  16. Bolzano, B.; Wissenschaftslehre, B.d., II. Sulzbach 1837 (based on the edition: Leipzig 1929).
  17. Bozhich, E.S. On arithmetic with the notion of `attainable number. Math. USSR-Izv. 1987, 29, 477–510. [Google Scholar] [CrossRef]
  18. Brewka, G.; Thimm, M.; Ulbricht, M. Strong inconsistency. Artif. Intellig. 2019, 267, 78–117. [Google Scholar] [CrossRef]
  19. Brown, B. Approximate Truth. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 81–104. [Google Scholar]
  20. Bueno, O. Mathematical Change and Inconsistency. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 59–80. [Google Scholar]
  21. Carbone, A.; Semmes, S. Making Proofs without Modus Ponens: An Introduction to the Combinatorics and Complexity of Cut Elimination. Bull. (New Ser. ) Amer. Math. Soc. 1997, 34, 131–159. [Google Scholar] [CrossRef]
  22. Carnielli, W.; Coniglio, M.E.; Marcos, J. Logics of Formal Inconsistency. In Handbook of Philosophical Logic; Gabbay, D.M., Guenther, F., Eds.; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
  23. Carnielli, W.; Rodrigues, A. On the Philosophy and Mathematics of the Logics of Formal Inconsistency. In New Directions in Paraconsistent Logic; Beziau, J.Y., Chakraborty, M., Dutta, S., Eds.; Springer Proceedings in Mathematics & Statistics; Springer: New Delhi, India, 2015; Volume 152. [Google Scholar]
  24. Carnielli, W. and Mariano, H.L. and Matulovic, M. “Reconciling First-Order Logic to Algebra. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Switzerland, 2018. [Google Scholar]
  25. Carnielli, W.; Malinowski, J. Contradictions, from Consistency to Inconsistency. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Cham, Switzerland, 2018. [Google Scholar]
  26. Chow, T.Y. The Consistency of Arithmetic. Math. Intell. 2019, 41, 22. [Google Scholar] [CrossRef]
  27. Coniglio Marcelo, E.; Toledo Guilherme, V. From Inconsistency to Incompatibility. Log. Log. Philos. 2023, 32, 181–216. [Google Scholar] [CrossRef]
  28. Curry, H. The inconsistency of certain formal logics. J. Symb. Log. 1942, 7, 115–117. [Google Scholar] [CrossRef]
  29. Da Costa, N.C.A. On the Theory of Inconsistent Formal Systems. Notre Dame J. Form. Log. 1974, XV, 497–510. [Google Scholar] [CrossRef]
  30. da Costa, N.; French, S. Inconsistency in Science: A Partial Perspective. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 105–118. [Google Scholar]
  31. da Costa, N.C.A. , Krause, D. Physics, inconsistency, and quasi-truth. Synthese 2014, 191, 3041–3055. [Google Scholar] [CrossRef]
  32. Criscuolo, G.; Giunchiglia, F.; Serafini, L. A Foundation for Metareasoning Part I: The Proof Theory. J. Log. Comput. 2002, 12, 167–208. [Google Scholar] [CrossRef]
  33. Czelakowski, J.; Dziobak, W. On Truth-Schemes for Intensional Logics. Rep. Math. Log. 2006, 41, 151–171. [Google Scholar]
  34. D’Ottaviano, I.; Loffredo, M.; Milton, C. Augustinis, Analytical tableaux for da Costa’s hierarchy of paraconsistent logics Cn, 1 ≤ n < ω. J. Appl. Non-Class. Log. 2005, 15, 69–103. [Google Scholar]
  35. van Dalen, D. Logic and Structure; Springer: London, UK, 2013. [Google Scholar]
  36. Dubois, D.; Prade, H. Being Consistent About Inconsistency: Toward the Rational Fusing of Inconsistent Propositional Logic Bases. In The Road to Universal Logic. Studies in Universal Logic; Koslow, A., Buchsbaum, A., Eds.; Birkhäuser: Cham, Switzerland, 2015. [Google Scholar]
  37. Eklund, M. Inconsistency and Replacement. Inquiry 2019, 62, 387–402. [Google Scholar] [CrossRef]
  38. Engesser, K.; Gabbay, D.; Lehmann, D. Nonmonotonicity and Holicity in Quantum Logic. In Handbook of Quantum Logic and Quantum Structures: Quantum Logic; Engesser, K., Gabbay, D.M., Lehmann, D., Eds, *!!! REPLACE !!!*, Eds.; Elsevier: Amsterdam, The Netherlands, 2009; pp. 587–623. [Google Scholar]
  39. Ershov, Y.L.; Palyutin, E.A. Mathematical Logic. Translated by Shokurov V. Mir Publishers, Moscow 1984.
  40. Estrada-González, L.E. and del Rosario Martínez-Ordaz, M.R. The Possibility and Fruitfulness of a Debate on the Principle of Non-contradiction. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Chem, Switzerland, 2018. [Google Scholar]
  41. Finger, M. Quantitative Logic Reasoning. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Chem, Switzerland, 2018. [Google Scholar]
  42. Freire, R.A. Interpretation and Truth in Set Theory. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Chem, Switzerland, 2018. [Google Scholar]
  43. Friend, M.; del Rosario Martínez-Ordaz, M.R. Keeping Globally Inconsistent Scientific Theories Locally Consistent. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Chem, Switzerland, 2018. [Google Scholar]
  44. Gabbay, D.; Hunter, A. Making inconsistency respectable: A logical framework for inconsistency in reasoning, part I—A position paper. In Fundamentals of Artificial Intelligence Research; Jorrand, P., Kelemen, J., Eds.; FAIR 1991. Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 1991; Volume 535. [Google Scholar]
  45. Ganov, V.A. Inconsistency of Nudel’man’s metatheory. Algebra Log. 1995, 34, 41–43. [Google Scholar] [CrossRef]
  46. Gaytán, D.; D'Ottaviano, I.M.L.; Morado, R. Provided You’re not Trivial: Adding Defaults and Paraconsistency to a Formal Model of Explanation. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Chem, Switzerland, 2018. [Google Scholar]
  47. Geuvers, H. Inconsistency of classical logic in type theory.
  48. Available online:. Available online: www.researchgate.net/publication/2414853_Inconsistency_of_Classical_Logic_in_Type_Theory (accessed on 2007).
  49. Goddard, L. The inconsistency of traditional logic. Australas. J. Philos. 1998, 76, 152–164. [Google Scholar] [CrossRef]
  50. Goddard, L. The inconsistency of Aristotelian logic? Australas. J. Philos. 2000, 78, 434–437. [Google Scholar] [CrossRef]
  51. Grant, J. Measuring Inconsistency in Generalized Propositional Logic. Log. Universalis 2020, 14, 331–356. [Google Scholar] [CrossRef]
  52. Grygiel, W. Towards the consistency of an inconsistent mind. Philos. Probl. Sci. 2010, 47, 70–88. [Google Scholar]
  53. Grzegorczyk, A. An Outline of Mathematical Logic: Fundamental Results and Notions Explained with All Details; Wojtasiewicz, O., Zawadowski, W., Synthese Library, Translater, D. Reidel Publishing Company: Dordrecht, Holland, Boston, MA, USA, Eds.; PWN: Warszawa, Poland, 1974; Volume 70. [Google Scholar]
  54. Herbrand, J. Recherches sur la Théorie de la Demonstration; Warszawa, Poland, 1930.
  55. Hertrich-Woleński, J. Contradiction and what then? Zagadnienia Nauk. 2015, 3, 217–228. (In Polish) [Google Scholar]
  56. Hohol, M. Mind: between inconsistency and non-triviality. Philos. Probl. Sci. 2010, 47, 89–108. [Google Scholar]
  57. Hunter, G. METALOGIC. An Introduction to the Metatheory of Standard First Order Logic, PALGRAVE MACMILLAN 1971.
  58. Jaśkowski, S. Rachunek zdaή dla systemόw dedukcyjnych sprzecznych. Stud. Soc. Sci. Torun. A 1948, 1, 57–77. [Google Scholar]
  59. Kahle, R. Gentzen’s Consistency Proof in Context. In Gentzen’s Centenary; Kahle, R., Rathjen, M., Eds.; The Quest for Consistency Springer International Publishing: Switzerland, 2015. [Google Scholar]
  60. Kifer, M.; Lozinskii, E.L. A logic for reasoning with inconsistency. J. Autom. Reason. 1992, 9, 179–215. [Google Scholar] [CrossRef]
  61. Kleene, S.C.; Rosser, J.B. The Inconsistency of Certain Formal Logics. Ann. Math. 1935, 36, 630–636. [Google Scholar] [CrossRef]
  62. Konikowska, B. A decompositional deduction system for a logic featuring inconsistency and uncertainty. J. Appl. Non-Class. Log. 2005, 15, 25–44. [Google Scholar] [CrossRef]
  63. Krajewski, S. On Gödel’s Theorem and Mechanism: Inconsistency or Unsoundness is Unavoidable in any Attempt to ‘Out-Gödel’ the Mechanist. Fundam. Informaticae 2007, 81, 1–9. [Google Scholar]
  64. Lassaigne, R.; de Rougemont, M. Logic and Complexity; Springer: London, UK, 2004. [Google Scholar]
  65. Malinowski, G. Inferential Paraconsistency. Log. Log. Philos. 2000, 8, 83–89. [Google Scholar] [CrossRef]
  66. Marcelino, S.; Caleiro, C.; Rivieccio, U. Plug and Play Negations. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Chem, Switzerland, 2018. [Google Scholar]
  67. Mangraviti, F.; Tedder, A. Consistent Theories in Inconsistent Logics. J. Philos. Log. 2023, 52, 1133–1148. [Google Scholar] [CrossRef]
  68. Marciszewski, W. Logic, Modern, History Of. In Dictionary of Logic as Applied in the Study of Language; Marciszewski, W., Ed.; Concepts/Methods/Theories; Nijhoff International Philosophy Series; Springer Science + Business Media: Dordrecht, The Netherlands, 1981; Volume 9, pp. 183–200. [Google Scholar]
  69. Marciszewski, W. Sentence Logic. In Dictionary of Logic as Applied in the Study of Language; Marciszewski, W., Ed.; Concepts/Methods/Theories; Nijhoff International Philosophy Series; Springer Science + Business Media: Dordrecht, The Netherlands, 1981; Volume 9, pp. 334–342. [Google Scholar]
  70. Martin, B.J.L. The Logical and Philosophical Foundations for the Possibility of True Contradictions. Ph.D. Thesis, University of London, London, UK, 2014. [Google Scholar]
  71. Martinez, M.V.; Molinaro, C.; Subrahmanian, V.S.; Amgoud, L. A General Framework for Reasoning on Inconsistency; Springer: New York, NY, USA; Heidelberg, Germany, Dordrecht, The Netherlands; London, UK, 2013. [Google Scholar]
  72. Meheus, J. How to Reason Sensibly yet Naturally from Inconsistencies. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 1–33. [Google Scholar]
  73. Miller, A.I. Inconsistent Reasoning toward Consistent Theories. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 35–42. [Google Scholar]
  74. Mirek, R. Relevant paraconsistent logic. Argument 2021, 11, 287–292. [Google Scholar]
  75. Mortensen, C. Aristotle’s Thesis in Consistent and Inconsistent Logics. Stud. Log. 1981, XLIII, 107–116. [Google Scholar] [CrossRef]
  76. Nelson, E. Elements. arXiv 2015, arXiv:1510.00369. [Google Scholar]
  77. Nersessian, N.J. Inconsistency, Generic Modeling, and Conceptual Change in Science. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 1–33. [Google Scholar]
  78. Nickles, T. From Copernicus to Ptolemy: Inconsistency and Method. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity, Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 1–33. [Google Scholar]
  79. Norton, J.D. A Paradox in Newtonian Gravitation Theory II. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 1–33. [Google Scholar]
  80. Paleo, B.W. Para-Disagreement Logics and Their Implementation Through Embedding in Coq and SMT. In Contradictions, from Consistency to Inconsistency; Carnielli, W., Malinowski, J., Eds.; Springer Nature: Chem, Switzerland, 2018. [Google Scholar]
  81. Patterson, D. Inconsistency Theories: The Significance of Semantic Ascent. Inquiry 2007, 50, 575–589. [Google Scholar] [CrossRef]
  82. Perzanowski, J. Parainconsistency, or Inconsistency Tamed, Investigated and Exploited. Log. Log. Philos. 2001, 9, 5–24. [Google Scholar] [CrossRef]
  83. Picollo, L. Truth in a Logic of Formal Inconsistency: How classical can it get? Log. J. IGPL 2023, 28, 771–806. [Google Scholar] [CrossRef]
  84. Pogorzelski, W.A. The Classical Propositional Calculus; PWN: Warszawa, Poland, 1975. [Google Scholar]
  85. Pogorzelski, W.A.; Wojtylak, P. Completeness Theory for Propositional Logics; Birkhäuser-Verlag AG: Basel, Switzerland; Boston, MA, USA; Berlin, Germany, 2008. [Google Scholar]
  86. Post, E.L. Introduction to a General Theory of Elementary Propositions. Am. J. Math. 1921, 43, 163–185. [Google Scholar] [CrossRef]
  87. Priest, G. Inconsistency and the Empirical Sciences. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2, pp. 118–128. [Google Scholar]
  88. Priest, G. Wittgenstein’s remarks on Gödel’s theorem. In Wittgenstein’s Lasting Significance; Kölbel, M., Weiss, B., Eds.; Routledge Taylor & Francis Group: London, UK, New York, NY, USA, 2005. [Google Scholar]
  89. Raftery, J.G. Inconsistency lemmas in algebraic logic. Math. Log. Quaterly 2013, 59, 393–406. [Google Scholar] [CrossRef]
  90. Rasiowa, H. Introduction to Modern Mathematics, North-Holland Publishing Company Amsterdam; Wojtasiewicz, O., Translator, *!!! REPLACE !!!*, Eds.; PWN: Warszawa, Poland, 1973. [Google Scholar]
  91. Raspa, V. Łukasiewicz on the Principle of Contradiction. J. Philos. Res. 1999, 24, 57–112. [Google Scholar] [CrossRef]
  92. Rautenberg, W. A Concise Introduction to Mathematical Logic; Springer Science + Business Media, LLC 2010.
  93. Rozonoer, L.I. On identification of inconsistencies in formal theories I. Avtom. I Telemekhanika 1983, 6, 113–124. [Google Scholar]
  94. Rozonoer, L.I. On identification of inconsistencies in formal theories II. Avtom. I Telemekhanika 1983, 7, 97–104. [Google Scholar]
  95. Rudnicki, K. Humans do not reason from contradictory premises. Psychol. Asp. Paraconsistency 2020, submitted.
  96. Setzer, A. The Use of Trustworthy Principles in a Revised Hilbert’s Program. In Gentzen’s Centenary. The Quest for Consistency; Kahle, R., Rathjen, M., Eds.; Springer International Publishing: Chem, Switzerland 2015. [Google Scholar]
  97. Srivastava, S.M. A Course on Mathematical Logic; Springer Science + Business Media, LLC 2008.
  98. Stępień; TJ; Stępień; ŁT. On the Consistency of the Arithmetic System. J. Math. Syst. Sci. 2017, 7, 43–55. [Google Scholar]
  99. Stępień, T.J.; Stępień Ł, T. Theorem on Inconsistency of the Classical Logic. In Conference Proceedings, Paris, France, ; World Academy of Science, Engineering and Technology: 2017; Volume 19, Part XIII, p.1558. 21–22 September.
  100. Stępień, T.J.; Stępień, Ł.T. On the Inconsistency of Classical Propositional Calculus. J. Math. Syst. Sci. 2020, 10, 13–14. [Google Scholar]
  101. Stępień, T.J.; Stępień, Ł.T. On the Inconsistency of Classical Logic. a talk delivered at LXVI Cracow Logic Conference, 3–4 November.
  102. Surma, S.J. On a Sequence of Contradiction-Tolerating Logics. In Consciousness, Knowledge, and Truth; Poli, R., Ed.; Springer: Dordrecht, The Netherlands, 1993. [Google Scholar]
  103. Tanaka, K.; Girard, P. Against Classical Paraconsistent Metatheory. Analysis 2023, 3, anac093. [Google Scholar] [CrossRef]
  104. Tarski, A. Über einige fundamentalle Begriffe der Metamathematik. Comptes Rendus Des Séances De La Société De Sci. Et Des Lett. De Vars. 1930, 23.
  105. Tuziak, R. Logic of Inconsistency. Some remarks on paraconsistent logic. Atut Oficyna Wydawnicza.
  106. Urchs, M. SCRAPING HEAVENS. On Inconsistencies in Sciences. Log. Log. Philos. 1999, 7, 151–165. [Google Scholar] [CrossRef]
  107. Vidal-Rosset, J. Paraconsistent, Tennant’s Logic is Inconsistent. Available online: www.vidal-rosset.net 2021 (accessed on 2022).
  108. Voevodsky, V. “What if current foundations of mathematics are inconsistent?” Lecture at the celebration of the 80s anniversary of the IAS. 25 September.
  109. Weber, E.; De Clercq, K. Why the Logic of Explanation is Inconsistency-adaptive”. In Inconsistency in Science; Meheus, J., Ed.; ORIGINS: Studies in the sources of scientific creativity; Springer Science + Business Media: Dordrecht, The Netherlands, 2002; Volume 2. [Google Scholar]
  110. Weber, Z. Paradoxes and Inconsistent Mathematics; Cambridge University Press: 2021.
  111. Weber, Z.; Badia, G.; Girard, P. What is an Inconsistent Truth Table? Australas. J. Philos. 2016, 94, 533–548. [Google Scholar] [CrossRef]
  112. Woleński, J. Metalogical Properties, Being Logical and Being Formal. Log. Log. Philos. 2002, 10, 211–221. [Google Scholar] [CrossRef]
  113. Woods, J. How Robust Can Inconsistency Get? IFCoLog J. Log. Its Appl. 2014, 1, 177–216. [Google Scholar]
  114. Wójcicki, R. Lectures on Propositional Calculi, Ossolineum–The Publishing House of the Polish Academy of Sciences, 1984.
  115. Wybraniec-Skardowska, U. On the Mutual Definability of the Notions of Entailment, Rejection, and Inconsistency. Axioms 2016, 5, 1–19. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
Prerpints.org logo

Preprints.org is a free preprint server supported by MDPI in Basel, Switzerland.

facebook logotwitter logolinkedin logo
MDPI Initiatives
Important Links
Subscribe

Disclaimer

Terms of Use

Privacy Policy

© 2024 MDPI (Basel, Switzerland) unless otherwise stated