# Deduction in Non-Fregean Propositional Logic SCI

^{*}

## Abstract

**:**

## 1. Introduction

**F1**- All names and all sentences have meaning and denotation. Meaning is not the same as denotation.
**F2**- A name and a sentence are the proper names of their denotations.
**F3**- Only one logical value can be assigned to each sentence: true or false.
**F4**- If two expressions have the same denotation, then they are exchangeable in any propositional context of a sentence without changing the logical value of that sentence.
**F5**- If two sentences are exchangeable in any propositional context of a sentence without changing its logical value, then they have the same denotation.

## 2. The Non-Fregean Propositional Logic $\mathsf{SCI}$

- $\mathbb{V}=\{{p}_{1},{p}_{2},{p}_{3},\dots \}$—a countable infinite set of propositional variables;
- $\{\neg ,\vee ,\wedge ,\to ,\leftrightarrow ,\equiv \}$—the set of propositional operations of negation ¬, disjunction ∨, conjunction ∧, implication →, equivalence ↔, and identity ≡.

- (SCI1) $\sim a\in D\mathrm{iff}(a\notin D)$;
- (SCI2) $a\bigsqcup b\in D\mathrm{iff}(a\in D\mathrm{or}b\in D)$;
- (SCI3) $a\sqcap b\in D\mathrm{iff}(a\in D\mathrm{and}b\in D)$;
- (SCI4) $a\Rightarrow b\in D\mathrm{iff}(a\notin D\mathrm{or}b\in D)$;
- (SCI5) $a\iff b\in D\mathrm{iff}(a\in D\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}b\in D)$;
- (SCI6) $a\circ b\in D\mathrm{iff}a=b$.

$v(\neg \phi )=\sim v\left(\phi \right)$ | $v(\phi \to \psi )=v\left(\phi \right)\Rightarrow v\left(\psi \right)$ |

$v(\phi \vee \psi )=v\left(\phi \right)\bigsqcup v\left(\psi \right)$ | $v(\phi \wedge \psi )=v\left(\phi \right)\sqcap v\left(\psi \right)$ |

$v(\phi \leftrightarrow \psi )=v\left(\phi \right)\iff v\left(\psi \right)$ | $v(\phi \equiv \psi )=v\left(\phi \right)\circ v\left(\psi \right)$ |

**Proposition**

**1.**

**Theorem**

**1**

**.**The logic $\mathsf{SCI}$ has the finite model property, i.e., every satisfiable $\mathsf{SCI}$-formula is satisfiable in a finite $\mathsf{SCI}$-model. Furthermore, the logic $\mathsf{SCI}$ is decidable.

**Corollary**

**1.**

- (${\equiv}_{1}$)$\phi \equiv \phi $;
- (${\equiv}_{2}$) $(\phi \equiv \psi )\to (\neg \phi \equiv \neg \psi )$;
- (${\equiv}_{3}$) $(\phi \equiv \psi )\to (\phi \to \psi )$;
- (${\equiv}_{4}$) $\left[\right(\phi \equiv \psi )\wedge (\vartheta \equiv \xi \left)\right]\to \left[\right(\phi \#\vartheta )\equiv (\psi \#\xi \left)\right]$, for $\#\in \{\wedge ,\vee ,\to ,\leftrightarrow ,\equiv \}$.

**Fact**

**2.**

- 1.
- φ is provable in the classical propositional logic.
- 2.
- φ is $\mathsf{SCI}$-provable.

**Theorem**

**2**

- 1.
- φ is $\mathsf{SCI}$-provable.
- 2.
- φ is $\mathsf{SCI}$-valid.

$\sim a\stackrel{\mathrm{df}}{=}\left(\right)open="\{"\; close>\begin{array}{cc}0,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{i}\mathrm{f}a\ne 0,\hfill \\ 1,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\hfill \end{array}$ | $a\bigsqcup b\stackrel{\mathrm{df}}{=}\left(\right)open="\{"\; close>\begin{array}{cc}0,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{i}\mathrm{f}a=0\mathrm{a}\mathrm{n}\mathrm{d}b=0,\hfill \\ 1,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\hfill \end{array}$ |

$a\Rightarrow b\stackrel{\mathrm{df}}{=}\left(\right)open="\{"\; close>\begin{array}{cc}0,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{i}\mathrm{f}a\ne 0\mathrm{a}\mathrm{n}\mathrm{d}b=0,\hfill \\ 1,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\hfill \end{array}$ | $a\sqcap b\stackrel{\mathrm{df}}{=}\left(\right)open="\{"\; close>\begin{array}{cc}0,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{i}\mathrm{f}a=0,\mathrm{o}\mathrm{r}b=0,\hfill \\ 1,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{i}\mathrm{f}b=2\mathrm{a}\mathrm{n}\mathrm{d}a\ne 0,\hfill \\ 2,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\hfill \end{array}$ |

$a\iff b\stackrel{\mathrm{df}}{=}\left(\right)open="\{"\; close>\begin{array}{cc}0,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{i}\mathrm{f}a\ne 0,b=0\mathrm{o}\mathrm{r}a=0,b\ne 0,\hfill \\ 1,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\hfill \end{array}$ | |

$a\circ b\stackrel{\mathrm{df}}{=}\left(\right)open="\{"\; close>\begin{array}{cc}0,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{i}\mathrm{f}a\ne b,\hfill \\ a,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{i}\mathrm{f}a=b\mathrm{a}\mathrm{n}\mathrm{d}a\ne 0,\hfill \\ 1,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\hfill \end{array}$ |

## 3. Sequent-Style Formalizations for $\mathsf{SCI}$

**Theorem**

**3**

- 1.
- φ is $\mathsf{SCI}$-valid;
- 2.
- φ is ${\mathsf{G}}_{\mathsf{SCI}}$-provable.

## 4. Dual Tableau System ${\mathsf{DT}}_{\mathsf{SCI}}$

- the formula $\phi $ is at the root of this tree,
- each node except the root is obtained by an application of a ${\mathsf{DT}}_{\mathsf{SCI}}$-rule to its predecessor node,
- a node does not have successors whenever it is a ${\mathsf{DT}}_{\mathsf{SCI}}$-axiomatic set or none of the rules applies to its set of formulas.

**Theorem**

**4**

- 1.
- φ is $\mathsf{SCI}$-valid;
- 2.
- φ is ${\mathsf{DT}}_{\mathsf{SCI}}$-provable.

## 5. A Substitution-Free Dual Tableau for $\mathsf{SCI}$

$\left({\mathsf{Ax}}_{{\mathsf{DT}}_{\mathsf{SCI}}^{*}}^{1}\right)$$\{\phi \equiv \phi \}\cup X$, | $\left({\mathsf{Ax}}_{{\mathsf{DT}}_{\mathsf{SCI}}^{*}}^{2}\right)$$\{\phi ,\neg \phi \}\cup X$, |

$\left({\mathsf{Ax}}_{{\mathsf{DT}}_{\mathsf{SCI}}^{*}}^{3}\right)$$\{\phi ,\neg \psi ,\phi \not\equiv \psi \}\cup X$, | $\left({\mathsf{Ax}}_{{\mathsf{DT}}_{\mathsf{SCI}}^{*}}^{4}\right)$$\{\neg \phi ,\psi ,\phi \not\equiv \psi \}\cup X$. |

**Proposition**

**3**

**Proof.**

**Proposition**

**4**

**Proof.**

**Theorem**

**5**

**Proof.**

**Proposition**

**5**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

- ${U}^{b}=\{{\left[\phi \right]}_{{R}_{\circ}}:\phi \in \mathbb{FOR}\}$,
- ${D}^{b}=\{{\left[\phi \right]}_{{R}_{\circ}}:\phi \in {\bigcup}_{n\in \mathbb{N}}{D}_{n}\}$, where
- ${D}_{0}=\{\phi \in {\mathbb{FOR}}^{0}:\neg \phi \in b\}$,
- ${D}_{n+1}={D}_{n+1}^{1}\cup {D}_{n+1}^{2}$, for${D}_{n+1}^{1}=\{\neg \phi \in {\mathbb{FOR}}^{n+1}:\phi \notin {D}_{n}\}$${D}_{n+1}^{2}=\{\phi \to \psi \in {\mathbb{FOR}}^{n+1}:\phi \notin {\bigcup}_{k\le n}{D}_{k}or\psi \in {\bigcup}_{k\le n}{D}_{k})\}$,

- operations ${\sim}^{b}$, ${\Rightarrow}^{b}$, ${\circ}^{b}$ are defined as:${\sim}^{b}{\left[\phi \right]}_{{R}_{\circ}}\stackrel{\mathrm{df}}{=}{[\neg \phi ]}_{{R}_{\circ}}$${\left[\phi \right]}_{{R}_{\circ}}{\Rightarrow}^{b}{\left[\psi \right]}_{{R}_{\circ}}\stackrel{\mathrm{df}}{=}{[\phi \to \psi ]}_{{R}_{\circ}}$${\left[\phi \right]}_{{R}_{\circ}}{\circ}^{b}{\left[\psi \right]}_{{R}_{\circ}}=\stackrel{\mathrm{df}}{=}{[\phi \equiv \psi ]}_{{R}_{\circ}}$.

**Proposition**

**8**

**Proof.**

(*) | If $d\left(\phi \right)=n$, then ${\left[\phi \right]}_{{R}_{\circ}}\in {D}^{b}$ iff $\phi \in {D}_{n}$. |

(**) | If $\phi \in {D}_{n}$, then $d\left(\phi \right)=n$. |

(***) | If $d\left(\phi \right)=n$ and $k\ne n$, then $\phi \notin {D}_{k}$. |

**Proposition**

**9.**

**Proposition**

**10**

**Proof.**

**Theorem**

**6**

**Proof.**

**Theorem**

**7**

- 1.
- φ is $\mathsf{SCI}$-valid;
- 2.
- φ is ${\mathsf{DT}}_{\mathsf{SCI}}^{*}$-provable.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Suszko, R. Non-Fregean logic and theories. Analele Univ. Bucur. Acta Log.
**1968**, 11, 105–125. [Google Scholar] - Suszko, R. Abolition of the Fregean axiom. In Logic Colloquium: Symposium on Logic Held at Boston, 1972–73; Parikh, R., Ed.; Lecture Notes in Mathematics; Springer: Heidelberg, Germany, 1975; Volume 453, pp. 169–239. [Google Scholar]
- Bloom, S.; Suszko, R. Investigation into the sentential calculus with identity. Notre Dame J. Form. Log.
**1972**, 13, 289–308. [Google Scholar] [CrossRef] - Golińska-Pilarek, J.; Huuskonen, T. Number of extensions of non-Fregean logics. J. Philos. Log.
**2005**, 34, 193–206. [Google Scholar] [CrossRef] - Golińska-Pilarek, J.; Huuskonen, T. Logic of descriptions. A new approach to the foundations of mathematics and science. Stud. Log. Gramm. Rhetor.
**2012**, 40, 63–94. [Google Scholar] - Golińska-Pilarek, J.; Huuskonen, T. Grzegorczyk’s non-Fregean logics and their formal properties. In Applications of Formal Philosophy; Urbaniak, R., Payette, G., Eds.; Logic, Argumentation and Reasoning; Springer International Publishing: New York, NY, USA, 2017; Volume 14, pp. 243–263. [Google Scholar]
- Golińska-Pilarek, J.; Huuskonen, T. A mystery of Grzegorczyk’s logic of descriptions. In The Lvov-Warsaw School. Past and Present; Garrido, A., Wybraniec-Skardowska, U., Eds.; Studies in Universal Logic; Springer Nature: Stuttgart, Germany, 2018; pp. 731–745. [Google Scholar]
- Golińska-Pilarek, J.; Huuskonen, T. Non-Fregean propositional logic with quantifiers. Notre Dame J. Form. Log.
**2016**, 57, 249–279. [Google Scholar] [CrossRef] - Suszko, R. Identity connective and modality. Stud. Log.
**1971**, 27, 7–39. [Google Scholar] [CrossRef] - Malinowski, G. Identity, many-valuedness and referentiality. Log. Log. Philos.
**2013**, 22, 375–387. [Google Scholar] [CrossRef] - Golińska-Pilarek, J. On the minimal non-Fregean Grzegorczyk’s logic. Stud. Log.
**2016**, 104, 209–234. [Google Scholar] [CrossRef] - Michaels, A. A uniform proof proceduree for SCI tautologies. Stud. Log.
**1974**, 33, 299–310. [Google Scholar] [CrossRef] - Wasilewska, A. A sequence formalization for SCI. Stud. Log.
**1976**, 35, 213–217. [Google Scholar] [CrossRef] - Chlebowski, S. Sequent calculi for SCI. Stud. Log.
**2018**, 106, 541–563. [Google Scholar] [CrossRef] - Golińska-Pilarek, J. Rasiowa-Sikorski proof system for the non-Fregean sentential logic SCI. J. Appl.-Non-Class. Log.
**2007**, 17, 511–519. [Google Scholar] [CrossRef] - Orłowska, E.; Golińska-Pilarek, J. Dual Tableaux: Foundations, Methodology, Case Studies; Trends in Logic; Springer: Dordrecht Heidelberg London New York, NY, USA, 2011; Volume 33. [Google Scholar]
- Rasiowa, H.; Sikorski, R. On Gentzen theorem. Fundam. Math.
**1960**, 48, 57–69. [Google Scholar] [CrossRef]

**Figure 1.**A ${\mathsf{G}}_{\mathsf{SCI}}$-proof of a formula $({p}_{1}\equiv {p}_{2})\to ({p}_{1}\to {p}_{2})$.

**Figure 2.**A ${\mathsf{G}}_{\mathsf{SCI}}$-proof of a formula $({p}_{1}\equiv {p}_{2})\to [({p}_{2}\equiv {p}_{3})\to ({p}_{1}\equiv {p}_{3})]$.

**Figure 3.**A ${\mathsf{DT}}_{\mathsf{SCI}}$-proof for the formula $({p}_{1}\equiv {p}_{2})\to ({p}_{1}\to {p}_{2})$.

**Figure 4.**A ${\mathsf{DT}}_{\mathsf{SCI}}$-proof for $({p}_{1}\equiv {p}_{2})\to [({p}_{2}\equiv {p}_{3})\to ({p}_{1}\equiv {p}_{3})]$.

**Figure 5.**A ${\mathsf{DT}}_{\mathsf{SCI}}^{*}$-proof for the formula $({p}_{1}\equiv {p}_{2})\to ({p}_{1}\to {p}_{2})$.

**Figure 6.**A ${\mathsf{DT}}_{\mathsf{SCI}}^{*}$-proof for $({p}_{1}\equiv {p}_{2})\to [({p}_{2}\equiv {p}_{3})\to ({p}_{1}\equiv {p}_{3})]$.

(${\neg}_{L}$) $\frac{\mathsf{\Gamma}\u22a2\phi ,\Delta}{\mathsf{\Gamma},\neg \phi \u22a2\Delta}$ (${\neg}_{R}$) $\frac{\mathsf{\Gamma},\phi \u22a2\Delta}{\mathsf{\Gamma}\u22a2\neg \phi ,\Delta}$, |

(${\to}_{L}$) $\frac{\mathsf{\Gamma}\u22a2\phi ,\Delta \phantom{\rule{1.em}{0ex}}|\phantom{\rule{1.em}{0ex}}{\mathsf{\Gamma}}^{\prime},\psi \u22a2{\Delta}^{\prime}}{\mathsf{\Gamma},{\mathsf{\Gamma}}^{\prime},\phi \to \psi \u22a2\Delta ,{\Delta}^{\prime}}$ (${\to}_{R}$) $\frac{\mathsf{\Gamma},\phi \u22a2\psi ,\Delta}{\mathsf{\Gamma}\u22a2\phi \to \psi ,\Delta}$, |

where $\phi ,\psi $ are any $\mathsf{SCI}$-formulas |

$\mathsf{\Gamma},{\mathsf{\Gamma}}^{\prime},\Delta ,{\Delta}^{\prime}$ are any finite (possibly empty) sequences of $\mathsf{SCI}$-formulas |

(${\equiv}_{{\mathsf{G}}_{\mathsf{SCI}}}$) $\frac{\mathsf{\Gamma},\Sigma ,{\mathsf{\Gamma}}^{\prime}[\phi /\psi ]\u22a2\Delta ,{\Delta}^{\prime}[\phi /\psi ]\phantom{\rule{1.em}{0ex}}|\phantom{\rule{1.em}{0ex}}\mathsf{\Gamma},{\mathsf{\Gamma}}^{\prime}[\phi /\psi ]\u22a2\Delta ,\Sigma ,{\Delta}^{\prime}[\phi /\psi ]}{\mathsf{\Gamma},\phi \equiv \psi ,{\mathsf{\Gamma}}^{\prime}\u22a2\Delta ,{\Delta}^{\prime}}$, |

where $\phi ,\psi $ are any $\mathsf{SCI}$-formulas and $\mathsf{\Sigma}$ is the sequence $\phi ,\psi ,\phi \equiv \psi ,\psi \equiv \phi $ |

$\mathsf{\Gamma},\Delta $ are atomic sequences and ${\mathsf{\Gamma}}^{\prime},{\Delta}^{\prime}$ are identities sequences |

(${W}_{L}$) $\frac{\mathsf{\Gamma}\u22a2\Delta}{\mathsf{\Gamma},\phi \u22a2\Delta}$ (${W}_{R}$) $\frac{\mathsf{\Gamma}\u22a2\Delta}{\mathsf{\Gamma}\u22a2\phi ,\Delta}$ |

(${C}_{L}$) $\frac{\mathsf{\Gamma},\phi ,\phi \u22a2\Delta}{\mathsf{\Gamma},\phi \u22a2\Delta}$ (${C}_{R}$) $\frac{\mathsf{\Gamma}\u22a2\phi ,\phi ,\Delta}{\mathsf{\Gamma}\u22a2\phi ,\Delta}$ |

(${P}_{L}$) $\frac{\mathsf{\Gamma},\phi ,\psi ,{\mathsf{\Gamma}}^{\prime}\u22a2\Delta}{\mathsf{\Gamma},\psi ,\phi ,{\mathsf{\Gamma}}^{\prime}\u22a2\Delta}$ (${P}_{R}$) $\frac{\mathsf{\Gamma}\u22a2\Delta ,\phi ,\psi ,{\Delta}^{\prime}}{\mathsf{\Gamma}\u22a2\Delta ,\psi ,\phi ,{\Delta}^{\prime}}$ |

($\mathsf{cut}$) $\frac{\mathsf{\Gamma}\u22a2\phi ,\Delta \phantom{\rule{1.em}{0ex}}|\phantom{\rule{1.em}{0ex}}\phi ,{\mathsf{\Gamma}}^{\prime}\u22a2{\Delta}^{\prime}}{\mathsf{\Gamma},{\mathsf{\Gamma}}^{\prime}\u22a2\Delta ,{\Delta}^{\prime}}$ |

where $\phi ,\psi $ are any $\mathsf{SCI}$-formulas |

$\mathsf{\Gamma},{\mathsf{\Gamma}}^{\prime},\Delta ,{\Delta}^{\prime}$ are any finite (possibly empty) sequences of $\mathsf{SCI}$-formulas |

(→) $\frac{\{\phi \to \psi \}\cup X}{\{\neg \phi ,\psi \}\cup X}$ ($\neg \phantom{\rule{0.166667em}{0ex}}\to $) $\frac{\{\neg (\phi \to \psi \left)\right\}\cup X}{\left\{\phi \right\}\cup X\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}\right\{\neg \psi \}\cup X}$ |

(¬) $\frac{\{\neg \neg \phi \}\cup X}{\left\{\phi \right\}\cup X}$ |

where $\phi $ and $\psi $ are any $\mathsf{SCI}$-formulas, |

and X is a finite (possibly empty) set of $\mathsf{SCI}$-formulas. |

(≡) $\frac{\left\{\phi \right(\psi \left)\right\}\cup X}{\{\psi \equiv \vartheta ,\phi (\psi \left)\right\}\cup X\phantom{\rule{0.277778em}{0ex}}\left|\phantom{\rule{0.277778em}{0ex}}\right\{\phi (\psi /\vartheta ),\phi \left(\psi \right)\}\cup X}$ |

where $\phi $ and $\vartheta $ are any $\mathsf{SCI}$-formulas, $\psi $ is any subformula of $\phi $, |

$\phi \left(\vartheta \right)$ is obtained from $\phi \left(\psi \right)$ by replacing some occurrences of $\psi $ with $\vartheta $ |

and X is a finite (possibly empty) set of $\mathsf{SCI}$-formulas. |

($\mathsf{ref}$) $\frac{X}{\{\phi \not\equiv \phi \}\cup X}$ ($\mathsf{sym}$) $\frac{\{\phi \not\equiv \psi \}\cup X}{\{\psi \not\equiv \phi ,\phi \not\equiv \psi \}\cup X}$ |

($\mathsf{tran}$) $\frac{\{\phi \not\equiv \psi ,\psi \not\equiv \vartheta \}\cup X}{\{\phi \not\equiv \vartheta ,\phi \not\equiv \psi ,\psi \not\equiv \vartheta \}\cup X}$ |

(${\equiv}_{\neg}$) $\frac{\{\phi \not\equiv \psi \}\cup X}{\{\neg \phi \not\equiv \neg \psi ,\phi \not\equiv \psi \}\cup X}$ |

(${\equiv}_{\to}$) $\frac{\{\phi \not\equiv \psi ,\vartheta \not\equiv \chi \}\cup X}{\left\{\right(\phi \to \vartheta )\not\equiv (\psi \to \chi ),\phi \not\equiv \psi ,\vartheta \not\equiv \chi \}\cup X}$ |

(${\equiv}_{\equiv}$) $\frac{\{\phi \not\equiv \psi ,\vartheta \not\equiv \chi \}\cup X}{\left\{\right(\phi \equiv \vartheta )\not\equiv (\psi \equiv \chi ),\phi \not\equiv \psi ,\vartheta \not\equiv \chi \}\cup X}$ |

where $\phi ,\psi ,\vartheta ,\chi $ are any $\mathsf{SCI}$-formulas, |

and X is a finite (possibly empty) set of $\mathsf{SCI}$-formulas. |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Golińska-Pilarek, J.; Welle, M.
Deduction in Non-Fregean Propositional Logic SCI. *Axioms* **2019**, *8*, 115.
https://doi.org/10.3390/axioms8040115

**AMA Style**

Golińska-Pilarek J, Welle M.
Deduction in Non-Fregean Propositional Logic SCI. *Axioms*. 2019; 8(4):115.
https://doi.org/10.3390/axioms8040115

**Chicago/Turabian Style**

Golińska-Pilarek, Joanna, and Magdalena Welle.
2019. "Deduction in Non-Fregean Propositional Logic SCI" *Axioms* 8, no. 4: 115.
https://doi.org/10.3390/axioms8040115