Propositional calculus information

The propositional calculus^[a] is a branch of logic.^[1] It is also called propositional logic,^[2] statement logic,^[1] sentential calculus,^[3] sentential logic,^[1] or sometimes zeroth-order logic.^[4]^[5] It deals with propositions^[1] (which can be true or false)^[6] and relations between propositions,^[7] including the construction of arguments based on them.^[8] Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication, biconditional, and negation.^[9]^[10]^[11]^[12] Some sources include other connectives, as in the table below.

Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

Propositional logic is typically studied with a formal language, in which propositions are represented by letters, which are called propositional variables. These are then used, together with symbols for connectives, to make compound propositions. Because of this, the propositional variables are called atomic formulas of a formal zeroth-order language.^[10]^[2] While the atomic propositions are typically represented by letters of the alphabet,^[10] there is a variety of notations to represent the logical connectives. The following table shows the main notational variants for each of the connectives in propositional logic.

Notational variants of the connectives^[13]^[14]
Connective	Symbol
AND	$A\land B$ , $A\cdot B$ , $AB$ , $A\&B$ , $A\&\&B$
equivalent	$A\equiv B$ , $A\Leftrightarrow B$ , $A\leftrightharpoons B$
implies	$A\Rightarrow B$ , $A\supset B$ , $A\rightarrow B$
NAND	$A{\overline {\land }}B$ , $A\mid B$ , ${\overline {A\cdot B}}$
nonequivalent	$A\not \equiv B$ , $A\not \Leftrightarrow B$ , $A\nleftrightarrow B$
NOR	$A{\overline {\lor }}B$ , $A\downarrow B$ , ${\overline {A+B}}$
NOT	$\neg A$ , $-A$ , ${\overline {A}}$ , $\sim A$
OR	$A\lor B$ , $A+B$ , $A\mid B$ , $A\parallel B$
XNOR	$A$ XNOR $B$
XOR	$A{\underline {\lor }}B$ , $A\oplus B$

The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic,^[1] in which formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false.^[15] The principle of bivalence and the law of excluded middle are upheld. By comparison with first-order logic, truth-functional propositional logic is considered to be zeroth-order logic.^[4]^[5]

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^ ^a ^b ^c ^d ^e "Propositional Logic | Internet Encyclopedia of Philosophy". Retrieved 22 March 2024.
^ ^a ^b Franks, Curtis (2023), "Propositional Logic", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Fall 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 22 March 2024
^ Weisstein, Eric W. "Propositional Calculus". mathworld.wolfram.com. Retrieved 22 March 2024.
^ ^a ^b Bělohlávek, Radim; Dauben, Joseph Warren; Klir, George J. (2017). Fuzzy logic and mathematics: a historical perspective. New York, NY, United States of America: Oxford University Press. p. 463. ISBN 978-0-19-020001-5.
^ ^a ^b Manzano, María (2005). Extensions of first order logic. Cambridge tracts in theoretical computer science (Digitally printed first paperback version ed.). Cambridge: Cambridge University Press. p. 180. ISBN 978-0-521-35435-6.
^ McGrath, Matthew; Frank, Devin (2023), "Propositions", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 22 March 2024
^ "Predicate Logic". www3.cs.stonybrook.edu. Retrieved 22 March 2024.
^ "Philosophy 404: Lecture Five". www.webpages.uidaho.edu. Retrieved 22 March 2024.
^ "3.1 Propositional Logic". www.teach.cs.toronto.edu. Retrieved 22 March 2024.
^ ^a ^b ^c Davis, Steven; Gillon, Brendan S., eds. (2004). Semantics: a reader. New York: Oxford University Press. ISBN 978-0-19-513697-5.
^ Plato, Jan von (2013). Elements of logical reasoning (1. publ ed.). Cambridge: Cambridge University press. pp. 9, 32, 121. ISBN 978-1-107-03659-8.
^ "Propositional Logic". www.cs.miami.edu. Retrieved 22 March 2024.
^ Plato, Jan von (2013). Elements of logical reasoning (1. publ ed.). Cambridge: Cambridge University press. p. 9. ISBN 978-1-107-03659-8.
^ Weisstein, Eric W. "Connective". mathworld.wolfram.com. Retrieved 22 March 2024.
^ "Propositional Logic | Brilliant Math & Science Wiki". brilliant.org. Retrieved 20 August 2020.

[:1-2] "Propositional Logic | Internet Encyclopedia of Philosophy". Retrieved 22 March 2024.

[:2-3] Franks, Curtis (2023), "Propositional Logic", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Fall 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 22 March 2024

[:18-4] Weisstein, Eric W. "Propositional Calculus". mathworld.wolfram.com. Retrieved 22 March 2024.

[:10-5] Bělohlávek, Radim; Dauben, Joseph Warren; Klir, George J. (2017). Fuzzy logic and mathematics: a historical perspective. New York, NY, United States of America: Oxford University Press. p. 463. ISBN 978-0-19-020001-5.

[:11-6] Manzano, María (2005). Extensions of first order logic. Cambridge tracts in theoretical computer science (Digitally printed first paperback version ed.). Cambridge: Cambridge University Press. p. 180. ISBN 978-0-521-35435-6.

[:22-7] McGrath, Matthew; Frank, Devin (2023), "Propositions", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 22 March 2024

[8] "Predicate Logic". www3.cs.stonybrook.edu. Retrieved 22 March 2024.

[9] "Philosophy 404: Lecture Five". www.webpages.uidaho.edu. Retrieved 22 March 2024.

[:5-10] "3.1 Propositional Logic". www.teach.cs.toronto.edu. Retrieved 22 March 2024.

[:0-11] Davis, Steven; Gillon, Brendan S., eds. (2004). Semantics: a reader. New York: Oxford University Press. ISBN 978-0-19-513697-5.

[:3-12] Plato, Jan von (2013). Elements of logical reasoning (1. publ ed.). Cambridge: Cambridge University press. pp. 9, 32, 121. ISBN 978-1-107-03659-8.

[:12-13] "Propositional Logic". www.cs.miami.edu. Retrieved 22 March 2024.

[14] Plato, Jan von (2013). Elements of logical reasoning (1. publ ed.). Cambridge: Cambridge University press. p. 9. ISBN 978-1-107-03659-8.

[:23-15] Weisstein, Eric W. "Connective". mathworld.wolfram.com. Retrieved 22 March 2024.

[16] "Propositional Logic | Brilliant Math & Science Wiki". brilliant.org. Retrieved 20 August 2020.

Propositional calculus information

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Propositional calculus
Rules of inference
Implication introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive / hypothetical syllogism Constructive / destructive dilemma Absorption / modus tollens / modus ponendo tollens Negation introduction
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