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Propositional calculus

Logical study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives

Science Mathematics Foundations Mathematical logic

Propositional calculus, also known as propositional logic or sentential logic, studies propositions that are either true or false and their combinations using logical connectives such as conjunction, disjunction, implication, biconditional, and negation. Unlike first-order logic, it does not involve predicates or quantifiers, serving instead as the foundation of higher logics. Propositions are represented by letters called propositional variables, forming formulas in a formal language. The classical form, truth-functional propositional logic, adheres to the principle of bivalence and law of excluded middle, assigning each formula a truth value of true or false.

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History

Main article: History of logic

Although propositional logic had been hinted by earlier philosophers, Chrysippus is often credited with development of a deductive system for propositional logic as his main achievement in the 3rd century BC³³ which was expanded by his successor Stoics. The logic was focused on propositions. This was different from the traditional syllogistic logic, which focused on terms. However, most of the original writings were lost³⁴ and, at some time between the 3rd and 6th century CE, Stoic logic faded into oblivion, to be resurrected only in the 20th century, in the wake of the (re)-discovery of propositional logic.³⁵

Symbolic logic, which would come to be important to refine propositional logic, was first developed by the 17th/18th-century mathematician Gottfried Leibniz, whose calculus ratiocinator was, however, unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan, completely independent of Leibniz.³⁶

Gottlob Frege's predicate logic builds upon propositional logic, and has been described as combining "the distinctive features of syllogistic logic and propositional logic."³⁷ Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including natural deduction, truth trees and truth tables. Natural deduction was invented by Gerhard Gentzen and Stanisław Jaśkowski. Truth trees were invented by Evert Willem Beth.³⁸ The invention of truth tables, however, is of uncertain attribution.

Within works by Frege³⁹ and Bertrand Russell,⁴⁰ are ideas influential to the invention of truth tables. The actual tabular structure (being formatted as a table), itself, is generally credited to either Ludwig Wittgenstein or Emil Post (or both, independently).⁴¹ Besides Frege and Russell, others credited with having ideas preceding truth tables include Philo, Boole, Charles Sanders Peirce,⁴² and Ernst Schröder. Others credited with the tabular structure include Jan Łukasiewicz, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis.⁴³ Ultimately, some have concluded, like John Shosky, that "It is far from clear that any one person should be given the title of 'inventor' of truth-tables".⁴⁴

Sentences

Main article: Proposition

Propositional logic, as currently studied in universities, is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered in evaluating the conditions for the truth of a sentence, or whether a sentence logically follows from some other sentence or group of sentences.⁴⁵

Declarative sentences

Propositional logic deals with statements, which are defined as declarative sentences having truth value.⁴⁶ ⁴⁷ Examples of statements might include:

Wikipedia is a free online encyclopedia that anyone can edit.
London is the capital of England.
All Wikipedia editors speak at least three languages.

Declarative sentences are contrasted with questions, such as "What is Wikipedia?", and imperative statements, such as "Please add citations to support the claims in this article.".⁴⁸ ⁴⁹ Such non-declarative sentences have no truth value,⁵⁰ and are only dealt with in nonclassical logics, called erotetic and imperative logics.

Compounding sentences with connectives

See also: Atomic formula and Atomic sentence

In propositional logic, a statement can contain one or more other statements as parts.⁵¹ Compound sentences are formed from simpler sentences and express relationships among the constituent sentences.⁵² This is done by combining them with logical connectives:⁵³ ⁵⁴ the main types of compound sentences are negations, conjunctions, disjunctions, implications, and biconditionals,⁵⁵ which are formed by using the corresponding connectives to connect propositions.⁵⁶ ⁵⁷ In English, these connectives are expressed by the words "and" (conjunction), "or" (disjunction), "not" (negation), "if" (material conditional), and "if and only if" (biconditional).⁵⁸ ⁵⁹ Examples of such compound sentences might include:

Wikipedia is a free online encyclopedia that anyone can edit, and millions already have. (conjunction)
It is not true that all Wikipedia editors speak at least three languages. (negation)
Either London is the capital of England, or London is the capital of the United Kingdom, or both. (disjunction)⁶⁰

If sentences lack any logical connectives, they are called simple sentences,⁶¹ or atomic sentences;⁶² if they contain one or more logical connectives, they are called compound sentences,⁶³ or molecular sentences.⁶⁴

Sentential connectives are a broader category that includes logical connectives.⁶⁵ ⁶⁶ Sentential connectives are any linguistic particles that bind sentences to create a new compound sentence,⁶⁷ ⁶⁸ or that inflect a single sentence to create a new sentence.⁶⁹ A logical connective, or propositional connective, is a kind of sentential connective with the characteristic feature that, when the original sentences it operates on are (or express) propositions, the new sentence that results from its application also is (or expresses) a proposition.⁷⁰ Philosophers disagree about what exactly a proposition is,⁷¹ ⁷² as well as about which sentential connectives in natural languages should be counted as logical connectives.⁷³ ⁷⁴ Sentential connectives are also called sentence-functors,⁷⁵ and logical connectives are also called truth-functors.⁷⁶

Arguments

Main article: Argument

An argument is defined as a pair of things, namely a set of sentences, called the premises,⁷⁷ and a sentence, called the conclusion.⁷⁸ ⁷⁹ ⁸⁰ The conclusion is claimed to follow from the premises,⁸¹ and the premises are claimed to support the conclusion.⁸²

Example argument

The following is an example of an argument within the scope of propositional logic:

Premise 1: If it's raining, then it's cloudy. Premise 2: It's raining. Conclusion: It's cloudy.

The logical form of this argument is known as modus ponens,⁸³ which is a classically valid form.⁸⁴ So, in classical logic, the argument is valid, although it may or may not be sound, depending on the meteorological facts in a given context. This example argument will be reused when explaining § Formalization.

Validity and soundness

Main articles: Validity (logic) and Soundness

An argument is valid if, and only if, it is necessary that, if all its premises are true, its conclusion is true.⁸⁵ ⁸⁶ ⁸⁷ Alternatively, an argument is valid if, and only if, it is impossible for all the premises to be true while the conclusion is false.⁸⁸ ⁸⁹

Validity is contrasted with soundness.⁹⁰ An argument is sound if, and only if, it is valid and all its premises are true.⁹¹ ⁹² Otherwise, it is unsound.⁹³

Logic, in general, aims to precisely specify valid arguments.⁹⁴ This is done by defining a valid argument as one in which its conclusion is a logical consequence of its premises,⁹⁵ which, when this is understood as semantic consequence, means that there is no case in which the premises are true but the conclusion is not true⁹⁶ – see § Semantics below.

Formalization

Propositional logic is typically studied through a formal system in which formulas of a formal language are interpreted to represent propositions. This formal language is the basis for proof systems, which allow a conclusion to be derived from premises if, and only if, it is a logical consequence of them. This section will show how this works by formalizing the § Example argument. The formal language for a propositional calculus will be fully specified in § Language, and an overview of proof systems will be given in § Proof systems.

Propositional variables

Main article: Propositional variable

Since propositional logic is not concerned with the structure of propositions beyond the point where they cannot be decomposed any more by logical connectives,⁹⁷ ⁹⁸ it is typically studied by replacing such atomic (indivisible) statements with letters of the alphabet, which are interpreted as variables representing statements (propositional variables).⁹⁹ With propositional variables, the § Example argument would then be symbolized as follows:

Premise 1: P → Q {\displaystyle P\to Q} Premise 2: P {\displaystyle P} Conclusion: Q {\displaystyle Q}

When P is interpreted as "It's raining" and Q as "it's cloudy" these symbolic expressions correspond exactly with the original expression in natural language. Not only that, but they will also correspond with any other inference with the same logical form.

When a formal system is used to represent formal logic, only statement letters (usually capital roman letters such as P {\displaystyle P} , Q {\displaystyle Q} and R {\displaystyle R} ) are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself.

Gentzen notation

If we assume that the validity of modus ponens has been accepted as an axiom, then the same § Example argument can also be depicted like this:

P → Q , P Q {\displaystyle {\frac {P\to Q,P}{Q}}}

This method of displaying it is Gentzen's notation for natural deduction and sequent calculus.¹⁰⁰ The premises are shown above a line, called the inference line,¹⁰¹ separated by a comma, which indicates combination of premises.¹⁰² The conclusion is written below the inference line.¹⁰³ The inference line represents syntactic consequence,¹⁰⁴ sometimes called deductive consequence,¹⁰⁵> which is also symbolized with ⊢.¹⁰⁶ ¹⁰⁷ So the above can also be written in one line as P → Q , P ⊢ Q {\displaystyle P\to Q,P\vdash Q} .¹⁰⁸

Syntactic consequence is contrasted with semantic consequence,¹⁰⁹ which is symbolized with ⊧.¹¹⁰ ¹¹¹ In this case, the conclusion follows syntactically because the natural deduction inference rule of modus ponens has been assumed. For more on inference rules, see the sections on proof systems below.

Language

The language (commonly called L {\displaystyle {\mathcal {L}}} )¹¹² ¹¹³ ¹¹⁴ of a propositional calculus is defined in terms of:¹¹⁵ ¹¹⁶

a set of primitive symbols, called atomic formulas, atomic sentences,¹¹⁷ ¹¹⁸ atoms,¹¹⁹ placeholders, prime formulas,¹²⁰ proposition letters, sentence letters,¹²¹ or variables, and
a set of operator symbols, called connectives,¹²² ¹²³ ¹²⁴ logical connectives,¹²⁵ logical operators,¹²⁶ truth-functional connectives,¹²⁷ truth-functors,¹²⁸ or propositional connectives.¹²⁹

A well-formed formula is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar. The language L {\displaystyle {\mathcal {L}}} , then, is defined either as being identical to its set of well-formed formulas,¹³⁰ or as containing that set (together with, for instance, its set of connectives and variables).¹³¹ ¹³²

Usually the syntax of L {\displaystyle {\mathcal {L}}} is defined recursively by just a few definitions, as seen next; some authors explicitly include parentheses as punctuation marks when defining their language's syntax,¹³³ ¹³⁴ while others use them without comment.¹³⁵ ¹³⁶

Syntax

Given a set of atomic propositional variables p 1 {\displaystyle p_{1}} , p 2 {\displaystyle p_{2}} , p 3 {\displaystyle p_{3}} , ..., and a set of propositional connectives c 1 1 {\displaystyle c_{1}^{1}} , c 2 1 {\displaystyle c_{2}^{1}} , c 3 1 {\displaystyle c_{3}^{1}} , ..., c 1 2 {\displaystyle c_{1}^{2}} , c 2 2 {\displaystyle c_{2}^{2}} , c 3 2 {\displaystyle c_{3}^{2}} , ..., c 1 3 {\displaystyle c_{1}^{3}} , c 2 3 {\displaystyle c_{2}^{3}} , c 3 3 {\displaystyle c_{3}^{3}} , ..., a formula of propositional logic is defined recursively by these definitions:¹³⁷ ¹³⁸ ¹³⁹ ¹⁴⁰

Definition 1: Atomic propositional variables are formulas. Definition 2: If c n m {\displaystyle c_{n}^{m}} is a propositional connective, and ⟨ {\displaystyle \langle } A, B, C, … ⟩ {\displaystyle \rangle } is a sequence of m, possibly but not necessarily atomic, possibly but not necessarily distinct, formulas, then the result of applying c n m {\displaystyle c_{n}^{m}} to ⟨ {\displaystyle \langle } A, B, C, … ⟩ {\displaystyle \rangle } is a formula. Definition 3: Nothing else is a formula.

Writing the result of applying c n m {\displaystyle c_{n}^{m}} to ⟨ {\displaystyle \langle } A, B, C, … ⟩ {\displaystyle \rangle } in functional notation, as c n m {\displaystyle c_{n}^{m}} (A, B, C, …), we have the following as examples of well-formed formulas:

p 5 {\displaystyle p_{5}}
c 3 2 ( p 2 , p 9 ) {\displaystyle c_{3}^{2}(p_{2},p_{9})}
c 3 2 ( p 1 , c 2 1 ( p 3 ) ) {\displaystyle c_{3}^{2}(p_{1},c_{2}^{1}(p_{3}))}
c 1 3 ( p 4 , p 6 , c 2 2 ( p 1 , p 2 ) ) {\displaystyle c_{1}^{3}(p_{4},p_{6},c_{2}^{2}(p_{1},p_{2}))}
c 4 2 ( c 1 1 ( p 7 ) , c 3 1 ( p 8 ) ) {\displaystyle c_{4}^{2}(c_{1}^{1}(p_{7}),c_{3}^{1}(p_{8}))}
c 2 3 ( c 1 2 ( p 3 , p 4 ) , c 2 1 ( p 5 ) , c 3 2 ( p 6 , p 7 ) ) {\displaystyle c_{2}^{3}(c_{1}^{2}(p_{3},p_{4}),c_{2}^{1}(p_{5}),c_{3}^{2}(p_{6},p_{7}))}
c 3 1 ( c 1 3 ( p 2 , p 3 , c 2 2 ( p 4 , p 5 ) ) ) {\displaystyle c_{3}^{1}(c_{1}^{3}(p_{2},p_{3},c_{2}^{2}(p_{4},p_{5})))}

What was given as Definition 2 above, which is responsible for the composition of formulas, is referred to by Colin Howson as the principle of composition.¹⁴¹ ¹⁴² It is this recursion in the definition of a language's syntax which justifies the use of the word "atomic" to refer to propositional variables, since all formulas in the language L {\displaystyle {\mathcal {L}}} are built up from the atoms as ultimate building blocks.¹⁴³ Composite formulas (all formulas besides atoms) are called molecules,¹⁴⁴ or molecular sentences.¹⁴⁵ (This is an imperfect analogy with chemistry, since a chemical molecule may sometimes have only one atom, as in monatomic gases.)¹⁴⁶

The definition that "nothing else is a formula", given above as Definition 3, excludes any formula from the language which is not specifically required by the other definitions in the syntax.¹⁴⁷ In particular, it excludes infinitely long formulas from being well-formed.¹⁴⁸ It is sometimes called the Closure Clause.¹⁴⁹

CF grammar in BNF

An alternative to the syntax definitions given above is to write a context-free (CF) grammar for the language L {\displaystyle {\mathcal {L}}} in Backus-Naur form (BNF).¹⁵⁰ ¹⁵¹ This is more common in computer science than in philosophy.¹⁵² It can be done in many ways,¹⁵³ of which a particularly brief one, for the common set of five connectives, is this single clause:¹⁵⁴ ¹⁵⁵

ϕ ::= a 1 , a 2 , … | ¬ ϕ | ϕ & ψ | ϕ ∨ ψ | ϕ → ψ | ϕ ↔ ψ {\displaystyle \phi ::=a_{1},a_{2},\ldots ~|~\neg \phi ~|~\phi ~\&~\psi ~|~\phi \vee \psi ~|~\phi \rightarrow \psi ~|~\phi \leftrightarrow \psi }

This clause, due to its self-referential nature (since ϕ {\displaystyle \phi } is in some branches of the definition of ϕ {\displaystyle \phi } ), also acts as a recursive definition, and therefore specifies the entire language. To expand it to add modal operators, one need only add … | ◻ ϕ | ◊ ϕ {\displaystyle |~\Box \phi ~|~\Diamond \phi } to the end of the clause.¹⁵⁶

Constants and schemata

Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition,¹⁵⁷ while propositional variables range over the set of all atomic propositions.¹⁵⁸ Schemata, or schematic letters, however, range over all formulas.¹⁵⁹ ¹⁶⁰ (Schematic letters are also called metavariables.)¹⁶¹ It is common to represent propositional constants by A, B, and C, propositional variables by P, Q, and R, and schematic letters are often Greek letters, most often φ, ψ, and χ.¹⁶² ¹⁶³

However, some authors recognize only two "propositional constants" in their formal system: the special symbol ⊤ {\displaystyle \top } , called "truth", which always evaluates to True, and the special symbol ⊥ {\displaystyle \bot } , called "falsity", which always evaluates to False.¹⁶⁴ ¹⁶⁵ ¹⁶⁶ Other authors also include these symbols, with the same meaning, but consider them to be "zero-place truth-functors",¹⁶⁷ or equivalently, "nullary connectives".¹⁶⁸

Semantics

Main articles: Semantics of logic and Model theory

To serve as a model of the logic of a given natural language, a formal language must be semantically interpreted.¹⁶⁹ In classical logic, all propositions evaluate to exactly one of two truth-values: True or False.¹⁷⁰ ¹⁷¹ For example, "Wikipedia is a free online encyclopedia that anyone can edit" evaluates to True,¹⁷² while "Wikipedia is a paper encyclopedia" evaluates to False.¹⁷³

In other respects, the following formal semantics can apply to the language of any propositional logic, but the assumptions that there are only two semantic values (bivalence), that only one of the two is assigned to each formula in the language (noncontradiction), and that every formula gets assigned a value (excluded middle), are distinctive features of classical logic.¹⁷⁴ ¹⁷⁵ ¹⁷⁶ To learn about nonclassical logics with more than two truth-values, and their unique semantics, one may consult the articles on "Many-valued logic", "Three-valued logic", "Finite-valued logic", and "Infinite-valued logic".

Interpretation (case) and argument

Main article: Interpretation (logic)

For a given language L {\displaystyle {\mathcal {L}}} , an interpretation,¹⁷⁷ valuation,¹⁷⁸ Boolean valuation,¹⁷⁹ or case,¹⁸⁰ ¹⁸¹ is an assignment of semantic values to each formula of L {\displaystyle {\mathcal {L}}} .¹⁸² For a formal language of classical logic, a case is defined as an assignment, to each formula of L {\displaystyle {\mathcal {L}}} , of one or the other, but not both, of the truth values, namely truth (T, or 1) and falsity (F, or 0).¹⁸³ ¹⁸⁴ An interpretation that follows the rules of classical logic is sometimes called a Boolean valuation.¹⁸⁵ ¹⁸⁶ An interpretation of a formal language for classical logic is often expressed in terms of truth tables.¹⁸⁷ ¹⁸⁸ Since each formula is only assigned a single truth-value, an interpretation may be viewed as a function, whose domain is L {\displaystyle {\mathcal {L}}} , and whose range is its set of semantic values V = { T , F } {\displaystyle {\mathcal {V}}=\{{\mathsf {T}},{\mathsf {F}}\}} ,¹⁸⁹ or V = { 1 , 0 } {\displaystyle {\mathcal {V}}=\{1,0\}} .¹⁹⁰

For n {\displaystyle n} distinct propositional symbols there are 2 n {\displaystyle 2^{n}} distinct possible interpretations. For any particular symbol a {\displaystyle a} , for example, there are 2 1 = 2 {\displaystyle 2^{1}=2} possible interpretations: either a {\displaystyle a} is assigned T, or a {\displaystyle a} is assigned F. And for the pair a {\displaystyle a} , b {\displaystyle b} there are 2 2 = 4 {\displaystyle 2^{2}=4} possible interpretations: either both are assigned T, or both are assigned F, or a {\displaystyle a} is assigned T and b {\displaystyle b} is assigned F, or a {\displaystyle a} is assigned F and b {\displaystyle b} is assigned T.¹⁹¹ Since L {\displaystyle {\mathcal {L}}} has ℵ 0 {\displaystyle \aleph _{0}} , that is, denumerably many propositional symbols, there are 2 ℵ 0 = c {\displaystyle 2^{\aleph _{0}}={\mathfrak {c}}} , and therefore uncountably many distinct possible interpretations of L {\displaystyle {\mathcal {L}}} as a whole.¹⁹²

Where I {\displaystyle {\mathcal {I}}} is an interpretation and φ {\displaystyle \varphi } and ψ {\displaystyle \psi } represent formulas, the definition of an argument, given in § Arguments, may then be stated as a pair ⟨ { φ 1 , φ 2 , φ 3 , . . . , φ n } , ψ ⟩ {\displaystyle \langle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\},\psi \rangle } , where { φ 1 , φ 2 , φ 3 , . . . , φ n } {\displaystyle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\}} is the set of premises and ψ {\displaystyle \psi } is the conclusion. The definition of an argument's validity, i.e. its property that { φ 1 , φ 2 , φ 3 , . . . , φ n } ⊨ ψ {\displaystyle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\}\models \psi } , can then be stated as its absence of a counterexample, where a counterexample is defined as a case I {\displaystyle {\mathcal {I}}} in which the argument's premises { φ 1 , φ 2 , φ 3 , . . . , φ n } {\displaystyle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\}} are all true but the conclusion ψ {\displaystyle \psi } is not true.¹⁹³ ¹⁹⁴ As will be seen in § Semantic truth, validity, consequence, this is the same as to say that the conclusion is a semantic consequence of the premises.

Propositional connective semantics

Main articles: Logical connective and Truth function

An interpretation assigns semantic values to atomic formulas directly.¹⁹⁵ ¹⁹⁶ Molecular formulas are assigned a function of the value of their constituent atoms, according to the connective used;¹⁹⁷ ¹⁹⁸ the connectives are defined in such a way that the truth-value of a sentence formed from atoms with connectives depends on the truth-values of the atoms that they're applied to, and only on those.¹⁹⁹ ²⁰⁰ This assumption is referred to by Colin Howson as the assumption of the truth-functionality of the connectives.²⁰¹

Semantics via. truth tables

Since logical connectives are defined semantically only in terms of the truth values that they take when the propositional variables that they're applied to take either of the two possible truth values,²⁰² ²⁰³ the semantic definition of the connectives is usually represented as a truth table for each of the connectives,²⁰⁴ ²⁰⁵ ²⁰⁶ as seen below:

p {\displaystyle p}	q {\displaystyle q}	p ∧ q {\displaystyle p\land q}	p ∨ q {\displaystyle p\lor q}	p → q {\displaystyle p\rightarrow q}	p ⇔ q {\displaystyle p\Leftrightarrow q}	¬ p {\displaystyle \neg p}	¬ q {\displaystyle \neg q}
T	T	T	T	T	T	F	F
T	F	F	T	F	F	F	T
F	T	F	T	T	F	T	F
F	F	F	F	T	T	T	T

This table covers each of the main five logical connectives:²⁰⁷ ²⁰⁸ ²⁰⁹ ²¹⁰ conjunction (here notated p ∧ q {\displaystyle p\land q} ), disjunction (p ∨ q), implication (p → q), biconditional (p ↔ q) and negation, (¬p, or ¬q, as the case may be). It is sufficient for determining the semantics of each of these operators.²¹¹ ²¹² ²¹³ For more truth tables for more different kinds of connectives, see the article "Truth table".

Semantics via assignment expressions

Some authors (viz., all the authors cited in this subsection) write out the connective semantics using a list of statements instead of a table. In this format, where I ( φ ) {\displaystyle {\mathcal {I}}(\varphi )} is the interpretation of φ {\displaystyle \varphi } , the five connectives are defined as:²¹⁴ ²¹⁵

I ( ¬ P ) = T {\displaystyle {\mathcal {I}}(\neg P)={\mathsf {T}}} if, and only if, I ( P ) = F {\displaystyle {\mathcal {I}}(P)={\mathsf {F}}}
I ( P ∧ Q ) = T {\displaystyle {\mathcal {I}}(P\land Q)={\mathsf {T}}} if, and only if, I ( P ) = T {\displaystyle {\mathcal {I}}(P)={\mathsf {T}}} and I ( Q ) = T {\displaystyle {\mathcal {I}}(Q)={\mathsf {T}}}
I ( P ∨ Q ) = T {\displaystyle {\mathcal {I}}(P\lor Q)={\mathsf {T}}} if, and only if, I ( P ) = T {\displaystyle {\mathcal {I}}(P)={\mathsf {T}}} or I ( Q ) = T {\displaystyle {\mathcal {I}}(Q)={\mathsf {T}}}
I ( P → Q ) = T {\displaystyle {\mathcal {I}}(P\to Q)={\mathsf {T}}} if, and only if, it is true that, if I ( P ) = T {\displaystyle {\mathcal {I}}(P)={\mathsf {T}}} , then I ( Q ) = T {\displaystyle {\mathcal {I}}(Q)={\mathsf {T}}}
I ( P ↔ Q ) = T {\displaystyle {\mathcal {I}}(P\leftrightarrow Q)={\mathsf {T}}} if, and only if, it is true that I ( P ) = T {\displaystyle {\mathcal {I}}(P)={\mathsf {T}}} if, and only if, I ( Q ) = T {\displaystyle {\mathcal {I}}(Q)={\mathsf {T}}}

Instead of I ( φ ) {\displaystyle {\mathcal {I}}(\varphi )} , the interpretation of φ {\displaystyle \varphi } may be written out as | φ | {\displaystyle |\varphi |} ,²¹⁶ ²¹⁷ or, for definitions such as the above, I ( φ ) = T {\displaystyle {\mathcal {I}}(\varphi )={\mathsf {T}}} may be written simply as the English sentence " φ {\displaystyle \varphi } is given the value T {\displaystyle {\mathsf {T}}} ".²¹⁸ Yet other authors²¹⁹ ²²⁰ may prefer to speak of a Tarskian model M {\displaystyle {\mathfrak {M}}} for the language, so that instead they'll use the notation M ⊨ φ {\displaystyle {\mathfrak {M}}\models \varphi } , which is equivalent to saying I ( φ ) = T {\displaystyle {\mathcal {I}}(\varphi )={\mathsf {T}}} , where I {\displaystyle {\mathcal {I}}} is the interpretation function for M {\displaystyle {\mathfrak {M}}} .²²¹

Connective definition methods

Some of these connectives may be defined in terms of others: for instance, implication, p → q {\displaystyle p\rightarrow q} , may be defined in terms of disjunction and negation, as ¬ p ∨ q {\displaystyle \neg p\lor q} ;²²² and disjunction may be defined in terms of negation and conjunction, as ¬ ( ¬ p ∧ ¬ q {\displaystyle \neg (\neg p\land \neg q} .²²³ In fact, a truth-functionally complete system,²²⁴ in the sense that all and only the classical propositional tautologies are theorems, may be derived using only disjunction and negation (as Russell, Whitehead, and Hilbert did), or using only implication and negation (as Frege did), or using only conjunction and negation, or even using only a single connective for "not and" (the Sheffer stroke),²²⁵ as Jean Nicod did.²²⁶ A joint denial connective (logical NOR) will also suffice, by itself, to define all other connectives. Besides NOR and NAND, no other connectives have this property.²²⁷ ²²⁸

Some authors, namely Howson ²²⁹ and Cunningham,²³⁰ distinguish equivalence from the biconditional. (As to equivalence, Howson calls it "truth-functional equivalence", while Cunningham calls it "logical equivalence".) Equivalence is symbolized with ⇔ and is a metalanguage symbol, while a biconditional is symbolized with ↔ and is a logical connective in the object language L {\displaystyle {\mathcal {L}}} . Regardless, an equivalence or biconditional is true if, and only if, the formulas connected by it are assigned the same semantic value under every interpretation. Other authors often do not make this distinction, and may use the word "equivalence",²³¹ and/or the symbol ⇔,²³² to denote their object language's biconditional connective.

Semantic truth, validity, consequence

Given φ {\displaystyle \varphi } and ψ {\displaystyle \psi } as formulas (or sentences) of a language L {\displaystyle {\mathcal {L}}} , and I {\displaystyle {\mathcal {I}}} as an interpretation (or case)²³³ of L {\displaystyle {\mathcal {L}}} , then the following definitions apply:²³⁴ ²³⁵

Truth-in-a-case:²³⁶ A sentence φ {\displaystyle \varphi } of L {\displaystyle {\mathcal {L}}} is true under an interpretation I {\displaystyle {\mathcal {I}}} if I {\displaystyle {\mathcal {I}}} assigns the truth value T to φ {\displaystyle \varphi } .²³⁷ ²³⁸ If φ {\displaystyle \varphi } is true under I {\displaystyle {\mathcal {I}}} , then I {\displaystyle {\mathcal {I}}} is called a model of φ {\displaystyle \varphi } .²³⁹
Falsity-in-a-case:²⁴⁰ φ {\displaystyle \varphi } is false under an interpretation I {\displaystyle {\mathcal {I}}} if, and only if, ¬ φ {\displaystyle \neg \varphi } is true under I {\displaystyle {\mathcal {I}}} .²⁴¹ ²⁴² ²⁴³ This is the "truth of negation" definition of falsity-in-a-case.²⁴⁴ Falsity-in-a-case may also be defined by the "complement" definition: φ {\displaystyle \varphi } is false under an interpretation I {\displaystyle {\mathcal {I}}} if, and only if, φ {\displaystyle \varphi } is not true under I {\displaystyle {\mathcal {I}}} .²⁴⁵ ²⁴⁶ In classical logic, these definitions are equivalent, but in nonclassical logics, they are not.²⁴⁷
Semantic consequence: A sentence ψ {\displaystyle \psi } of L {\displaystyle {\mathcal {L}}} is a semantic consequence ( φ ⊨ ψ {\displaystyle \varphi \models \psi } ) of a sentence φ {\displaystyle \varphi } if there is no interpretation under which φ {\displaystyle \varphi } is true and ψ {\displaystyle \psi } is not true.²⁴⁸ ²⁴⁹ ²⁵⁰
Valid formula (tautology): A sentence φ {\displaystyle \varphi } of L {\displaystyle {\mathcal {L}}} is logically valid ( ⊨ φ {\displaystyle \models \varphi } ),²⁵¹ or a tautology,²⁵² ²⁵³ref name="ms32²⁵⁴ if it is true under every interpretation,²⁵⁵ ²⁵⁶ or true in every case.²⁵⁷
Consistent sentence: A sentence of L {\displaystyle {\mathcal {L}}} is consistent if it is true under at least one interpretation. It is inconsistent if it is not consistent.²⁵⁸ ²⁵⁹ An inconsistent formula is also called self-contradictory,²⁶⁰ and said to be a self-contradiction,²⁶¹ or simply a contradiction,²⁶² ²⁶³ ²⁶⁴ although this latter name is sometimes reserved specifically for statements of the form ( p ∧ ¬ p ) {\displaystyle (p\land \neg p)} .²⁶⁵

For interpretations (cases) I {\displaystyle {\mathcal {I}}} of L {\displaystyle {\mathcal {L}}} , these definitions are sometimes given:

Complete case: A case I {\displaystyle {\mathcal {I}}} is complete if, and only if, either φ {\displaystyle \varphi } is true-in- I {\displaystyle {\mathcal {I}}} or ¬ φ {\displaystyle \neg \varphi } is true-in- I {\displaystyle {\mathcal {I}}} , for any φ {\displaystyle \varphi } in L {\displaystyle {\mathcal {L}}} .²⁶⁶ ²⁶⁷
Consistent case: A case I {\displaystyle {\mathcal {I}}} is consistent if, and only if, there is no φ {\displaystyle \varphi } in L {\displaystyle {\mathcal {L}}} such that both φ {\displaystyle \varphi } and ¬ φ {\displaystyle \neg \varphi } are true-in- I {\displaystyle {\mathcal {I}}} .²⁶⁸ ²⁶⁹

For classical logic, which assumes that all cases are complete and consistent,²⁷⁰ the following theorems apply:

For any given interpretation, a given formula is either true or false under it.²⁷¹ ²⁷²
No formula is both true and false under the same interpretation.²⁷³ ²⁷⁴
φ {\displaystyle \varphi } is true under I {\displaystyle {\mathcal {I}}} if, and only if, ¬ φ {\displaystyle \neg \varphi } is false under I {\displaystyle {\mathcal {I}}} ;²⁷⁵ ²⁷⁶ ¬ φ {\displaystyle \neg \varphi } is true under I {\displaystyle {\mathcal {I}}} if, and only if, φ {\displaystyle \varphi } is not true under I {\displaystyle {\mathcal {I}}} .²⁷⁷
If φ {\displaystyle \varphi } and ( φ → ψ ) {\displaystyle (\varphi \to \psi )} are both true under I {\displaystyle {\mathcal {I}}} , then ψ {\displaystyle \psi } is true under I {\displaystyle {\mathcal {I}}} .²⁷⁸ ²⁷⁹
If ⊨ φ {\displaystyle \models \varphi } and ⊨ ( φ → ψ ) {\displaystyle \models (\varphi \to \psi )} , then ⊨ ψ {\displaystyle \models \psi } .²⁸⁰
( φ → ψ ) {\displaystyle (\varphi \to \psi )} is true under I {\displaystyle {\mathcal {I}}} if, and only if, either φ {\displaystyle \varphi } is not true under I {\displaystyle {\mathcal {I}}} , or ψ {\displaystyle \psi } is true under I {\displaystyle {\mathcal {I}}} .²⁸¹
φ ⊨ ψ {\displaystyle \varphi \models \psi } if, and only if, ( φ → ψ ) {\displaystyle (\varphi \to \psi )} is logically valid, that is, φ ⊨ ψ {\displaystyle \varphi \models \psi } if, and only if, ⊨ ( φ → ψ ) {\displaystyle \models (\varphi \to \psi )} .²⁸² ²⁸³

Proof systems

See also: Proof theory and Proof calculus

Proof systems in propositional logic can be broadly classified into semantic proof systems and syntactic proof systems,²⁸⁴ ²⁸⁵ ²⁸⁶ according to the kind of logical consequence that they rely on: semantic proof systems rely on semantic consequence ( φ ⊨ ψ {\displaystyle \varphi \models \psi } ),²⁸⁷ whereas syntactic proof systems rely on syntactic consequence ( φ ⊢ ψ {\displaystyle \varphi \vdash \psi } ).²⁸⁸ Semantic consequence deals with the truth values of propositions in all possible interpretations, whereas syntactic consequence concerns the derivation of conclusions from premises based on rules and axioms within a formal system.²⁸⁹ This section gives a very brief overview of the kinds of proof systems, with anchors to the relevant sections of this article on each one, as well as to the separate Wikipedia articles on each one.

Semantic proof systems

Semantic proof systems rely on the concept of semantic consequence, symbolized as φ ⊨ ψ {\displaystyle \varphi \models \psi } , which indicates that if φ {\displaystyle \varphi } is true, then ψ {\displaystyle \psi } must also be true in every possible interpretation.²⁹⁰

Truth tables

Main article: Truth table

A truth table is a semantic proof method used to determine the truth value of a propositional logic expression in every possible scenario.²⁹¹ By exhaustively listing the truth values of its constituent atoms, a truth table can show whether a proposition is true, false, tautological, or contradictory.²⁹² See § Semantic proof via truth tables.

Semantic tableaux

Main article: Method of analytic tableaux

A semantic tableau is another semantic proof technique that systematically explores the truth of a proposition.²⁹³ It constructs a tree where each branch represents a possible interpretation of the propositions involved.²⁹⁴ If every branch leads to a contradiction, the original proposition is considered to be a contradiction, and its negation is considered a tautology.²⁹⁵ See § Semantic proof via tableaux.

Syntactic proof systems

Syntactic proof systems, in contrast, focus on the formal manipulation of symbols according to specific rules. The notion of syntactic consequence, φ ⊢ ψ {\displaystyle \varphi \vdash \psi } , signifies that ψ {\displaystyle \psi } can be derived from φ {\displaystyle \varphi } using the rules of the formal system.²⁹⁶

Axiomatic systems

Main article: Axiomatic system (logic)

An axiomatic system is a set of axioms or assumptions from which other statements (theorems) are logically derived.²⁹⁷ In propositional logic, axiomatic systems define a base set of propositions considered to be self-evidently true, and theorems are proved by applying deduction rules to these axioms.²⁹⁸ See § Syntactic proof via axioms.

Natural deduction

Main article: Natural deduction

Natural deduction is a syntactic method of proof that emphasizes the derivation of conclusions from premises through the use of intuitive rules reflecting ordinary reasoning.²⁹⁹ Each rule reflects a particular logical connective and shows how it can be introduced or eliminated.³⁰⁰ See § Syntactic proof via natural deduction.

Sequent calculus

Main article: Sequent calculus

The sequent calculus is a formal system that represents logical deductions as sequences or "sequents" of formulas.³⁰¹ Developed by Gerhard Gentzen, this approach focuses on the structural properties of logical deductions and provides a powerful framework for proving statements within propositional logic.³⁰² ³⁰³

Semantic proof via truth tables

p	q	r	q ∨ r	r → ¬p	q ∨ r → (r → ¬p)	p → (q ∨ r → (r → ¬p))
T	T	T	T	F	F	F
T	T	F	T	T	T	T
T	F	T	T	F	F	F
T	F	F	F	T	T	T
F	T	T	T	T	T	T
F	T	F	T	T	T	T
F	F	T	T	T	T	T
F	F	F	F	T	T	T

p	→	(q	∨	r	→	(r	→	¬	p))
T	→	(F	∨	T	→	(T	→	¬	T))
T	→	(	T		→	(T	→	F	))
T	→	(	T		→		F		)
T	→				F
	F
T	F	F	T	T	F	T	F	F	T

Semantic proof via tableaux

Main article: Method of analytic tableaux

Since truth tables have 2n lines for n variables, they can be tiresomely long for large values of n.³¹⁸ Analytic tableaux are a more efficient, but nevertheless mechanical,³¹⁹ semantic proof method; they take advantage of the fact that "we learn nothing about the validity of the inference from examining the truth-value distributions which make either the premises false or the conclusion true: the only relevant distributions when considering deductive validity are clearly just those which make the premises true or the conclusion false."³²⁰

Analytic tableaux for propositional logic are fully specified by the rules that are stated in schematic form below.³²¹ These rules use "signed formulas", where a signed formula is an expression T X {\displaystyle TX} or F X {\displaystyle FX} , where X {\displaystyle X} is a (unsigned) formula of the language L {\displaystyle {\mathcal {L}}} .³²² (Informally, T X {\displaystyle TX} is read " X {\displaystyle X} is true", and F X {\displaystyle FX} is read " X {\displaystyle X} is false".)³²³ Their formal semantic definition is that "under any interpretation, a signed formula T X {\displaystyle TX} is called true if X {\displaystyle X} is true, and false if X {\displaystyle X} is false, whereas a signed formula F X {\displaystyle FX} is called false if X {\displaystyle X} is true, and true if X {\displaystyle X} is false."³²⁴

1 ) T ∼ X F X F ∼ X T X s p a c e r 2 ) T ( X ∧ Y ) T X T Y F ( X ∧ Y ) F X | F Y s p a c e r 3 ) T ( X ∨ Y ) T X | T Y F ( X ∨ Y ) F X F Y s p a c e r 4 ) T ( X ⊃ Y ) F X | T Y F ( X ⊃ Y ) T X F Y {\displaystyle {\begin{aligned}&1)\quad {\frac {T\sim X}{FX}}\quad &&{\frac {F\sim X}{TX}}\\{\phantom {spacer}}\\&2)\quad {\frac {T(X\land Y)}{\begin{matrix}TX\\TY\end{matrix}}}\quad &&{\frac {F(X\land Y)}{FX|FY}}\\{\phantom {spacer}}\\&3)\quad {\frac {T(X\lor Y)}{TX|TY}}\quad &&{\frac {F(X\lor Y)}{\begin{matrix}FX\\FY\end{matrix}}}\\{\phantom {spacer}}\\&4)\quad {\frac {T(X\supset Y)}{FX|TY}}\quad &&{\frac {F(X\supset Y)}{\begin{matrix}TX\\FY\end{matrix}}}\end{aligned}}}

In this notation, rule 2 means that T ( X ∧ Y ) {\displaystyle T(X\land Y)} yields both T X , T Y {\displaystyle TX,TY} , whereas F ( X ∧ Y ) {\displaystyle F(X\land Y)} branches into F X , F Y {\displaystyle FX,FY} . The notation is to be understood analogously for rules 3 and 4.³²⁵ Often, in tableaux for classical logic, the signed formula notation is simplified so that T φ {\displaystyle T\varphi } is written simply as φ {\displaystyle \varphi } , and F φ {\displaystyle F\varphi } as ¬ φ {\displaystyle \neg \varphi } , which accounts for naming rule 1 the "Rule of Double Negation".³²⁶ ³²⁷

One constructs a tableau for a set of formulas by applying the rules to produce more lines and tree branches until every line has been used, producing a complete tableau. In some cases, a branch can come to contain both T X {\displaystyle TX} and F X {\displaystyle FX} for some X {\displaystyle X} , which is to say, a contradiction. In that case, the branch is said to close.³²⁸ If every branch in a tree closes, the tree itself is said to close.³²⁹ In virtue of the rules for construction of tableaux, a closed tree is a proof that the original formula, or set of formulas, used to construct it was itself self-contradictory, and therefore false.³³⁰ Conversely, a tableau can also prove that a logical formula is tautologous: if a formula is tautologous, its negation is a contradiction, so a tableau built from its negation will close.³³¹

To construct a tableau for an argument ⟨ { φ 1 , φ 2 , φ 3 , . . . , φ n } , ψ ⟩ {\displaystyle \langle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\},\psi \rangle } , one first writes out the set of premise formulas, { φ 1 , φ 2 , φ 3 , . . . , φ n } {\displaystyle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\}} , with one formula on each line, signed with T {\displaystyle T} (that is, T φ {\displaystyle T\varphi } for each T φ {\displaystyle T\varphi } in the set);³³² and together with those formulas (the order is unimportant), one also writes out the conclusion, ψ {\displaystyle \psi } , signed with F {\displaystyle F} (that is, F ψ {\displaystyle F\psi } ).³³³ One then produces a truth tree (analytic tableau) by using all those lines according to the rules.³³⁴ A closed tree will be proof that the argument was valid, in virtue of the fact that φ ⊨ ψ {\displaystyle \varphi \models \psi } if, and only if, { φ , ∼ ψ } {\displaystyle \{\varphi ,\sim \psi \}} is inconsistent (also written as φ , ∼ ψ ⊨ {\displaystyle \varphi ,\sim \psi \models } ).³³⁵

List of classically valid argument forms

Using semantic checking methods, such as truth tables or semantic tableaux, to check for tautologies and semantic consequences, it can be shown that, in classical logic, the following classical argument forms are semantically valid, i.e., these tautologies and semantic consequences hold.³³⁶ We use φ {\displaystyle \varphi } ⟚ ψ {\displaystyle \psi } to denote equivalence of φ {\displaystyle \varphi } and ψ {\displaystyle \psi } , that is, as an abbreviation for both φ ⊨ ψ {\displaystyle \varphi \models \psi } and ψ ⊨ φ {\displaystyle \psi \models \varphi } ;³³⁷ as an aid to reading the symbols, a description of each formula is given. The description reads the symbol ⊧ (called the "double turnstile") as "therefore", which is a common reading of it,³³⁸ ³³⁹ although many authors prefer to read it as "entails",³⁴⁰ ³⁴¹ or as "models".³⁴²

Name	Sequent	Description
Modus Ponens	( ( p → q ) ∧ p ) ⊨ q {\displaystyle ((p\to q)\land p)\models q} ³⁴³	If p then q; p; therefore q
Modus Tollens	( ( p → q ) ∧ ¬ q ) ⊨ ¬ p {\displaystyle ((p\to q)\land \neg q)\models \neg p} ³⁴⁴	If p then q; not q; therefore not p
Hypothetical Syllogism	( ( p → q ) ∧ ( q → r ) ) ⊨ ( p → r ) {\displaystyle ((p\to q)\land (q\to r))\models (p\to r)} ³⁴⁵	If p then q; if q then r; therefore, if p then r
Disjunctive Syllogism	( ( p ∨ q ) ∧ ¬ p ) ⊨ q {\displaystyle ((p\lor q)\land \neg p)\models q} ³⁴⁶	Either p or q, or both; not p; therefore, q
Constructive Dilemma	( ( p → q ) ∧ ( r → s ) ∧ ( p ∨ r ) ) ⊨ ( q ∨ s ) {\displaystyle ((p\to q)\land (r\to s)\land (p\lor r))\models (q\lor s)} ³⁴⁷	If p then q; and if r then s; but p or r; therefore q or s
Destructive Dilemma	( ( p → q ) ∧ ( r → s ) ∧ ( ¬ q ∨ ¬ s ) ) ⊨ ( ¬ p ∨ ¬ r ) {\displaystyle ((p\to q)\land (r\to s)\land (\neg q\lor \neg s))\models (\neg p\lor \neg r)}	If p then q; and if r then s; but not q or not s; therefore not p or not r
Bidirectional Dilemma	( ( p → q ) ∧ ( r → s ) ∧ ( p ∨ ¬ s ) ) ⊨ ( q ∨ ¬ r ) {\displaystyle ((p\to q)\land (r\to s)\land (p\lor \neg s))\models (q\lor \neg r)}	If p then q; and if r then s; but p or not s; therefore q or not r
Simplification	( p ∧ q ) ⊨ p {\displaystyle (p\land q)\models p} ³⁴⁸	p and q are true; therefore p is true
Conjunction	p , q ⊨ ( p ∧ q ) {\displaystyle p,q\models (p\land q)} ³⁴⁹	p and q are true separately; therefore they are true conjointly
Addition	p ⊨ ( p ∨ q ) {\displaystyle p\models (p\lor q)} ³⁵⁰ ³⁵¹	p is true; therefore the disjunction (p or q) is true
Composition of conjunction	( ( p → q ) ∧ ( p → r ) ) {\displaystyle ((p\to q)\land (p\to r))} ⟚ ( p → ( q ∧ r ) ) {\displaystyle (p\to (q\land r))}	If p then q; and if p then r; therefore if p is true then q and r are true
Composition of disjunction	( ( p → q ) ∨ ( p → r ) ) {\displaystyle ((p\to q)\lor (p\to r))} ⟚ ( p → ( q ∨ r ) ) {\displaystyle (p\to (q\lor r))}	If p then q; or if p then r; therefore if p is true then q or r is true
De Morgan's Theorem (1)	¬ ( p ∧ q ) {\displaystyle \neg (p\land q)} ⟚ ( ¬ p ∨ ¬ q ) {\displaystyle (\neg p\lor \neg q)} ³⁵²	The negation of (p and q) is equiv. to (not p or not q)
De Morgan's Theorem (2)	¬ ( p ∨ q ) {\displaystyle \neg (p\lor q)} ⟚ ( ¬ p ∧ ¬ q ) {\displaystyle (\neg p\land \neg q)} ³⁵³	The negation of (p or q) is equiv. to (not p and not q)
Commutation (1)	( p ∨ q ) {\displaystyle (p\lor q)} ⟚ ( q ∨ p ) {\displaystyle (q\lor p)} ³⁵⁴	(p or q) is equiv. to (q or p)
Commutation (2)	( p ∧ q ) {\displaystyle (p\land q)} ⟚ ( q ∧ p ) {\displaystyle (q\land p)} ³⁵⁵	(p and q) is equiv. to (q and p)
Commutation (3)	( p ↔ q ) {\displaystyle (p\leftrightarrow q)} ⟚ ( q ↔ p ) {\displaystyle (q\leftrightarrow p)} ³⁵⁶	(p iff q) is equiv. to (q iff p)
Association (1)	( p ∨ ( q ∨ r ) ) {\displaystyle (p\lor (q\lor r))} ⟚ ( ( p ∨ q ) ∨ r ) {\displaystyle ((p\lor q)\lor r)} ³⁵⁷	p or (q or r) is equiv. to (p or q) or r
Association (2)	( p ∧ ( q ∧ r ) ) {\displaystyle (p\land (q\land r))} ⟚ ( ( p ∧ q ) ∧ r ) {\displaystyle ((p\land q)\land r)} ³⁵⁸	p and (q and r) is equiv. to (p and q) and r
Distribution (1)	( p ∧ ( q ∨ r ) ) {\displaystyle (p\land (q\lor r))} ⟚ ( ( p ∧ q ) ∨ ( p ∧ r ) ) {\displaystyle ((p\land q)\lor (p\land r))} ³⁵⁹	p and (q or r) is equiv. to (p and q) or (p and r)
Distribution (2)	( p ∨ ( q ∧ r ) ) {\displaystyle (p\lor (q\land r))} ⟚ ( ( p ∨ q ) ∧ ( p ∨ r ) ) {\displaystyle ((p\lor q)\land (p\lor r))} ³⁶⁰	p or (q and r) is equiv. to (p or q) and (p or r)
Double Negation	p {\displaystyle p} ⟚ ¬ ¬ p {\displaystyle \neg \neg p} ³⁶¹ ³⁶²	p is equivalent to the negation of not p
Transposition	( p → q ) {\displaystyle (p\to q)} ⟚ ( ¬ q → ¬ p ) {\displaystyle (\neg q\to \neg p)} ³⁶³	If p then q is equiv. to if not q then not p
Material Implication	( p → q ) {\displaystyle (p\to q)} ⟚ ( ¬ p ∨ q ) {\displaystyle (\neg p\lor q)} ³⁶⁴	If p then q is equiv. to not p or q
Material Equivalence (1)	( p ↔ q ) {\displaystyle (p\leftrightarrow q)} ⟚ ( ( p → q ) ∧ ( q → p ) ) {\displaystyle ((p\to q)\land (q\to p))} ³⁶⁵	(p iff q) is equiv. to (if p is true then q is true) and (if q is true then p is true)
Material Equivalence (2)	( p ↔ q ) {\displaystyle (p\leftrightarrow q)} ⟚ ( ( p ∧ q ) ∨ ( ¬ p ∧ ¬ q ) ) {\displaystyle ((p\land q)\lor (\neg p\land \neg q))} ³⁶⁶	(p iff q) is equiv. to either (p and q are true) or (both p and q are false)
Material Equivalence (3)	( p ↔ q ) {\displaystyle (p\leftrightarrow q)} ⟚ ( ( p ∨ ¬ q ) ∧ ( ¬ p ∨ q ) ) {\displaystyle ((p\lor \neg q)\land (\neg p\lor q))}	(p iff q) is equiv to., both (p or not q is true) and (not p or q is true)
Exportation	( ( p ∧ q ) → r ) ⊨ ( p → ( q → r ) ) {\displaystyle ((p\land q)\to r)\models (p\to (q\to r))} ³⁶⁷	from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true)
Importation	( p → ( q → r ) ) ⊨ ( ( p ∧ q ) → r ) {\displaystyle (p\to (q\to r))\models ((p\land q)\to r)} ³⁶⁸	If p then (if q then r) is equivalent to if p and q then r
Idempotence of disjunction	p {\displaystyle p} ⟚ ( p ∨ p ) {\displaystyle (p\lor p)} ³⁶⁹	p is true is equiv. to p is true or p is true
Idempotence of conjunction	p {\displaystyle p} ⟚ ( p ∧ p ) {\displaystyle (p\land p)} ³⁷⁰	p is true is equiv. to p is true and p is true
Tertium non datur (Law of Excluded Middle)	⊨ ( p ∨ ¬ p ) {\displaystyle \models (p\lor \neg p)} ³⁷¹ ³⁷²	p or not p is true
Law of Non-Contradiction	⊨ ¬ ( p ∧ ¬ p ) {\displaystyle \models \neg (p\land \neg p)} ³⁷³ ³⁷⁴	p and not p is false, is a true statement
Explosion	( p ∧ ¬ p ) ⊨ q {\displaystyle (p\land \neg p)\models q} ³⁷⁵	p and not p; therefore q

Syntactic proof via natural deduction

Main article: Natural deduction

Natural deduction, since it is a method of syntactical proof, is specified by providing inference rules (also called rules of proof)³⁷⁶ for a language with the typical set of connectives { − , & , ∨ , → , ↔ } {\displaystyle \{-,\&,\lor ,\to ,\leftrightarrow \}} ; no axioms are used other than these rules.³⁷⁷ The rules are covered below, and a proof example is given afterwards.

Notation styles

Different authors vary to some extent regarding which inference rules they give, which will be noted. More striking to the look and feel of a proof, however, is the variation in notation styles. The § Gentzen notation, which was covered earlier for a short argument, can actually be stacked to produce large tree-shaped natural deduction proofs³⁷⁸ ³⁷⁹—not to be confused with "truth trees", which is another name for analytic tableaux.³⁸⁰ There is also a style due to Stanisław Jaśkowski, where the formulas in the proof are written inside various nested boxes,³⁸¹ and there is a simplification of Jaśkowski's style due to Fredric Fitch (Fitch notation), where the boxes are simplified to simple horizontal lines beneath the introductions of suppositions, and vertical lines to the left of the lines that are under the supposition.³⁸² Lastly, there is the only notation style which will actually be used in this article, which is due to Patrick Suppes,³⁸³ but was much popularized by E.J. Lemmon and Benson Mates.³⁸⁴ This method has the advantage that, graphically, it is the least intensive to produce and display, which made it a natural choice for the editor who wrote this part of the article, who did not understand the complex LaTeX commands that would be required to produce proofs in the other methods.

A proof, then, laid out in accordance with the Suppes–Lemmon notation style,³⁸⁵ is a sequence of lines containing sentences,³⁸⁶ where each sentence is either an assumption, or the result of applying a rule of proof to earlier sentences in the sequence.³⁸⁷ Each line of proof is made up of a sentence of proof, together with its annotation, its assumption set, and the current line number.³⁸⁸ The assumption set lists the assumptions on which the given sentence of proof depends, which are referenced by the line numbers.³⁸⁹ The annotation specifies which rule of proof was applied, and to which earlier lines, to yield the current sentence.³⁹⁰ See the § Natural deduction proof example.

Inference rules

Natural deduction inference rules, due ultimately to Gentzen, are given below.³⁹¹ There are ten primitive rules of proof, which are the rule assumption, plus four pairs of introduction and elimination rules for the binary connectives, and the rule reductio ad adbsurdum.³⁹² Disjunctive Syllogism can be used as an easier alternative to the proper ∨-elimination,³⁹³ and MTT and DN are commonly given rules,³⁹⁴ although they are not primitive.³⁹⁵

List of Inference Rules

Rule Name	Alternative names	Annotation	Assumption set	Statement
Rule of Assumptions³⁹⁶	Assumption³⁹⁷	A³⁹⁸ ³⁹⁹	The current line number.⁴⁰⁰	At any stage of the argument, introduce a proposition as an assumption of the argument.⁴⁰¹ ⁴⁰²
Conjunction introduction	Ampersand introduction,⁴⁰³ ⁴⁰⁴ conjunction (CONJ)⁴⁰⁵ ⁴⁰⁶	m, n &I⁴⁰⁷ ⁴⁰⁸	The union of the assumption sets at lines m and n.⁴⁰⁹	From φ {\displaystyle \varphi } and ψ {\displaystyle \psi } at lines m and n, infer φ & ψ {\displaystyle \varphi ~\&~\psi } .⁴¹⁰ ⁴¹¹
Conjunction elimination	Simplification (S),⁴¹² ampersand elimination⁴¹³ ⁴¹⁴	m &E⁴¹⁵ ⁴¹⁶	The same as at line m.⁴¹⁷	From φ & ψ {\displaystyle \varphi ~\&~\psi } at line m, infer φ {\displaystyle \varphi } and ψ {\displaystyle \psi } .⁴¹⁸ ⁴¹⁹
Disjunction introduction⁴²⁰	Addition (ADD)⁴²¹	m ∨I⁴²² ⁴²³	The same as at line m.⁴²⁴	From φ {\displaystyle \varphi } at line m, infer φ ∨ ψ {\displaystyle \varphi \lor \psi } , whatever ψ {\displaystyle \psi } may be.⁴²⁵ ⁴²⁶
Disjunction elimination	Wedge elimination,⁴²⁷ dilemma (DL)⁴²⁸	j,k,l,m,n ∨E⁴²⁹	The lines j,k,l,m,n.⁴³⁰	From φ ∨ ψ {\displaystyle \varphi \lor \psi } at line j, and an assumption of φ {\displaystyle \varphi } at line k, and a derivation of χ {\displaystyle \chi } from φ {\displaystyle \varphi } at line l, and an assumption of ψ {\displaystyle \psi } at line m, and a derivation of χ {\displaystyle \chi } from ψ {\displaystyle \psi } at line n, infer χ {\displaystyle \chi } .⁴³¹
Disjunctive Syllogism	Wedge elimination (∨E),⁴³² modus tollendo ponens (MTP)⁴³³	m,n DS⁴³⁴	The union of the assumption sets at lines m and n.⁴³⁵	From φ ∨ ψ {\displaystyle \varphi \lor \psi } at line m and − φ {\displaystyle -\varphi } at line n, infer ψ {\displaystyle \psi } ; from φ ∨ ψ {\displaystyle \varphi \lor \psi } at line m and − ψ {\displaystyle -\psi } at line n, infer φ {\displaystyle \varphi } .⁴³⁶
Arrow elimination⁴³⁷	Modus ponendo ponens (MPP),⁴³⁸ ⁴³⁹ modus ponens (MP),⁴⁴⁰ ⁴⁴¹ conditional elimination	m, n →E⁴⁴² ⁴⁴³	The union of the assumption sets at lines m and n.⁴⁴⁴	From φ → ψ {\displaystyle \varphi \to \psi } at line m, and φ {\displaystyle \varphi } at line n, infer ψ {\displaystyle \psi } .⁴⁴⁵
Arrow introduction⁴⁴⁶	Conditional proof (CP),⁴⁴⁷ ⁴⁴⁸ ⁴⁴⁹ conditional introduction	n, →I (m)⁴⁵⁰ ⁴⁵¹	Everything in the assumption set at line n, excepting m, the line where the antecedent was assumed.⁴⁵²	From ψ {\displaystyle \psi } at line n, following from the assumption of φ {\displaystyle \varphi } at line m, infer φ → ψ {\displaystyle \varphi \to \psi } .⁴⁵³
Reductio ad absurdum⁴⁵⁴	Indirect Proof (IP),⁴⁵⁵ negation introduction (−I),⁴⁵⁶ negation elimination (−E)⁴⁵⁷	m, n RAA (k)⁴⁵⁸	The union of the assumption sets at lines m and n, excluding k (the denied assumption).⁴⁵⁹	From a sentence and its denial⁴⁶⁰ at lines m and n, infer the denial of any assumption appearing in the proof (at line k).⁴⁶¹
Double arrow introduction⁴⁶²	Biconditional definition (Df ↔),⁴⁶³ biconditional introduction	m, n ↔ I⁴⁶⁴	The union of the assumption sets at lines m and n.⁴⁶⁵	From φ → ψ {\displaystyle \varphi \to \psi } and ψ → φ {\displaystyle \psi \to \varphi } at lines m and n, infer φ ↔ ψ {\displaystyle \varphi \leftrightarrow \psi } .⁴⁶⁶
Double arrow elimination⁴⁶⁷	Biconditional definition (Df ↔),⁴⁶⁸ biconditional elimination	m ↔ E⁴⁶⁹	The same as at line m.⁴⁷⁰	From φ ↔ ψ {\displaystyle \varphi \leftrightarrow \psi } at line m, infer either φ → ψ {\displaystyle \varphi \to \psi } or ψ → φ {\displaystyle \psi \to \varphi } .⁴⁷¹
Double negation⁴⁷² ⁴⁷³	Double negation elimination	m DN⁴⁷⁴	The same as at line m.⁴⁷⁵	From − − φ {\displaystyle --\varphi } at line m, infer φ {\displaystyle \varphi } .⁴⁷⁶
Modus tollendo tollens⁴⁷⁷	Modus tollens (MT)⁴⁷⁸	m, n MTT⁴⁷⁹	The union of the assumption sets at lines m and n.⁴⁸⁰	From φ → ψ {\displaystyle \varphi \to \psi } at line m, and − ψ {\displaystyle -\psi } at line n, infer − φ {\displaystyle -\varphi } .⁴⁸¹

Natural deduction proof example

The proof below⁴⁸² derives − P {\displaystyle -P} from P → Q {\displaystyle P\to Q} and − Q {\displaystyle -Q} using only MPP and RAA, which shows that MTT is not a primitive rule, since it can be derived from those two other rules.

Derivation of MTT from MPP and RAA

Assumption set	Line number	Sentence of proof	Annotation
1	1	P → Q {\displaystyle P\to Q}	A
2	2	− Q {\displaystyle -Q}	A
3	3	P {\displaystyle P}	A
1, 3	4	Q {\displaystyle Q}	1, 3 →E
1, 2	5	− P {\displaystyle -P}	2, 4 RAA

Syntactic proof via axioms

Main article: Hilbert system

It is possible to perform proofs axiomatically, which means that certain tautologies are taken as self-evident and various others are deduced from them using modus ponens as an inference rule, as well as a rule of substitution, which permits replacing any well-formed formula with any substitution-instance of it.⁴⁸³ Alternatively, one uses axiom schemas instead of axioms, and no rule of substitution is used.⁴⁸⁴

This section gives the axioms of some historically notable axiomatic systems for propositional logic. For more examples, as well as metalogical theorems that are specific to such axiomatic systems (such as their completeness and consistency), see the article Axiomatic system (logic).

Frege's Begriffsschrift

Although axiomatic proof has been used since the famous Ancient Greek textbook, Euclid's Elements of Geometry, in propositional logic it dates back to Gottlob Frege's 1879 Begriffsschrift.⁴⁸⁵ ⁴⁸⁶ Frege's system used only implication and negation as connectives.⁴⁸⁷ It had six axioms:⁴⁸⁸ ⁴⁸⁹ ⁴⁹⁰

Proposition 1: a → ( b → a ) {\displaystyle a\to (b\to a)}
Proposition 2: ( c → ( b → a ) ) → ( ( c → b ) → ( c → a ) ) {\displaystyle (c\to (b\to a))\to ((c\to b)\to (c\to a))}
Proposition 8: ( d → ( b → a ) ) → ( b → ( d → a ) ) {\displaystyle (d\to (b\to a))\to (b\to (d\to a))}
Proposition 28: ( b → a ) → ( ¬ a → ¬ b ) {\displaystyle (b\to a)\to (\neg a\to \neg b)}
Proposition 31: ¬ ¬ a → a {\displaystyle \neg \neg a\to a}
Proposition 41: a → ¬ ¬ a {\displaystyle a\to \neg \neg a}

These were used by Frege together with modus ponens and a rule of substitution (which was used but never precisely stated) to yield a complete and consistent axiomatization of classical truth-functional propositional logic.⁴⁹¹

Łukasiewicz's P2

Jan Łukasiewicz showed that, in Frege's system, "the third axiom is superfluous since it can be derived from the preceding two axioms, and that the last three axioms can be replaced by the single sentence C C N p N q C p q {\displaystyle CCNpNqCpq} ".⁴⁹² Which, taken out of Łukasiewicz's Polish notation into modern notation, means ( ¬ p → ¬ q ) → ( p → q ) {\displaystyle (\neg p\rightarrow \neg q)\rightarrow (p\rightarrow q)} . Hence, Łukasiewicz is credited⁴⁹³ with this system of three axioms:

p → ( q → p ) {\displaystyle p\to (q\to p)}
( p → ( q → r ) ) → ( ( p → q ) → ( p → r ) ) {\displaystyle (p\to (q\to r))\to ((p\to q)\to (p\to r))}
( ¬ p → ¬ q ) → ( q → p ) {\displaystyle (\neg p\to \neg q)\to (q\to p)}

Just like Frege's system, this system uses a substitution rule and uses modus ponens as an inference rule.⁴⁹⁴ The exact same system was given (with an explicit substitution rule) by Alonzo Church,⁴⁹⁵ who referred to it as the system P2⁴⁹⁶ ⁴⁹⁷ and helped popularize it.⁴⁹⁸

Schematic form of P2

One may avoid using the rule of substitution by giving the axioms in schematic form, using them to generate an infinite set of axioms. Hence, using Greek letters to represent schemata (metalogical variables that may stand for any well-formed formulas), the axioms are given as:⁴⁹⁹ ⁵⁰⁰

φ → ( ψ → φ ) {\displaystyle \varphi \to (\psi \to \varphi )}
( φ → ( ψ → χ ) ) → ( ( φ → ψ ) → ( φ → χ ) ) {\displaystyle (\varphi \to (\psi \to \chi ))\to ((\varphi \to \psi )\to (\varphi \to \chi ))}
( ¬ φ → ¬ ψ ) → ( ψ → φ ) {\displaystyle (\neg \varphi \to \neg \psi )\to (\psi \to \varphi )}

The schematic version of P2 is attributed to John von Neumann,⁵⁰¹ and is used in the Metamath "set.mm" formal proof database.⁵⁰² It has also been attributed to Hilbert,⁵⁰³ and named H {\displaystyle {\mathcal {H}}} in this context.⁵⁰⁴

Proof example in P2

As an example, a proof of A → A {\displaystyle A\to A} in P2 is given below. First, the axioms are given names:

(A1) ( p → ( q → p ) ) {\displaystyle (p\to (q\to p))} (A2) ( ( p → ( q → r ) ) → ( ( p → q ) → ( p → r ) ) ) {\displaystyle ((p\to (q\to r))\to ((p\to q)\to (p\to r)))} (A3) ( ( ¬ p → ¬ q ) → ( q → p ) ) {\displaystyle ((\neg p\to \neg q)\to (q\to p))}

And the proof is as follows:

A → ( ( B → A ) → A ) {\displaystyle A\to ((B\to A)\to A)} (instance of (A1))
( A → ( ( B → A ) → A ) ) → ( ( A → ( B → A ) ) → ( A → A ) ) {\displaystyle (A\to ((B\to A)\to A))\to ((A\to (B\to A))\to (A\to A))} (instance of (A2))
( A → ( B → A ) ) → ( A → A ) {\displaystyle (A\to (B\to A))\to (A\to A)} (from (1) and (2) by modus ponens)
A → ( B → A ) {\displaystyle A\to (B\to A)} (instance of (A1))
A → A {\displaystyle A\to A} (from (4) and (3) by modus ponens)

Solvers

One notable difference between propositional calculus and predicate calculus is that satisfiability of a propositional formula is decidable.⁵⁰⁵: 81 Deciding satisfiability of propositional logic formulas is an NP-complete problem. However, practical methods exist (e.g., DPLL algorithm, 1962; Chaff algorithm, 2001) that are very fast for many useful cases. Recent work has extended the SAT solver algorithms to work with propositions containing arithmetic expressions; these are the SMT solvers.

Notes

External links

Wikimedia Commons has media related to Propositional logic.

Klement, Kevin C. "Propositional Logic". In Fieser, James; Dowden, Bradley (eds.). Internet Encyclopedia of Philosophy. Retrieved 7 April 2025.
Franks, Curtis (2024). "Propositional Logic". In Zalta, Edward N.; Nodelman, Uri (eds.). Stanford Encyclopedia of Philosophy (Winter 2024 ed.). Metaphysics Research Lab, Stanford University. Retrieved 7 April 2025.

Formal Predicate Calculus, contains a systematic formal development with axiomatic proof
forall x: an introduction to formal logic, by P.D. Magnus, covers formal semantics and proof theory for sentential logic.
Chapter 2 / Propositional Logic from Logic In Action
Propositional sequent calculus prover on Project Nayuki. (note: implication can be input in the form !X|Y, and a sequent can be a single formula prefixed with > and having no commas)
Propositional Logic - A Generative Grammar
A Propositional Calculator that helps to understand simple expressions

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The turnstile, for syntactic consequence, is of lower precedence than the comma, which represents premise combination, which in turn is of lower precedence than the arrow, used for material implication; so no parentheses are needed to interpret this formula.[44] /wiki/Order_of_operations ↩
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A very general and abstract syntax is given here, following the notation in the SEP,[2] but including the third definition, which is very commonly given explicitly by other sources, such as Gillon,[14] Bostock,[37] Allen & Hand,[38] and many others. As noted elsewhere in the article, languages variously compose their set of atomic propositional variables from uppercase or lowercase letters (often focusing on P/p, Q/q, and R/r), with or without subscript numerals; and in their set of connectives, they may include either the full set of five typical connectives, { ¬ , ∧ , ∨ , → , ↔ } {\displaystyle \{\neg ,\land ,\lor ,\to ,\leftrightarrow \}} , or any of the truth-functionally complete subsets of it. (And, of course, they may also use any of the notational variants of these connectives.) ↩
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Note that the phrase "principle of composition" has referred to other things in other contexts, and even in the context of logic, since Bertrand Russell used it to refer to the principle that "a proposition which implies each of two propositions implies them both."[52] /wiki/Bertrand_Russell ↩
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The name "interpretation" is used by some authors and the name "case" by other authors. This article will be indifferent and use either, since it is collaboratively edited and there is no consensus about which terminology to adopt. ↩
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Hunter, Geoffrey (1971). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Press. ISBN 0-520-02356-0. 0-520-02356-0 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. pp. ix, x, 5–6, 15–16, 20, 24–29, 38, 42–43, 47. ISBN 978-0-415-13342-5. 978-0-415-13342-5 ↩
Landman, Fred (1991). "Structures for Semantics". Studies in Linguistics and Philosophy. 45: 127. doi:10.1007/978-94-011-3212-1. ISBN 978-0-7923-1240-6. ISSN 0924-4662. 978-0-7923-1240-6 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Landman, Fred (1991). "Structures for Semantics". Studies in Linguistics and Philosophy. 45: 127. doi:10.1007/978-94-011-3212-1. ISBN 978-0-7923-1240-6. ISSN 0924-4662. 978-0-7923-1240-6 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Landman, Fred (1991). "Structures for Semantics". Studies in Linguistics and Philosophy. 45: 127. doi:10.1007/978-94-011-3212-1. ISBN 978-0-7923-1240-6. ISSN 0924-4662. 978-0-7923-1240-6 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. pp. ix, x, 5–6, 15–16, 20, 24–29, 38, 42–43, 47. ISBN 978-0-415-13342-5. 978-0-415-13342-5 ↩
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A truth-functionally complete set of connectives[2] is also called simply functionally complete, or adequate for truth-functional logic,[39] or expressively adequate,[77] or simply adequate.[39][77] ↩
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Conventionally ⊨ φ {\displaystyle \models \varphi } , with nothing to the left of the turnstile, is used to symbolize a tautology. It may be interpreted as saying that φ {\displaystyle \varphi } is a semantic consequence of the empty set of formulae, i.e., { } ⊨ φ {\displaystyle \{\}\models \varphi } , but with the empty brackets omitted for simplicity;[37] which is just the same as to say that it is a tautology, i.e., that there is no interpretation under which it is false.[37] ↩
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Restall, Greg (2010). Logic: an introduction. Fundamentals of philosophy. London: Routledge. pp. 5, 36–41, 55–60, 69. ISBN 978-0-415-40068-8. 978-0-415-40068-8 ↩
Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. pp. ix, x, 5–6, 15–16, 20, 24–29, 38, 42–43, 47. ISBN 978-0-415-13342-5. 978-0-415-13342-5 ↩
Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. pp. ix, x, 5–6, 15–16, 20, 24–29, 38, 42–43, 47. ISBN 978-0-415-13342-5. 978-0-415-13342-5 ↩
Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. pp. ix, x, 5–6, 15–16, 20, 24–29, 38, 42–43, 47. ISBN 978-0-415-13342-5. 978-0-415-13342-5 ↩
Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. pp. ix, x, 5–6, 15–16, 20, 24–29, 38, 42–43, 47. ISBN 978-0-415-13342-5. 978-0-415-13342-5 ↩
Restall, Greg (2010). Logic: an introduction. Fundamentals of philosophy. London: Routledge. pp. 5, 36–41, 55–60, 69. ISBN 978-0-415-40068-8. 978-0-415-40068-8 ↩
Restall, Greg (2010). Logic: an introduction. Fundamentals of philosophy. London: Routledge. pp. 5, 36–41, 55–60, 69. ISBN 978-0-415-40068-8. 978-0-415-40068-8 ↩
Restall, Greg (2010). Logic: an introduction. Fundamentals of philosophy. London: Routledge. pp. 5, 36–41, 55–60, 69. ISBN 978-0-415-40068-8. 978-0-415-40068-8 ↩
Restall, Greg (2010). Logic: an introduction. Fundamentals of philosophy. London: Routledge. pp. 5, 36–41, 55–60, 69. ISBN 978-0-415-40068-8. 978-0-415-40068-8 ↩
Bostock, David (1997). Intermediate logic. Oxford : New York: Clarendon Press; Oxford University Press. pp. 4–5, 8–13, 18–19, 22, 27, 29, 191, 194. ISBN 978-0-19-875141-0. 978-0-19-875141-0 ↩
Bostock, David (1997). Intermediate logic. Oxford : New York: Clarendon Press; Oxford University Press. pp. 4–5, 8–13, 18–19, 22, 27, 29, 191, 194. ISBN 978-0-19-875141-0. 978-0-19-875141-0 ↩
Bostock, David (1997). Intermediate logic. Oxford : New York: Clarendon Press; Oxford University Press. pp. 4–5, 8–13, 18–19, 22, 27, 29, 191, 194. ISBN 978-0-19-875141-0. 978-0-19-875141-0 ↩
Lawson, Mark V. (2019). A first course in logic. Boca Raton: CRC Press, Taylor & Francis Group. pp. example 1.58. ISBN 978-0-8153-8664-3. 978-0-8153-8664-3 ↩
Bostock, David (1997). Intermediate logic. Oxford : New York: Clarendon Press; Oxford University Press. pp. 4–5, 8–13, 18–19, 22, 27, 29, 191, 194. ISBN 978-0-19-875141-0. 978-0-19-875141-0 ↩
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Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. pp. ix, x, 5–6, 15–16, 20, 24–29, 38, 42–43, 47. ISBN 978-0-415-13342-5. 978-0-415-13342-5 ↩
Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. pp. ix, x, 5–6, 15–16, 20, 24–29, 38, 42–43, 47. ISBN 978-0-415-13342-5. 978-0-415-13342-5 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Toida, Shunichi (2 August 2009). "Proof of Implications". CS381 Discrete Structures/Discrete Mathematics Web Course Material. Department of Computer Science, Old Dominion University. Retrieved 10 March 2010. http://www.cs.odu.edu/~toida/nerzic/content/logic/prop_logic/implications/implication_proof.html ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Hodges, Wilfrid (2001). Logic (2 ed.). London: Penguin Books. pp. 130–131. ISBN 978-0-14-100314-6. 978-0-14-100314-6 ↩
Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. pp. 6, 8, 14–16, 19–20, 44–48, 50–53, 56. ISBN 978-0-203-85155-5. 978-0-203-85155-5 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Pelletier, Francis Jeffry; Hazen, Allen (2024), "Natural Deduction Systems in Logic", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 22 March 2024 https://plato.stanford.edu/archives/spr2024/entries/natural-deduction/ ↩
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Restall, Greg (2010). Logic: an introduction. Fundamentals of philosophy. London: Routledge. pp. 5, 36–41, 55–60, 69. ISBN 978-0-415-40068-8. 978-0-415-40068-8 ↩
Pelletier, Francis Jeffry; Hazen, Allen (2024), "Natural Deduction Systems in Logic", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 22 March 2024 https://plato.stanford.edu/archives/spr2024/entries/natural-deduction/ ↩
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Pelletier, Francis Jeffry; Hazen, Allen (2024), "Natural Deduction Systems in Logic", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 22 March 2024 https://plato.stanford.edu/archives/spr2024/entries/natural-deduction/ ↩
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Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Arthur, Richard T. W. (2017). An introduction to logic: using natural deduction, real arguments, a little history, and some humour (2nd ed.). Peterborough, Ontario: Broadview Press. ISBN 978-1-55481-332-2. OCLC 962129086. 978-1-55481-332-2 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Arthur, Richard T. W. (2017). An introduction to logic: using natural deduction, real arguments, a little history, and some humour (2nd ed.). Peterborough, Ontario: Broadview Press. ISBN 978-1-55481-332-2. OCLC 962129086. 978-1-55481-332-2 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Arthur, Richard T. W. (2017). An introduction to logic: using natural deduction, real arguments, a little history, and some humour (2nd ed.). Peterborough, Ontario: Broadview Press. ISBN 978-1-55481-332-2. OCLC 962129086. 978-1-55481-332-2 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Arthur, Richard T. W. (2017). An introduction to logic: using natural deduction, real arguments, a little history, and some humour (2nd ed.). Peterborough, Ontario: Broadview Press. ISBN 978-1-55481-332-2. OCLC 962129086. 978-1-55481-332-2 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
To simplify the statement of the rule, the word "denial" here is used in this way: the denial of a formula φ {\displaystyle \varphi } that is not a negation is − φ {\displaystyle -\varphi } , whereas a negation, − φ {\displaystyle -\varphi } , has two denials, viz., φ {\displaystyle \varphi } and − − φ {\displaystyle --\varphi } .[38] ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Arthur, Richard T. W. (2017). An introduction to logic: using natural deduction, real arguments, a little history, and some humour (2nd ed.). Peterborough, Ontario: Broadview Press. ISBN 978-1-55481-332-2. OCLC 962129086. 978-1-55481-332-2 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Arthur, Richard T. W. (2017). An introduction to logic: using natural deduction, real arguments, a little history, and some humour (2nd ed.). Peterborough, Ontario: Broadview Press. ISBN 978-1-55481-332-2. OCLC 962129086. 978-1-55481-332-2 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39–40. ISBN 978-0-412-38090-7. 978-0-412-38090-7 ↩
Allen, Colin; Hand, Michael (2022). Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-54364-4. 978-0-262-54364-4 ↩
Smullyan, Raymond M. (23 July 2014). A Beginner's Guide to Mathematical Logic. Courier Corporation. pp. 102–103. ISBN 978-0-486-49237-7. 978-0-486-49237-7 ↩
Smullyan, Raymond M. (23 July 2014). A Beginner's Guide to Mathematical Logic. Courier Corporation. pp. 102–103. ISBN 978-0-486-49237-7. 978-0-486-49237-7 ↩
Bostock, David (1997). Intermediate logic. Oxford : New York: Clarendon Press; Oxford University Press. pp. 4–5, 8–13, 18–19, 22, 27, 29, 191, 194. ISBN 978-0-19-875141-0. 978-0-19-875141-0 ↩
Smullyan, Raymond M. (23 July 2014). A Beginner's Guide to Mathematical Logic. Courier Corporation. pp. 102–103. ISBN 978-0-486-49237-7. 978-0-486-49237-7 ↩
Franks, Curtis (2024). "Propositional Logic". In Zalta, Edward N.; Nodelman, Uri (eds.). Stanford Encyclopedia of Philosophy (Winter 2024 ed.). Metaphysics Research Lab, Stanford University. Retrieved 7 April 2025. https://plato.stanford.edu/archives/win2024/entries/logic-propositional/ ↩
Smullyan, Raymond M. (23 July 2014). A Beginner's Guide to Mathematical Logic. Courier Corporation. pp. 102–103. ISBN 978-0-486-49237-7. 978-0-486-49237-7 ↩
Mendelsohn, Richard L. (10 January 2005). The Philosophy of Gottlob Frege. Cambridge University Press. p. 185. ISBN 978-1-139-44403-3. 978-1-139-44403-3 ↩
Łukasiewicz, Jan (1970). Jan Lukasiewicz: Selected Works. North-Holland. p. 136. https://books.google.com/books?id=Jb_zOwAACAAJ ↩
Mendelsohn, Richard L. (10 January 2005). The Philosophy of Gottlob Frege. Cambridge University Press. p. 185. ISBN 978-1-139-44403-3. 978-1-139-44403-3 ↩
Łukasiewicz, Jan (1970). Jan Lukasiewicz: Selected Works. North-Holland. p. 136. https://books.google.com/books?id=Jb_zOwAACAAJ ↩
Smullyan, Raymond M. (23 July 2014). A Beginner's Guide to Mathematical Logic. Courier Corporation. pp. 102–103. ISBN 978-0-486-49237-7. 978-0-486-49237-7 ↩
Smullyan, Raymond M. (23 July 2014). A Beginner's Guide to Mathematical Logic. Courier Corporation. pp. 102–103. ISBN 978-0-486-49237-7. 978-0-486-49237-7 ↩
Church, Alonzo (1996). Introduction to Mathematical Logic. Princeton University Press. p. 119. ISBN 978-0-691-02906-1. 978-0-691-02906-1 ↩
Church, Alonzo (1996). Introduction to Mathematical Logic. Princeton University Press. p. 119. ISBN 978-0-691-02906-1. 978-0-691-02906-1 ↩
"Proof Explorer - Home Page - Metamath". us.metamath.org. Retrieved 2 July 2024. https://us.metamath.org/mpegif/mmset.html#scaxioms ↩
"Proof Explorer - Home Page - Metamath". us.metamath.org. Retrieved 2 July 2024. https://us.metamath.org/mpegif/mmset.html#scaxioms ↩
Bostock, David (1997). Intermediate logic. Oxford : New York: Clarendon Press; Oxford University Press. pp. 4–5, 8–13, 18–19, 22, 27, 29, 191, 194. ISBN 978-0-19-875141-0. 978-0-19-875141-0 ↩
"Proof Explorer - Home Page - Metamath". us.metamath.org. Retrieved 2 July 2024. https://us.metamath.org/mpegif/mmset.html#scaxioms ↩
Smullyan, Raymond M. (23 July 2014). A Beginner's Guide to Mathematical Logic. Courier Corporation. pp. 102–103. ISBN 978-0-486-49237-7. 978-0-486-49237-7 ↩
"Proof Explorer - Home Page - Metamath". us.metamath.org. Retrieved 2 July 2024. https://us.metamath.org/mpegif/mmset.html#scaxioms ↩
Walicki, Michał (2017). Introduction to mathematical logic (Extended ed.). New Jersey: World Scientific. p. 126. ISBN 978-981-4719-95-7. 978-981-4719-95-7 ↩
Walicki, Michał (2017). Introduction to mathematical logic (Extended ed.). New Jersey: World Scientific. p. 126. ISBN 978-981-4719-95-7. 978-981-4719-95-7 ↩
Quine, W. V. O. (1980). Mathematical Logic. Harvard University Press. ISBN 0-674-55451-5. 0-674-55451-5 ↩

p	→	(q	∨	r	→	(r	→	¬	p))
T	→	(F	∨	T	→	(T	→	¬	T))
T	→	(	T		→	(T	→	F	))
T	→	(	T		→		F		)
T	→				F
	F
T	F	F	T	T	F	T	F	F	T

p	→	(q	∨	r	→	(r	→	¬	p))
T	→	(F	∨	T	→	(T	→	¬	T))
T	→	(	T		→	(T	→	F	))
T	→	(	T		→		F		)
T	→				F
	F
T	F	F	T	T	F	T	F	F	T

History

Sentences

Declarative sentences

Compounding sentences with connectives

Arguments

Example argument

Validity and soundness

Formalization

Propositional variables

Gentzen notation

Language

Syntax

CF grammar in BNF

Constants and schemata

Semantics

Interpretation (case) and argument

Propositional connective semantics

Semantics via. truth tables

Semantics via assignment expressions

Connective definition methods

Semantic truth, validity, consequence

Proof systems

Semantic proof systems

Truth tables

Semantic tableaux

Syntactic proof systems

Axiomatic systems

Natural deduction

Sequent calculus

Semantic proof via truth tables

Semantic proof via tableaux

List of classically valid argument forms

Syntactic proof via natural deduction

Notation styles

Inference rules

Natural deduction proof example

Syntactic proof via axioms

Frege's Begriffsschrift

Łukasiewicz's P2

Schematic form of P2

Proof example in P2

Solvers

See also

Higher logical levels

Related topics

Notes

Further reading

Related works

External links

References

p	→	(q	∨	r	→	(r	→	¬	p))
T	→	(F	∨	T	→	(T	→	¬	T))
T	→	(	T		→	(T	→	F	))
T	→	(	T		→		F		)
T	→				F
	F
T	F	F	T	T	F	T	F	F	T