In mathematical logic, a propositional calculus or logic (also called sentential calculus or sentential logic) is a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of inference rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions. Such a constructed sequence of formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem.
Usually in Truthfunctional propositional logic, formulas are interpreted as having either a truth value of true or a truth value of false. Truthfunctional propositional logic and systems isomorphic to it, are considered to be zerothorder logic.
History
Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus^{[1]} and expanded by the Stoics. The logic was focused on propositions. This advancement was different from the traditional syllogistic logic which was focused on terms. However, later in antiquity, the propositional logic developed by the Stoics was no longer understood. Consequently, the system was essentially reinvented by Peter Abelard.^{[2]}
Propositional logic was eventually refined using symbolic logic. Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were reachieved by logicians like George Boole and Augustus De Morgan completely independent of Leibniz.^{[3]}
Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic was an advancement from the earlier propositional logic. Predicate logic has been described to be combining "the distinctive features of syllogistic logic and propositional logic."^{[4]} Consequently, it ushered a new era in the history of logic. However, advances in propositional logic were still made after Frege. These include Natural Deduction, TruthTrees and TruthTables. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz. TruthTrees were invented by Evert Willem Beth.^{[5]} The invention of truthtables, however, is of controversial attribution.
The ideas preceding truth tables have been found in both Frege^{[6]} and Bertrand Russell^{[7]} whereas the actual 'tabular structure' (i.e. being formed in a table format) is generally credited to either Ludwig Wittgenstein, Emil Post or both (independently of one another).^{[6]} Besides Frege and Russell, others credited for having preceding ideas of truthtables include Philo, Boole, Charles Sanders Peirce, and Ernst Schröder. And besides Post and Wittgenstein, others credited with the tabular structure include Łukasiewicz, Schröder, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis.^{[7]} Ultimately, some, like John Shosky, have concluded "It is far from clear that any one person should be given the title of 'inventor' of truthtables.".^{[7]}
Terminology
In general terms, a calculus is a formal system that consists of a set of syntactic expressions (wellformed formulæ or wffs), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation, intended to be interpreted to be logical equivalence, on the space of expressions.
When the formal system is intended to be a logical system, the expressions are meant to be interpreted to be statements, and the rules, known to be inference rules, are typically intended to be truthpreserving. In this setting, the rules (which may include axioms) can then be used to derive ("infer") formulæ representing true statements from given formulæ representing true statements.
The set of axioms may be empty, a nonempty finite set, a countably infinite set, or be given by axiom schemata. A formal grammar recursively defines the expressions and wellformed formulæ (wffs) of the language. In addition a semantics may be given which defines truth and valuations (or interpretations).
The language of a propositional calculus consists of
 a set of primitive symbols, variously referred to be atomic formulae, placeholders, proposition letters, or variables, and
 a set of operator symbols, variously interpreted to be logical operators or logical connectives.
A wellformed formula (wff) is any atomic formula, or any formula that can be built up from atomic formulæ by means of operator symbols according to the rules of the grammar.
Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition, while propositional variables range over the set of all atomic propositions. Schemata, however, range over all propositions. 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 $\backslash varphi\; \backslash ,\backslash !$, $\backslash psi$, and $\backslash chi$.
Basic concepts
The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in the details of
 their language, that is, the particular collection of primitive symbols and operator symbols,
 the set of axioms, or distinguished formulæ, and
 the set of inference rules.
We may represent any given proposition with a letter which we call a propositional constant, analogous to representing a number by a letter in mathematics, for instance, $a\; =\; 5$. We require that all propositions have exactly one of two truthvalues: true or false. To take an example, let $P$ be the proposition that it is raining outside. This will be true if it is raining outside and false otherwise.
 We then define truthfunctional operators, beginning with negation. We write $\backslash neg\; P$ to represent the negation of $P$, which can be thought of to be the denial of $P$. In the example above, $\backslash neg\; P$ expresses that it is not raining outside, or by a more standard reading: "It is not the case that it is raining outside." When $P$ is true, $\backslash neg\; P$ is false; and when $P$ is false, $\backslash neg\; P$ is true. $\backslash neg\; \backslash neg\; P$ always has the same truthvalue like$P$.
 Conjunction is a truthfunctional connective which forms a proposition out of two simpler propositions, for example, $P$ and $Q$. The conjunction of $P$ and $Q$ is written $P\; \backslash and\; Q$, and expresses that each are true. We read $P\; \backslash and\; Q$ for "$P$ and $Q$". For any two propositions, there are four possible assignments of truth values:
 $P$ is true and $Q$ is true
 $P$ is true and $Q$ is false
 $P$ is false and $Q$ is true
 $P$ is false and $Q$ is false
 The conjunction of $P$ and $Q$ is true in case 1 and is false otherwise. Where $P$ is the proposition that it is raining outside and $Q$ is the proposition that a coldfront is over Kansas, $P\; \backslash and\; Q$ is true when it is raining outside and there is a coldfront over Kansas. If it is not raining outside, then $P\; \backslash and\; Q$ is false; and if there is no coldfront over Kansas, then $P\; \backslash and\; Q$ is false.
 Disjunction resembles conjunction in that it forms a proposition out of two simpler propositions. We write it $P\; \backslash vee\; Q$, and it is read "$P$ or $Q$". It expresses that either $P$ or $Q$ is true. Thus, in the cases listed above, the disjunction of $P$ and $Q$ is true in all cases except 4. Using the example above, the disjunction expresses that it is either raining outside or there is a cold front over Kansas. (Note, this use of disjunction is supposed to resemble the use of the English word "or". However, it is most like the English inclusive "or", which can be used to express the truth of at least one of two propositions. It is not like the English exclusive "or", which expresses the truth of exactly one of two propositions. That is to say, the exclusive "or" is false when both $P$ and $Q$ are true (case 1). An example of the exclusive or is: You may have a bagel or a pastry, but not both. Often in natural language, given the appropriate context, the addendum "but not both" is omitted but implied. In mathematics, however, "or" is always inclusive or; if exclusive or is meant it will be specified, possibly by "xor".)
 Material conditional also joins two simpler propositions, and we write $P\; \backslash rightarrow\; Q$, which is read "if $P$ then $Q$". The proposition to the left of the arrow is called the antecedent and the proposition to the right is called the consequent. (There is no such designation for conjunction or disjunction, since they are commutative operations.) It expresses that $Q$ is true whenever $P$ is true. Thus it is true in every case above except case 2, because this is the only case when $P$ is true but $Q$ is not. Using the example, if $P$ then $Q$ expresses that if it is raining outside then there is a coldfront over Kansas. The material conditional is often confused with physical causation. The material conditional, however, only relates two propositions by their truthvalues—which is not the relation of cause and effect. It is contentious in the literature whether the material implication represents logical causation.
 Biconditional joins two simpler propositions, and we write $P\; \backslash leftrightarrow\; Q$, which is read "$P$ if and only if $Q$". It expresses that $P$ and $Q$ have the same truthvalue, thus $P$ if and only if $Q$ is true in cases 1 and 4, and false otherwise.
It is extremely helpful to look at the truth tables for these different operators, as well as the method of analytic tableaux.
Closure under operations
Propositional logic is closed under truthfunctional connectives. That is to say, for any proposition $\backslash varphi\; \backslash ,\backslash !$, $\backslash neg\; \backslash varphi\; \backslash ,\backslash !$ is also a proposition. Likewise, for any propositions $\backslash varphi\; \backslash ,\backslash !$ and $\backslash psi\; \backslash ,\backslash !$, $\backslash varphi\; \backslash and\; \backslash psi\; \backslash ,\backslash !$ is a proposition, and similarly for disjunction, conditional, and biconditional. This implies that, for instance, $P\; \backslash and\; Q$ is a proposition, and so it can be conjoined with another proposition. In order to represent this, we need to use parentheses to indicate which proposition is conjoined with which. For instance, $P\; \backslash and\; Q\; \backslash and\; R$ is not a wellformed formula, because we do not know if we are conjoining $P\; \backslash and\; Q$ with $R$ or if we are conjoining $P$ with $Q\; \backslash and\; R$. Thus we must write either $(P\; \backslash and\; Q)\; \backslash and\; R$ to represent the former, or $P\; \backslash and\; (Q\; \backslash and\; R)$ to represent the latter. By evaluating the truth conditions, we see that both expressions have the same truth conditions (will be true in the same cases), and moreover that any proposition formed by arbitrary conjunctions will have the same truth conditions, regardless of the location of the parentheses. This means that conjunction is associative, however, one should not assume that parentheses never serve a purpose. For instance, the sentence $P\; \backslash and\; (Q\; \backslash vee\; R)$ does not have the same truth conditions of $(P\; \backslash and\; Q)\; \backslash vee\; R$, so they are different sentences distinguished only by the parentheses. One can verify this by the truthtable method referenced above.
Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truthvalues. A simple way to generate this is by truthtables, in which one writes $P$, $Q$, …, $Z$ for any list of $k$ propositional constants—that is to say, any list of propositional constants with $k$ entries. Below this list, one writes $2^k$ rows, and below $P$ one fills in the first half of the rows with true (or T) and the second half with false (or F). Below $Q$ one fills in onequarter of the rows with T, then onequarter with F, then onequarter with T and the last quarter with F. The next column alternates between true and false for each eighth of the rows, then sixteenths, and so on, until the last propositional constant varies between T and F for each row. This will give a complete listing of cases or truthvalue assignments possible for those propositional constants.
Argument
The propositional calculus then defines an argument to be a set of propositions. A valid argument is a set of propositions, the last of which follows from—or is implied by—the rest. All other arguments are invalid. The simplest valid argument is modus ponens, one instance of which is the following set of propositions:
 $$
\begin{array}{rl}
1. & P \rightarrow Q \\
2. & P \\
\hline
\therefore & Q
\end{array}
This is a set of three propositions, each line is a proposition, and the last follows from the rest. The first two lines are called premises, and the last line the conclusion. We say that any proposition $C$ follows from any set of propositions $(P\_1,\; ...,\; P\_n)$, if $C$ must be true whenever every member of the set $(P\_1,\; ...,\; P\_n)$ is true. In the argument above, for any $P$ and $Q$, whenever $P\; \backslash rightarrow\; Q$ and $P$ are true, necessarily $Q$ is true. Notice that, when $P$ is true, we cannot consider cases 3 and 4 (from the truth table). When $P\; \backslash rightarrow\; Q$ is true, we cannot consider case 2. This leaves only case 1, in which Q is also true. Thus Q is implied by the premises.
This generalizes schematically. Thus, where $\backslash varphi\; \backslash ,\backslash !$ and $\backslash psi$ may be any propositions at all,
 $$
\begin{array}{rl}
1. & \varphi \rightarrow \psi \\
2. & \varphi \\
\hline
\therefore & \psi
\end{array}
Other argument forms are convenient, but not necessary. Given a complete set of axioms (see below for one such set), modus ponens is sufficient to prove all other argument forms in propositional logic, thus they may be considered to be a derivative. Note, this is not true of the extension of propositional logic to other logics like firstorder logic. Firstorder logic requires at least one additional rule of inference in order to obtain completeness.
The significance of argument in formal logic is that one may obtain new truths from established truths. In the first example above, given the two premises, the truth of $Q$ is not yet known or stated. After the argument is made, $Q$ is deduced. In this way, we define a deduction system to be a set of all propositions that may be deduced from another set of propositions. For instance, given the set of propositions $A\; =\; \backslash \{\; P\; \backslash or\; Q,\; \backslash neg\; Q\; \backslash and\; R,\; (P\; \backslash or\; Q)\; \backslash rightarrow\; R\; \backslash \}$, we can define a deduction system, $\backslash Gamma$, which is the set of all propositions which follow from $A$. Reiteration is always assumed, so $P\; \backslash or\; Q,\; \backslash neg\; Q\; \backslash and\; R,\; (P\; \backslash or\; Q)\; \backslash rightarrow\; R\; \backslash in\; \backslash Gamma$. Also, from the first element of $A$, last element, as well as modus ponens, $R$ is a consequence, and so $R\; \backslash in\; \backslash Gamma$. Because we have not included sufficiently complete axioms, though, nothing else may be deduced. Thus, even though most deduction systems studied in propositional logic are able to deduce $(P\; \backslash or\; Q)\; \backslash leftrightarrow\; (\backslash neg\; P\; \backslash rightarrow\; Q)$, this one is too weak to prove such a proposition.
Generic description of a propositional calculus
A propositional calculus is a formal system $\backslash mathcal\{L\}\; =\; \backslash mathcal\{L\}\; \backslash left(\; \backslash Alpha,\backslash \; \backslash Omega,\backslash \; \backslash Zeta,\backslash \; \backslash Iota\; \backslash right)$, where:
 The alpha set $\backslash Alpha$ is a finite set of elements called proposition symbols or propositional variables. Syntactically speaking, these are the most basic elements of the formal language $\backslash mathcal\{L\}$, otherwise referred to as atomic formulæ or terminal elements. In the examples to follow, the elements of $\backslash Alpha$ are typically the letters $p$, $q$, $r$, and so on.
 $\backslash Omega\; =\; \backslash Omega\_0\; \backslash cup\; \backslash Omega\_1\; \backslash cup\; \backslash ldots\; \backslash cup\; \backslash Omega\_j\; \backslash cup\; \backslash ldots\; \backslash cup\; \backslash Omega\_m.$
 In this partition, $\backslash Omega\_j$ is the set of operator symbols of arity $j$.
 In the more familiar propositional calculi, $\backslash Omega$ is typically partitioned as follows:
 $\backslash Omega\_1\; =\; \backslash \{\; \backslash lnot\; \backslash \},$
 $\backslash Omega\_2\; \backslash subseteq\; \backslash \{\; \backslash land,\; \backslash lor,\; \backslash rightarrow,\; \backslash leftrightarrow\; \backslash \}.$
 A frequently adopted convention treats the constant logical values as operators of arity zero, thus:
 $\backslash Omega\_0\; =\; \backslash \{\; 0,\; 1\; \backslash \}.\backslash ,\backslash !$
 Some writers use the tilde (~), or N, instead of $\backslash neg$; and some use the ampersand (&), the prefixed K, or $\backslash cdot$ instead of $\backslash wedge$. Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, or $\backslash \{\; \backslash bot,\; \backslash top\; \backslash \}$ all being seen in various contexts instead of {0, 1}.
 The zeta set $\backslash Zeta$ is a finite set of transformation rules that are called inference rules when they acquire logical applications.
 The iota set $\backslash Iota$ is a finite set of initial points that are called axioms when they receive logical interpretations.
The language of $\backslash mathcal\{L\}$, also known as its set of formulæ, wellformed formulas or wffs, is inductively defined by the following rules:
 Base: Any element of the alpha set $\backslash Alpha$ is a formula of $\backslash mathcal\{L\}$.
 If $p\_1,\; p\_2,\; \backslash ldots,\; p\_j$ are formulæ and $f$ is in $\backslash Omega\_j$, then $\backslash left(\; f(p\_1,\; p\_2,\; \backslash ldots,\; p\_j)\; \backslash right)$ is a formula.
 Closed: Nothing else is a formula of $\backslash mathcal\{L\}$.
Repeated applications of these rules permits the construction of complex formulæ. For example:
 By rule 1, $p$ is a formula.
 By rule 2, $\backslash neg\; p$ is a formula.
 By rule 1, $q$ is a formula.
 By rule 2, $(\; \backslash neg\; p\; \backslash lor\; q\; )$ is a formula.
Example 1. Simple axiom system
Let $\backslash mathcal\{L\}\_1\; =\; \backslash mathcal\{L\}(\backslash Alpha,\backslash Omega,\backslash Zeta,\backslash Iota)$, where $\backslash Alpha$, $\backslash Omega$, $\backslash Zeta$, $\backslash Iota$ are defined as follows:
 The alpha set $\backslash Alpha$, is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:
 $\backslash Alpha\; =\; \backslash \{p,\; q,\; r,\; s,\; t,\; u\; \backslash \}.\backslash ,\backslash !$
 Of the three connectives for conjunction, disjunction, and implication ($\backslash wedge$, $\backslash lor$, and $\backslash rightarrow$), one can be taken as primitive and the other two can be defined in terms of it and negation ($\backslash neg$).^{[8]} Indeed, all of the logical connectives can be defined in terms of a sole sufficient operator. The biconditional ($\backslash leftrightarrow$) can of course be defined in terms of conjunction and implication, with $a\; \backslash leftrightarrow\; b$ defined as $(a\; \backslash to\; b)\; \backslash land\; (b\; \backslash to\; a)$.
 Adopting negation and implication as the two primitive operations of a propositional calculus is tantamount to having the omega set $\backslash Omega\; =\; \backslash Omega\_1\; \backslash cup\; \backslash Omega\_2$ partition as follows:
 $\backslash Omega\_1\; =\; \backslash \{\; \backslash lnot\; \backslash \},$
 $\backslash Omega\_2\; =\; \backslash \{\; \backslash rightarrow\; \backslash \}.$
 $(p\; \backslash to\; (q\; \backslash to\; p))$
 $((p\; \backslash to\; (q\; \backslash to\; r))\; \backslash to\; ((p\; \backslash to\; q)\; \backslash to\; (p\; \backslash to\; r)))$
 $((\backslash neg\; p\; \backslash to\; \backslash neg\; q)\; \backslash to\; (q\; \backslash to\; p))$
 The rule of inference is modus ponens (i.e., from $p$ and $(p\; \backslash to\; q)$, infer $q$). Then $a\; \backslash lor\; b$ is defined as $\backslash neg\; a\; \backslash to\; b$, and $a\; \backslash land\; b$ is defined as $\backslash neg(a\; \backslash to\; \backslash neg\; b)$.
Example 2. Natural deduction system
Let $\backslash mathcal\{L\}\_2\; =\; \backslash mathcal\{L\}(\backslash Alpha,\; \backslash Omega,\; \backslash Zeta,\; \backslash Iota)$, where $\backslash Alpha$, $\backslash Omega$, $\backslash Zeta$, $\backslash Iota$ are defined as follows:
 The alpha set $\backslash Alpha$, is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:
 $\backslash Alpha\; =\; \backslash \{p,\; q,\; r,\; s,\; t,\; u\; \backslash \}.\backslash ,\backslash !$
 The omega set $\backslash Omega\; =\; \backslash Omega\_1\; \backslash cup\; \backslash Omega\_2$ partitions as follows:
 $\backslash Omega\_1\; =\; \backslash \{\; \backslash lnot\; \backslash \},$
 $\backslash Omega\_2\; =\; \backslash \{\; \backslash land,\; \backslash lor,\; \backslash rightarrow,\; \backslash leftrightarrow\; \backslash \}.$
In the following example of a propositional calculus, the transformation rules are intended to be interpreted as the inference rules of a socalled natural deduction system. The particular system presented here has no initial points, which means that its interpretation for logical applications derives its theorems from an empty axiom set.
 The set of initial points is empty, that is, $\backslash Iota\; =\; \backslash varnothing$.
 The set of transformation rules, $\backslash Zeta$, is described as follows:
Our propositional calculus has ten inference rules. These rules allow us to derive other true formulae given a set of formulae that are assumed to be true. The first nine simply state that we can infer certain wffs from other wffs. The last rule however uses hypothetical reasoning in the sense that in the premise of the rule we temporarily assume an (unproven) hypothesis to be part of the set of inferred formulae to see if we can infer a certain other formula. Since the first nine rules don't do this they are usually described as nonhypothetical rules, and the last one as a hypothetical rule.
In describing the transformation rules, we may introduce a metalanguage symbol $\backslash vdash$. It is basically a convenient shorthand for saying "infer that". The format is $\backslash Gamma\; \backslash vdash\; \backslash psi$, in which $\backslash Gamma$ is a (possibly empty) set of formulae called premises, and $\backslash psi$ is a formula called conclusion. The transformation rule $\backslash Gamma\; \backslash vdash\; \backslash psi$ means that if every proposition in $\backslash Gamma$ is a theorem (or has the same truth value as the axioms), then $\backslash psi$ is also a theorem. Note that considering the following rule Conjunction introduction, we will know whenever $\backslash Gamma$ has more than one formula, we can always safely reduce it into one formula using conjunction. So for short, from that time on we may represent $\backslash Gamma$ as one formula instead of a set. Another omission for convenience is when $\backslash Gamma$ is an empty set, in which case $\backslash Gamma$ may not appear.
 Negation introduction
 From $(p\; \backslash to\; q)$ and $(p\; \backslash to\; \backslash neg\; q)$, infer $\backslash neg\; p$.
 That is, $\backslash \{\; (p\; \backslash to\; q),\; (p\; \backslash to\; \backslash neg\; q)\; \backslash \}\; \backslash vdash\; \backslash neg\; p$.
 Negation elimination
 From $\backslash neg\; p$, infer $(p\; \backslash to\; r)$.
 That is, $\backslash \{\; \backslash neg\; p\; \backslash \}\; \backslash vdash\; (p\; \backslash to\; r)$.
 Double negative elimination
 From $\backslash neg\; \backslash neg\; p$, infer $p$.
 That is, $\backslash neg\; \backslash neg\; p\; \backslash vdash\; p$.
 Conjunction introduction
 From $p$ and $q$, infer $(p\; \backslash land\; q)$.
 That is, $\backslash \{\; p,\; q\; \backslash \}\; \backslash vdash\; (p\; \backslash land\; q)$.
 Conjunction elimination
 From $(p\; \backslash land\; q)$, infer $p$.
 From $(p\; \backslash land\; q)$, infer $q$.
 That is, $(p\; \backslash land\; q)\; \backslash vdash\; p$ and $(p\; \backslash land\; q)\; \backslash vdash\; q$.
 Disjunction introduction
 From $p$, infer $(p\; \backslash lor\; q)$.
 From $q$, infer $(p\; \backslash lor\; q)$.
 That is, $p\; \backslash vdash\; (p\; \backslash lor\; q)$ and $q\; \backslash vdash\; (p\; \backslash lor\; q)$.
 Disjunction elimination
 From $(p\; \backslash lor\; q)$ and $(p\; \backslash to\; r)$ and $(q\; \backslash to\; r)$, infer $r$.
 That is, $\backslash \{p\; \backslash lor\; q,\; p\; \backslash to\; r,\; q\; \backslash to\; r\backslash \}\; \backslash vdash\; r$.
 Biconditional introduction
 From $(p\; \backslash to\; q)$ and $(q\; \backslash to\; p)$, infer $(p\; \backslash leftrightarrow\; q)$.
 That is, $\backslash \{p\; \backslash to\; q,\; q\; \backslash to\; p\backslash \}\; \backslash vdash\; (p\; \backslash leftrightarrow\; q)$.
 Biconditional elimination
 From $(p\; \backslash leftrightarrow\; q)$, infer $(p\; \backslash to\; q)$.
 From $(p\; \backslash leftrightarrow\; q)$, infer $(q\; \backslash to\; p)$.
 That is, $(p\; \backslash leftrightarrow\; q)\; \backslash vdash\; (p\; \backslash to\; q)$ and $(p\; \backslash leftrightarrow\; q)\; \backslash vdash\; (q\; \backslash to\; p)$.
 Modus ponens (conditional elimination)
 From $p$ and $(p\; \backslash to\; q)$, infer $q$.
 That is, $\backslash \{\; p,\; p\; \backslash to\; q\backslash \}\; \backslash vdash\; q$.
 Conditional proof (conditional introduction)
 From [accepting $p$ allows a proof of $q$], infer $(p\; \backslash to\; q)$.
 That is, $(p\; \backslash vdash\; q)\; \backslash vdash\; (\backslash vdash\; (p\; \backslash to\; q))$.
Basic and derived argument forms
Basic and Derived Argument Forms

Name

Sequent

Description

Modus Ponens

$((p\; \backslash to\; q)\; \backslash land\; p)\; \backslash vdash\; q$

If $p$ then $q$; $p$; therefore $q$

Modus Tollens

$((p\; \backslash to\; q)\; \backslash land\; \backslash neg\; q)\; \backslash vdash\; \backslash neg\; p$

If $p$ then $q$; not $q$; therefore not $p$

Hypothetical Syllogism

$((p\; \backslash to\; q)\; \backslash land\; (q\; \backslash to\; r))\; \backslash vdash\; (p\; \backslash to\; r)$

If $p$ then $q$; if $q$ then $r$; therefore, if $p$ then $r$

Disjunctive Syllogism

$((p\; \backslash lor\; q)\; \backslash land\; \backslash neg\; p)\; \backslash vdash\; q$

Either $p$ or $q$, or both; not $p$; therefore, $q$

Constructive Dilemma

$((p\; \backslash to\; q)\; \backslash land\; (r\; \backslash to\; s)\; \backslash land\; (p\; \backslash lor\; r))\; \backslash vdash\; (q\; \backslash lor\; s)$

If $p$ then $q$; and if $r$ then $s$; but $p$ or $r$; therefore $q$ or $s$

Destructive Dilemma

$((p\; \backslash to\; q)\; \backslash land\; (r\; \backslash to\; s)\; \backslash land(\backslash neg\; q\; \backslash lor\; \backslash neg\; s))\; \backslash vdash\; (\backslash neg\; p\; \backslash lor\; \backslash neg\; r)$

If $p$ then $q$; and if $r$ then $s$; but not $q$ or not $s$; therefore not $p$ or not $r$

Bidirectional Dilemma

$((p\; \backslash to\; q)\; \backslash land\; (r\; \backslash to\; s)\; \backslash land(p\; \backslash lor\; \backslash neg\; s))\; \backslash vdash\; (q\; \backslash lor\; \backslash neg\; r)$

If $p$ then $q$; and if $r$ then $s$; but $p$ or not $s$; therefore $q$ or not $r$

Simplification

$(p\; \backslash land\; q)\; \backslash vdash\; p$

$p$ and $q$ are true; therefore $p$ is true

Conjunction

$p,\; q\; \backslash vdash\; (p\; \backslash land\; q)$

$p$ and $q$ are true separately; therefore they are true conjointly

Addition

$p\; \backslash vdash\; (p\; \backslash lor\; q)$

$p$ is true; therefore the disjunction ($p$ or $q$) is true

Composition

$((p\; \backslash to\; q)\; \backslash land\; (p\; \backslash to\; r))\; \backslash vdash\; (p\; \backslash to\; (q\; \backslash land\; r))$

If $p$ then $q$; and if $p$ then $r$; therefore if $p$ is true then $q$ and $r$ are true

De Morgan's Theorem (1)

$\backslash neg\; (p\; \backslash land\; q)\; \backslash vdash\; (\backslash neg\; p\; \backslash lor\; \backslash neg\; q)$

The negation of ($p$ and $q$) is equiv. to (not $p$ or not $q$)

De Morgan's Theorem (2)

$\backslash neg\; (p\; \backslash lor\; q)\; \backslash vdash\; (\backslash neg\; p\; \backslash land\; \backslash neg\; q)$

The negation of ($p$ or $q$) is equiv. to (not $p$ and not $q$)

Commutation (1)

$(p\; \backslash lor\; q)\; \backslash vdash\; (q\; \backslash lor\; p)$

($p$ or $q$) is equiv. to ($q$ or $p$)

Commutation (2)

$(p\; \backslash land\; q)\; \backslash vdash\; (q\; \backslash land\; p)$

($p$ and $q$) is equiv. to ($q$ and $p$)

Commutation (3)

$(p\; \backslash leftrightarrow\; q)\; \backslash vdash\; (q\; \backslash leftrightarrow\; p)$

($p$ is equiv. to $q$) is equiv. to ($q$ is equiv. to $p$)

Association (1)

$(p\; \backslash lor\; (q\; \backslash lor\; r))\; \backslash vdash\; ((p\; \backslash lor\; q)\; \backslash lor\; r)$

$p$ or ($q$ or $r$) is equiv. to ($p$ or $q$) or $r$

Association (2)

$(p\; \backslash land\; (q\; \backslash land\; r))\; \backslash vdash\; ((p\; \backslash land\; q)\; \backslash land\; r)$

$p$ and ($q$ and $r$) is equiv. to ($p$ and $q$) and $r$

Distribution (1)

$(p\; \backslash land\; (q\; \backslash lor\; r))\; \backslash vdash\; ((p\; \backslash land\; q)\; \backslash lor\; (p\; \backslash land\; r))$

$p$ and ($q$ or $r$) is equiv. to ($p$ and $q$) or ($p$ and $r$)

Distribution (2)

$(p\; \backslash lor\; (q\; \backslash land\; r))\; \backslash vdash\; ((p\; \backslash lor\; q)\; \backslash land\; (p\; \backslash lor\; r))$

$p$ or ($q$ and $r$) is equiv. to ($p$ or $q$) and ($p$ or $r$)

Double Negation

$p\; \backslash vdash\; \backslash neg\; \backslash neg\; p$

$p$ is equivalent to the negation of not $p$

Transposition

$(p\; \backslash to\; q)\; \backslash vdash\; (\backslash neg\; q\; \backslash to\; \backslash neg\; p)$

If $p$ then $q$ is equiv. to if not $q$ then not $p$

Material Implication

$(p\; \backslash to\; q)\; \backslash vdash\; (\backslash neg\; p\; \backslash lor\; q)$

If $p$ then $q$ is equiv. to not $p$ or $q$

Material Equivalence (1)

$(p\; \backslash leftrightarrow\; q)\; \backslash vdash\; ((p\; \backslash to\; q)\; \backslash land\; (q\; \backslash to\; p))$

($p$ 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)

$(p\; \backslash leftrightarrow\; q)\; \backslash vdash\; ((p\; \backslash land\; q)\; \backslash lor\; (\backslash neg\; p\; \backslash land\; \backslash neg\; q))$

($p$ iff $q$) is equiv. to either ($p$ and $q$ are true) or (both $p$ and $q$ are false)

Material Equivalence (3)

$(p\; \backslash leftrightarrow\; q)\; \backslash vdash\; ((p\; \backslash lor\; \backslash neg\; q)\; \backslash land\; (\backslash neg\; p\; \backslash lor\; q))$

($p$ iff $q$) is equiv to., both ($p$ or not $q$ is true) and (not $p$ or $q$ is true)

Exportation^{[9]}

$((p\; \backslash land\; q)\; \backslash to\; r)\; \backslash vdash\; (p\; \backslash to\; (q\; \backslash to\; r))$

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

$(p\; \backslash to\; (q\; \backslash to\; r))\; \backslash vdash\; ((p\; \backslash land\; q)\; \backslash to\; r)$

If $p$ then (if $q$ then $r$) is equivalent to if $p$ and $q$ then $r$

Tautology (1)

$p\; \backslash vdash\; (p\; \backslash lor\; p)$

$p$ is true is equiv. to $p$ is true or $p$ is true

Tautology (2)

$p\; \backslash vdash\; (p\; \backslash land\; p)$

$p$ is true is equiv. to $p$ is true and $p$ is true

Tertium non datur (Law of Excluded Middle)

$\backslash vdash\; (p\; \backslash lor\; \backslash neg\; p)$

$p$ or not $p$ is true

Law of NonContradiction

$\backslash vdash\; \backslash neg\; (p\; \backslash land\; \backslash neg\; p)$

$p$ and not $p$ is false, is a true statement

Proofs in propositional calculus
One of the main uses of a propositional calculus, when interpreted for logical applications, is to determine relations of logical equivalence between propositional formulæ. These relationships are determined by means of the available transformation rules, sequences of which are called derivations or proofs.
In the discussion to follow, a proof is presented as a sequence of numbered lines, with each line consisting of a single formula followed by a reason or justification for introducing that formula. Each premise of the argument, that is, an assumption introduced as an hypothesis of the argument, is listed at the beginning of the sequence and is marked as a "premise" in lieu of other justification. The conclusion is listed on the last line. A proof is complete if every line follows from the previous ones by the correct application of a transformation rule. (For a contrasting approach, see prooftrees).
Example of a proof
 To be shown that $A\; \backslash to\; A$.
 One possible proof of this (which, though valid, happens to contain more steps than are necessary) may be arranged as follows:
Example of a Proof

Number

Formula

Reason

1 
$A\backslash ,\backslash !$ 
premise

2 
$A\; \backslash lor\; A$ 
From (1) by disjunction introduction

3 
$(A\; \backslash lor\; A)\; \backslash land\; A$ 
From (1) and (2) by conjunction introduction

4 
$A\backslash ,\backslash !$ 
From (3) by conjunction elimination

5 
$A\; \backslash vdash\; A$ 
Summary of (1) through (4)

6 
$\backslash vdash\; A\; \backslash to\; A$ 
From (5) by conditional proof

Interpret $A\; \backslash vdash\; A$ as "Assuming $A$, infer $A$". Read $\backslash vdash\; A\; \backslash to\; A$ as "Assuming nothing, infer that $A$ implies $A$", or "It is a tautology that $A$ implies $A$", or "It is always true that $A$ implies $A$".
Soundness and completeness of the rules
The crucial properties of this set of rules are that they are sound and complete. Informally this means that the rules are correct and that no other rules are required. These claims can be made more formal as follows.
We define a truth assignment as a function that maps propositional variables to true or false. Informally such a truth assignment can be understood as the description of a possible state of affairs (or possible world) where certain statements are true and others are not. The semantics of formulae can then be formalized by defining for which "state of affairs" they are considered to be true, which is what is done by the following definition.
We define when such a truth assignment $A\; \backslash ,$ satisfies a certain wff with the following rules:
 $A\; \backslash ,$ satisfies the propositional variable $P\; \backslash ,$ if and only if $A(P)\; =\; \backslash text\{true\}\; \backslash ,$
 $A\; \backslash ,$ satisfies $\backslash neg\; \backslash phi\; \backslash ,$ if and only if $A\; \backslash ,$ does not satisfy $\backslash phi\; \backslash ,$
 $A\; \backslash ,$ satisfies $(\backslash phi\; \backslash land\; \backslash psi)\; \backslash ,$ if and only if $A\; \backslash ,$ satisfies both $\backslash phi\; \backslash ,$ and $\backslash psi\; \backslash ,$
 $A\; \backslash ,$ satisfies $(\backslash phi\; \backslash lor\; \backslash psi)\; \backslash ,$ if and only if $A\; \backslash ,$ satisfies at least one of either $\backslash phi\; \backslash ,$ or $\backslash psi\; \backslash ,$
 $A\; \backslash ,$ satisfies $(\backslash phi\; \backslash to\; \backslash psi)\; \backslash ,$ if and only if it is not the case that $A\; \backslash ,$ satisfies $\backslash phi\; \backslash ,$ but not $\backslash psi\; \backslash ,$
 $A\; \backslash ,$ satisfies $(\backslash phi\; \backslash leftrightarrow\; \backslash psi)\; \backslash ,$ if and only if $A\; \backslash ,$ satisfies both $\backslash phi\; \backslash ,$ and $\backslash psi\; \backslash ,$ or satisfies neither one of them
With this definition we can now formalize what it means for a formula $\backslash phi\; \backslash ,$ to be implied by a certain set $S\; \backslash ,$ of formulae. Informally this is true if in all worlds that are possible given the set of formulae $S\; \backslash ,$ the formula $\backslash phi\; \backslash ,$ also holds. This leads to the following formal definition: We say that a set $S\; \backslash ,$ of wffs semantically entails (or implies) a certain wff $\backslash phi\; \backslash ,$ if all truth assignments that satisfy all the formulae in $S\; \backslash ,$ also satisfy $\backslash phi\; \backslash ,.$
Finally we define syntactical entailment such that $\backslash phi\; \backslash ,$ is syntactically entailed by $S\; \backslash ,$ if and only if we can derive it with the inference rules that were presented above in a finite number of steps. This allows us to formulate exactly what it means for the set of inference rules to be sound and complete:
 Soundness
 If the set of wffs $S\; \backslash ,$ syntactically entails wff $\backslash phi\; \backslash ,$ then $S\; \backslash ,$ semantically entails $\backslash phi\; \backslash ,$
 Completeness
 If the set of wffs $S\; \backslash ,$ semantically entails wff $\backslash phi\; \backslash ,$ then $S\; \backslash ,$ syntactically entails $\backslash phi\; \backslash ,$
For the above set of rules this is indeed the case.
Sketch of a soundness proof
(For most logical systems, this is the comparatively "simple" direction of proof)
Notational conventions: Let $G$ be a variable ranging over sets of sentences. Let $A$, $B$, and $C$ range over sentences. For "$G$ syntactically entails $A$" we write "$G$ proves $A$". For "$G$ semantically entails $A$" we write "$G$ implies $A$".
We want to show: ($A$)($G$)(if $G$ proves $A$, then $G$ implies $A$).
We note that "$G$ proves $A$" has an inductive definition, and that gives us the immediate resources for demonstrating claims of the form "If $G$ proves $A$, then ...". So our proof proceeds by induction.
 Basis. Show: If $A$ is a member of $G$, then $G$ implies $A$.
 Basis. Show: If $A$ is an axiom, then $G$ implies $A$.
 Inductive step (induction on $n$, the length of the proof):
 Assume for arbitrary $G$ and $A$ that if $G$ proves $A$ in $n$ or fewer steps, then $G$ implies $A$.
 For each possible application of a rule of inference at step $n+1$, leading to a new theorem $B$, show that $G$ implies $B$.
Notice that Basis Step II can be omitted for natural deduction systems because they have no axioms. When used, Step II involves showing that each of the axioms is a (semantic) logical truth.
The Basis step(s) demonstrate(s) that the simplest provable sentences from $G$ are also implied by $G$, for any $G$. (The is simple, since the semantic fact that a set implies any of its members, is also trivial.) The Inductive step will systematically cover all the further sentences that might be provable—by considering each case where we might reach a logical conclusion using an inference rule—and shows that if a new sentence is provable, it is also logically implied. (For example, we might have a rule telling us that from "$A$" we can derive "$A$ or $B$". In III.a We assume that if $A$ is provable it is implied. We also know that if $A$ is provable then "$A$ or $B$" is provable. We have to show that then "$A$ or $B$" too is implied. We do so by appeal to the semantic definition and the assumption we just made. $A$ is provable from $G$, we assume. So it is also implied by $G$. So any semantic valuation making all of $G$ true makes $A$ true. But any valuation making $A$ true makes "$A$ or $B$" true, by the defined semantics for "or". So any valuation which makes all of $G$ true makes "$A$ or $B$" true. So "$A$ or $B$" is implied.) Generally, the Inductive step will consist of a lengthy but simple casebycase analysis of all the rules of inference, showing that each "preserves" semantic implication.
By the definition of provability, there are no sentences provable other than by being a member of $G$, an axiom, or following by a rule; so if all of those are semantically implied, the deduction calculus is sound.
Sketch of completeness proof
(This is usually the much harder direction of proof.)
We adopt the same notational conventions as above.
We want to show: If $G$ implies $A$, then $G$ proves $A$. We proceed by contraposition: We show instead that if $G$ does not prove $A$ then $G$ does not imply $A$.
 $G$ does not prove $A$. (Assumption)
 If $G$ does not prove $A$, then we can construct an (infinite) "Maximal Set", $G^*$, which is a superset of $G$ and which also does not prove $A$.
 Place an "ordering" on all the sentences in the language (e.g., shortest first, and equally long ones in extended alphabetical ordering), and number them $E\_1$, $E\_2$, …
 Define a series $G\_n$ of sets ($G\_0$, $G\_1$, …) inductively:
 $G\_0\; =\; G$
 If $G\_k\; \backslash cup\; \backslash \{\; E\_\{k+1\}\; \backslash \}$ proves $A$, then $G\_\{k+1\}\; =\; G\_k$
 If $G\_k\; \backslash cup\; \backslash \{\; E\_\{k+1\}\; \backslash \}$ does not prove $A$, then $G\_\{k+1\}\; =\; G\_k\; \backslash cup\; \backslash \{\; E\_\{k+1\}\; \backslash \}$
 Define $G^*$ as the union of all the $G\_n$. (That is, $G^*$ is the set of all the sentences that are in any $G\_n$.)
 It can be easily shown that
 $G^*$ contains (is a superset of) $G$ (by (b.i));
 $G^*$ does not prove $A$ (because if it proves $A$ then some sentence was added to some $G\_n$ which caused it to prove '$A$; but this was ruled out by definition); and
 $G^*$ is a "Maximal Set" (with respect to $A$): If any more sentences whatever were added to $G^*$, it would prove $A$. (Because if it were possible to add any more sentences, they should have been added when they were encountered during the construction of the $G\_n$, again by definition)

 If $G^*$ is a Maximal Set (wrt $A$), then it is "truthlike". This means that it contains the sentence "$C$" only if it does not contain the sentence not$C$; If it contains "$C$" and contains "If $C$ then $B$" then it also contains "$B$"; and so forth.
 If $G^*$ is truthlike there is a "$G^*$Canonical" valuation of the language: one that makes every sentence in $G^*$ true and everything outside $G^*$ false while still obeying the laws of semantic composition in the language.
 A $G^*$canonical valuation will make our original set $G$ all true, and make $A$ false.
 If there is a valuation on which $G$ are true and $A$ is false, then $G$ does not (semantically) imply $A$.
QED
Another outline for a completeness proof
If a formula is a tautology, then there is a truth table for it which shows that each valuation yields the value true for the formula. Consider such a valuation. By mathematical induction on the length of the subformulae, show that the truth or falsity of the subformula follows from the truth or falsity (as appropriate for the valuation) of each propositional variable in the subformula. Then combine the lines of the truth table together two at a time by using "(P is true implies S) implies ((P is false implies S) implies S)". Keep repeating this until all dependencies on propositional variables have been eliminated. The result is that we have proved the given tautology. Since every tautology is provable, the logic is complete.
Interpretation of a truthfunctional propositional calculus
An interpretation of a truthfunctional propositional calculus $\backslash mathcal\{P\}$ is an assignment to each propositional symbol of $\backslash mathcal\{P\}$ of one or the other (but not both) of the truth values truth (T) and falsity (F), and an assignment to the connective symbols of $\backslash mathcal\{P\}$ of their usual truthfunctional meanings. An interpretation of a truthfunctional propositional calculus may also be expressed in terms of truth tables.^{[10]}
For $n$ distinct propositional symbols there are $2^n$ distinct possible interpretations. For any particular symbol $a$, for example, there are $2^1=2$ possible interpretations:
 $a$ is assigned T, or
 $a$ is assigned F.
For the pair $a$, $b$ there are $2^2=4$ possible interpretations:
 both are assigned T,
 both are assigned F,
 $a$ is assigned T and $b$ is assigned F, or
 $a$ is assigned F and $b$ is assigned T.^{[10]}
Since $\backslash mathcal\{P\}$ has $\backslash aleph\_0$, that is, denumerably many propositional symbols, there are $2^\{\backslash aleph\_0\}=\backslash mathfrak\; c$, and therefore uncountably many distinct possible interpretations of $\backslash mathcal\{P\}$.^{[10]}
Interpretation of a sentence of truthfunctional propositional logic
If $\backslash phi\backslash ,\backslash !$ and $\backslash psi$ are formulae of $\backslash mathcal\{P\}$ and $\backslash mathcal\{I\}$ is an interpretation of $\backslash mathcal\{P\}$ then:
 A sentence of propositional logic is true under an interpretation $\backslash mathcal\{I\}$ iff $\backslash mathcal\{I\}$ assigns the truth value T to that sentence. If a sentence is true under an interpretation, then that interpretation is called a model of that sentence.
 $\backslash phi\backslash ,\backslash !$ is false under an interpretation $\backslash mathcal\{I\}$ iff $\backslash phi\backslash ,\backslash !$ is not true under $\backslash mathcal\{I\}$.^{[10]}
 A sentence of propositional logic is logically valid iff it is true under every interpretation
 $\backslash models\backslash phi$ means that $\backslash phi\backslash ,\backslash !$ is logically valid
 A sentence $\backslash psi$ of propositional logic is a semantic consequence of a sentence $\backslash phi\backslash ,\backslash !$ iff there is no interpretation under which $\backslash phi\backslash ,\backslash !$ is true and $\backslash psi$ is false.
 A sentence of propositional logic is consistent iff it is true under at least one interpretation. It is inconsistent if it is not consistent.
Some consequences of these definitions:
 For any given interpretation a given formula is either true or false.^{[10]}
 No formula is both true and false under the same interpretation.^{[10]}
 $\backslash phi\backslash ,\backslash !$ is false for a given interpretation iff $\backslash neg\backslash phi$ is true for that interpretation; and $\backslash phi\backslash ,\backslash !$ is true under an interpretation iff $\backslash neg\backslash phi$ is false under that interpretation.^{[10]}
 If $\backslash phi\backslash ,\backslash !$ and $(\backslash phi\; \backslash rightarrow\; \backslash psi)$ are both true under a given interpretation, then $\backslash psi$ is true under that interpretation.^{[10]}
 If $\backslash models\_\{\backslash mathrm\; P\}\backslash phi$ and $\backslash models\_\{\backslash mathrm\; P\}(\backslash phi\; \backslash rightarrow\; \backslash psi)$, then $\backslash models\_\{\backslash mathrm\; P\}\backslash psi$.^{[10]}
 $\backslash neg\backslash phi$ is true under $\backslash mathcal\{I\}$ iff $\backslash phi\backslash ,\backslash !$ is not true under $\backslash mathcal\{I\}$.
 $(\backslash phi\; \backslash rightarrow\; \backslash psi)$ is true under $\backslash mathcal\{I\}$ iff either $\backslash phi\backslash ,\backslash !$ is not true under $\backslash mathcal\{I\}$ or $\backslash psi$ is true under $\backslash mathcal\{I\}$.^{[10]}
 A sentence $\backslash psi$ of propositional logic is a semantic consequence of a sentence $\backslash phi\backslash ,\backslash !$ iff $(\backslash phi\; \backslash rightarrow\; \backslash psi)$ is logically valid, that is, $\backslash phi\; \backslash models\_\{\backslash mathrm\; P\}\; \backslash psi$ iff $\backslash models\_\{\backslash mathrm\; P\}(\backslash phi\; \backslash rightarrow\; \backslash psi)$.^{[10]}
Alternative calculus
It is possible to define another version of propositional calculus, which defines most of the syntax of the logical operators by means of axioms, and which uses only one inference rule.
Axioms
Let $\backslash phi\backslash ,\backslash !$, $\backslash chi$ and $\backslash psi$ stand for wellformed formulæ. (The wffs themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are as follows:
Axioms

Name

Axiom Schema

Description

THEN1

$\backslash phi\; \backslash to\; (\backslash chi\; \backslash to\; \backslash phi)$

Add hypothesis $\backslash chi$, implication introduction

THEN2

$(\backslash phi\; \backslash to\; (\backslash chi\; \backslash to\; \backslash psi))\; \backslash to\; ((\backslash phi\; \backslash to\; \backslash chi)\; \backslash to\; (\backslash phi\; \backslash to\; \backslash psi))$

Distribute hypothesis $\backslash phi\backslash ,\backslash !$ over implication

AND1

$\backslash phi\; \backslash land\; \backslash chi\; \backslash to\; \backslash phi$

Eliminate conjunction

AND2

$\backslash phi\; \backslash land\; \backslash chi\; \backslash to\; \backslash chi$


AND3

$\backslash phi\; \backslash to\; (\backslash chi\; \backslash to\; (\backslash phi\; \backslash land\; \backslash chi))$

Introduce conjunction

OR1

$\backslash phi\; \backslash to\; \backslash phi\; \backslash lor\; \backslash chi$

Introduce disjunction

OR2

$\backslash chi\; \backslash to\; \backslash phi\; \backslash lor\; \backslash chi$


OR3

$(\backslash phi\; \backslash to\; \backslash psi)\; \backslash to\; ((\backslash chi\; \backslash to\; \backslash psi)\; \backslash to\; (\backslash phi\; \backslash lor\; \backslash chi\; \backslash to\; \backslash psi))$

Eliminate disjunction

NOT1

$(\backslash phi\; \backslash to\; \backslash chi)\; \backslash to\; ((\backslash phi\; \backslash to\; \backslash neg\; \backslash chi)\; \backslash to\; \backslash neg\; \backslash phi)$

Introduce negation

NOT2

$\backslash phi\; \backslash to\; (\backslash neg\; \backslash phi\; \backslash to\; \backslash chi)$

Eliminate negation

NOT3

$\backslash phi\; \backslash lor\; \backslash neg\; \backslash phi$

Excluded middle, classical logic

IFF1

$(\backslash phi\; \backslash leftrightarrow\; \backslash chi)\; \backslash to\; (\backslash phi\; \backslash to\; \backslash chi)$

Eliminate equivalence

IFF2

$(\backslash phi\; \backslash leftrightarrow\; \backslash chi)\; \backslash to\; (\backslash chi\; \backslash to\; \backslash phi)$


IFF3

$(\backslash phi\; \backslash to\; \backslash chi)\; \backslash to\; ((\backslash chi\; \backslash to\; \backslash phi)\; \backslash to\; (\backslash phi\; \backslash leftrightarrow\; \backslash chi))$

Introduce equivalence

 Axiom THEN2 may be considered to be a "distributive property of implication with respect to implication."
 Axioms AND1 and AND2 correspond to "conjunction elimination". The relation between AND1 and AND2 reflects the commutativity of the conjunction operator.
 Axiom AND3 corresponds to "conjunction introduction."
 Axioms OR1 and OR2 correspond to "disjunction introduction." The relation between OR1 and OR2 reflects the commutativity of the disjunction operator.
 Axiom NOT1 corresponds to "reductio ad absurdum."
 Axiom NOT2 says that "anything can be deduced from a contradiction."
 Axiom NOT3 is called "tertium non datur" (Latin: "a third is not given") and reflects the semantic valuation of propositional formulae: a formula can have a truthvalue of either true or false. There is no third truthvalue, at least not in classical logic. Intuitionistic logicians do not accept the axiom NOT3.
Inference rule
The inference rule is modus ponens:
 $\backslash phi,\; \backslash \; \backslash phi\; \backslash rightarrow\; \backslash chi\; \backslash vdash\; \backslash chi$.
Metainference rule
Let a demonstration be represented by a sequence, with hypotheses to the left of the turnstile and the conclusion to the right of the turnstile. Then the deduction theorem can be stated as follows:
 If the sequence
 $\backslash phi\_1,\; \backslash \; \backslash phi\_2,\; \backslash \; ...\; ,\; \backslash \; \backslash phi\_n,\; \backslash \; \backslash chi\; \backslash vdash\; \backslash psi$
 has been demonstrated, then it is also possible to demonstrate the sequence
 $\backslash phi\_1,\; \backslash \; \backslash phi\_2,\; \backslash \; ...,\; \backslash \; \backslash phi\_n\; \backslash vdash\; \backslash chi\; \backslash rightarrow\; \backslash psi$.
This deduction theorem (DT) is not itself formulated with propositional calculus: it is not a theorem of propositional calculus, but a theorem about propositional calculus. In this sense, it is a metatheorem, comparable to theorems about the soundness or completeness of propositional calculus.
On the other hand, DT is so useful for simplifying the syntactical proof process that it can be considered and used as another inference rule, accompanying modus ponens. In this sense, DT corresponds to the natural conditional proof inference rule which is part of the first version of propositional calculus introduced in this article.
The converse of DT is also valid:
 If the sequence
 $\backslash phi\_1,\; \backslash \; \backslash phi\_2,\; \backslash \; ...,\; \backslash \; \backslash phi\_n\; \backslash vdash\; \backslash chi\; \backslash rightarrow\; \backslash psi$
 has been demonstrated, then it is also possible to demonstrate the sequence
 $\backslash phi\_1,\; \backslash \; \backslash phi\_2,\; \backslash \; ...\; ,\; \backslash \; \backslash phi\_n,\; \backslash \; \backslash chi\; \backslash vdash\; \backslash psi$
in fact, the validity of the converse of DT is almost trivial compared to that of DT:
 If
 $\backslash phi\_1,\; \backslash \; ...\; ,\; \backslash \; \backslash phi\_n\; \backslash vdash\; \backslash chi\; \backslash rightarrow\; \backslash psi$
 then
 1: $\backslash phi\_1,\; \backslash \; ...\; ,\; \backslash \; \backslash phi\_n,\; \backslash \; \backslash chi\; \backslash vdash\; \backslash chi\; \backslash rightarrow\; \backslash psi$
 2: $\backslash phi\_1,\; \backslash \; ...\; ,\; \backslash \; \backslash phi\_n,\; \backslash \; \backslash chi\; \backslash vdash\; \backslash chi$
 and from (1) and (2) can be deduced
 3: $\backslash phi\_1,\; \backslash \; ...\; ,\; \backslash \; \backslash phi\_n,\; \backslash \; \backslash chi\; \backslash vdash\; \backslash psi$
 by means of modus ponens, Q.E.D.
The converse of DT has powerful implications: it can be used to convert an axiom into an inference rule. For example, the axiom AND1,
 $\backslash vdash\; \backslash phi\; \backslash wedge\; \backslash chi\; \backslash rightarrow\; \backslash phi$
can be transformed by means of the converse of the deduction theorem into the inference rule
 $\backslash phi\; \backslash wedge\; \backslash chi\; \backslash vdash\; \backslash phi$
which is conjunction elimination, one of the ten inference rules used in the first version (in this article) of the propositional calculus.
Example of a proof
The following is an example of a (syntactical) demonstration, involving only axioms THEN1 and THEN2:
Prove: $A\; \backslash rightarrow\; A$ (Reflexivity of implication).
Proof:
 $(A\; \backslash rightarrow\; ((B\; \backslash rightarrow\; A)\; \backslash rightarrow\; A))\; \backslash rightarrow\; ((A\; \backslash rightarrow\; (B\; \backslash rightarrow\; A))\; \backslash rightarrow\; (A\; \backslash rightarrow\; A))$
 Axiom THEN2 with $\backslash phi\; =\; A\backslash ,\backslash !$, $\backslash chi\; =\; B\; \backslash rightarrow\; A$, $\backslash psi\; =\; A$
 $A\; \backslash rightarrow\; ((B\; \backslash rightarrow\; A)\; \backslash rightarrow\; A)$
 Axiom THEN1 with $\backslash phi\; =\; A\backslash ,\backslash !$, $\backslash chi\; =\; B\; \backslash rightarrow\; A$
 $(A\; \backslash rightarrow\; (B\; \backslash rightarrow\; A))\; \backslash rightarrow\; (A\; \backslash rightarrow\; A)$
 From (1) and (2) by modus ponens.
 $A\; \backslash rightarrow\; (B\; \backslash rightarrow\; A)$
 Axiom THEN1 with $\backslash phi\; =\; A\backslash ,\backslash !$, $\backslash chi\; =\; B$
 $A\; \backslash rightarrow\; A$
 From (3) and (4) by modus ponens.
Equivalence to equational logics
The preceding alternative calculus is an example of a Hilbertstyle deduction system. In the case of propositional systems the axioms are terms built with logical connectives and the only inference rule is modus ponens. Equational logic as standardly used informally in high school algebra is a different kind of calculus from Hilbert systems. Its theorems are equations and its inference rules express the properties of equality, namely that it is a congruence on terms that admits substitution.
Classical propositional calculus as described above is equivalent to Boolean algebra, while intuitionistic propositional calculus is equivalent to Heyting algebra. The equivalence is shown by translation in each direction of the theorems of the respective systems. Theorems $\backslash phi\backslash ,\backslash !$ of classical or intuitionistic propositional calculus are translated as equations $\backslash phi\; =\; 1\backslash ,\backslash !$ of Boolean or Heyting algebra respectively. Conversely theorems $x\; =\; y$ of Boolean or Heyting algebra are translated as theorems $(x\; \backslash to\; y)\; \backslash land\; (y\; \backslash to\; x)$ of classical or propositional calculus respectively, for which $x\; \backslash equiv\; y$ is a standard abbreviation. In the case of Boolean algebra $x\; =\; y$ can also be translated as $(x\; \backslash land\; y)\; \backslash lor\; (\backslash neg\; x\; \backslash land\; \backslash neg\; y)$, but this translation is incorrect intuitionistically.
In both Boolean and Heyting algebra, inequality $x\; \backslash le\; y$ can be used in place of equality. The equality $x\; =\; y$ is expressible as a pair of inequalities $x\; \backslash le\; y$ and $y\; \backslash le\; x$. Conversely the inequality $x\; \backslash le\; y$ is expressible as the equality $x\; \backslash land\; y\; =\; x$, or as $x\; \backslash lor\; y\; =\; y$. The significance of inequality for Hilbertstyle systems is that it corresponds to the latter's deduction or entailment symbol $\backslash vdash$. An entailment
 $\backslash phi\_1,\; \backslash \; \backslash phi\_2,\; \backslash \; \backslash dots,\; \backslash \; \backslash phi\_n\; \backslash vdash\; \backslash psi$
is translated in the inequality version of the algebraic framework as
 $\backslash phi\_1\backslash \; \backslash land\backslash \; \backslash phi\_2\backslash \; \backslash land\backslash \; \backslash dots\backslash \; \backslash land\; \backslash \; \backslash phi\_n\backslash \; \backslash \; \backslash le\backslash \; \backslash \; \backslash psi$
Conversely the algebraic inequality $x\; \backslash le\; y$ is translated as the entailment
 $x\backslash \; \backslash vdash\backslash \; y$.
The difference between implication $x\; \backslash to\; y$ and inequality or entailment $x\; \backslash le\; y$ or $x\backslash \; \backslash vdash\backslash \; y$ is that the former is internal to the logic while the latter is external. Internal implication between two terms is another term of the same kind. Entailment as external implication between two terms expresses a metatruth outside the language of the logic, and is considered part of the metalanguage. Even when the logic under study is intuitionistic, entailment is ordinarily understood classically as twovalued: either the left side entails, or is lessorequal to, the right side, or it is not.
Similar but more complex translations to and from algebraic logics are possible for natural deduction systems as described above and for the sequent calculus. The entailments of the latter can be interpreted as twovalued, but a more insightful interpretation is as a set, the elements of which can be understood as abstract proofs organized as the morphisms of a category. In this interpretation the cut rule of the sequent calculus corresponds to composition in the category. Boolean and Heyting algebras enter this picture as special categories having at most one morphism per homset, i.e., one proof per entailment, corresponding to the idea that existence of proofs is all that matters: any proof will do and there is no point in distinguishing them.
Graphical calculi
It is possible to generalize the definition of a formal language from a set of finite sequences over a finite basis to include many other sets of mathematical structures, so long as they are built up by finitary means from finite materials. What's more, many of these families of formal structures are especially wellsuited for use in logic.
For example, there are many families of graphs that are close enough analogues of formal languages that the concept of a calculus is quite easily and naturally extended to them. Indeed, many species of graphs arise as parse graphs in the syntactic analysis of the corresponding families of text structures. The exigencies of practical computation on formal languages frequently demand that text strings be converted into pointer structure renditions of parse graphs, simply as a matter of checking whether strings are wffs or not. Once this is done, there are many advantages to be gained from developing the graphical analogue of the calculus on strings. The mapping from strings to parse graphs is called parsing and the inverse mapping from parse graphs to strings is achieved by an operation that is called traversing the graph.
Other logical calculi
Propositional calculus is about the simplest kind of logical calculus in current use. It can be extended in several ways. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler – but in other ways more complex – than propositional calculus.) The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more finegrained details of the sentences being used.
Firstorder logic (aka firstorder predicate logic) results when the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, all keeping the rules of propositional logic with some new ones introduced. (For example, from "All dogs are mammals" we may infer "If Rover is a dog then Rover is a mammal".) With the tools of firstorder logic it is possible to formulate a number of theories, either with explicit axioms or by rules of inference, that can themselves be treated as logical calculi. Arithmetic is the best known of these; others include set theory and mereology. Secondorder logic and other higherorder logics are formal extensions of firstorder logic. Thus, it makes sense to refer to propositional logic as "zerothorder logic", when comparing it with these logics.
Modal logic also offers a variety of inferences that cannot be captured in propositional calculus. For example, from "Necessarily $p$" we may infer that $p$. From $p$ we may infer "It is possible that $p$". The translation between modal logics and algebraic logics concerns classical and intuitionistic logics but with the introduction of a unary operator on Boolean or Heyting algebras, different from the Boolean operations, interpreting the possibility modality, and in the case of Heyting algebra a second operator interpreting necessity (for Boolean algebra this is redundant since necessity is the De Morgan dual of possibility). The first operator preserves 0 and disjunction while the second preserves 1 and conjunction.
Manyvalued logics are those allowing sentences to have values other than true and false. (For example, neither and both are standard "extra values"; "continuum logic" allows each sentence to have any of an infinite number of "degrees of truth" between true and false.) These logics often require calculational devices quite distinct from propositional calculus. When the values form a Boolean algebra (which may have more than two or even infinitely many values), manyvalued logic reduces to classical logic; manyvalued logics are therefore only of independent interest when the values form an algebra that is not Boolean.
Solvers
Finding solutions to propositional logic formulae is an NPcomplete 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.
See also
Higher logical levels
Related topics
References
Further reading
 Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY.
 Chang, C.C. and Keisler, H.J. (1973), Model Theory, NorthHolland, Amsterdam, Netherlands.
 Kohavi, Zvi (1978), Switching and Finite Automata Theory, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.
 Korfhage, Robert R. (1974), Discrete Computational Structures, Academic Press, New York, NY.
 Lambek, J. and Scott, P.J. (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press, Cambridge, UK.
 Mendelson, Elliot (1964), Introduction to Mathematical Logic, D. Van Nostrand Company.
Related works
External links
 Klement, Kevin C. (2006), "Propositional Logic", in James Fieser and Bradley Dowden (eds.), Eprint.
 Introduction to Mathematical Logic by V. Detlovs and K. Podnieks
 Alternative calculus
 proof theory for sentential logic.
 Propositional Logic on PlanetMath (GFDLed)
 GFDLed)


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