# Difference between revisions of "Propositional calculus"

Jon Awbrey (talk | contribs) |
Jon Awbrey (talk | contribs) (update) |
||

(18 intermediate revisions by the same user not shown) | |||

Line 1: | Line 1: | ||

− | + | <font size="3">☞</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. | |

− | + | A '''propositional calculus''' (or a '''sentential calculus''') is a formal system that represents the materials and the principles of ''propositional logic'' (or ''sentential logic''). Propositional logic is a domain of formal subject matter that is, up to somorphism, constituted by the structural relationships of mathematical objects called ''propositions''. | |

− | + | In general terms, a calculus is a formal system that consists of a set of syntactic expressions (''well-formed formulas'' or ''wffs''), a distinguished subset of these expressions, plus a set of transformation rules that define a binary relation on the space of expressions. | |

− | + | When the expressions are interpreted for mathematical purposes, the transformation rules are typically intended to preserve some type of semantic equivalence relation among the expressions. In particular, when the expressions are interpreted as a logical system, the semantic equivalence is typically intended to be logical equivalence. In this setting, the transformation rules can be used to derive logically equivalent expressions from any given expression. These derivations include as special cases (1) the problem of ''simplifying'' expressions and (2) the problem of deciding whether a given expression is equivalent to an expression in the distinguished subset, typically interpreted as the subset of logical ''axioms''. | |

− | The | + | The set of axioms may be empty, a nonempty finite set, a countably infinite set, or given by axiom schemata. A formal grammar recursively defines the expressions and well-formed formulas (wffs) of the language. In addition a semantics is given which defines truth and valuations (or interpretations). It allows us to determine which wffs are valid, that is, are theorems. |

+ | |||

+ | The language of a propositional calculus consists of (1) a set of primitive symbols, variously referred to as ''atomic formulas'', ''placeholders'', ''proposition letters'', or ''variables'', and (2) a set of operator symbols, variously interpreted as ''logical operators'' or ''logical connectives''. A ''well-formed formula'' (''wff'') is any atomic formula or any formula that can be built up from atomic formulas by means of operator symbols. | ||

The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their language, that is, the particular collection of primitive symbols and operator symbols, (2) the set of axioms, or distingushed formulas, and (3) the set of transformation rules that are available. | The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their language, that is, the particular collection of primitive symbols and operator symbols, (2) the set of axioms, or distingushed formulas, and (3) the set of transformation rules that are available. | ||

Line 19: | Line 21: | ||

==Generic description of a propositional calculus== | ==Generic description of a propositional calculus== | ||

− | A '''propositional calculus''' is a | + | A '''propositional calculus''' is a formal system <math>\mathcal{L} = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, whose formulas are constructed in the following manner: |

− | |||

− | |||

− | |||

− | |||

+ | * The ''alpha set'' <math>\Alpha\!</math> is a finite set of elements called ''proposition symbols'' or ''propositional variables''. Syntactically speaking, these are the most basic elements of the formal language <math>\mathcal{L},</math> otherwise referred to as ''atomic formulas'' or ''terminal elements''. In the examples to follow, the elements of <math>\Alpha\!</math> are typically the letters ''p'', ''q'', ''r'', and so on. | ||

− | + | * The ''omega set'' <math>\Omega\!</math> is a finite set of elements called ''operator symbols'' or ''logical connectives''. The set <math>\Omega\!</math> is partitioned into disjoint subsets as follows: | |

+ | ::: <math>\Omega = \Omega_0 \cup \Omega_1 \cup \ldots \cup \Omega_j \cup \ldots \cup \Omega_m.\!</math> | ||

− | : In this partition, <math>\Omega_j\!</math> is the set of operator symbols of '' | + | : In this partition, <math>\Omega_j\!</math> is the set of operator symbols of ''arity'' <math>j.\!</math> |

: In the more familiar propositional calculi, <math>\Omega\!</math> is typically partitioned as follows: | : In the more familiar propositional calculi, <math>\Omega\!</math> is typically partitioned as follows: | ||

− | ::: <math>\Omega_1 = \{ \lnot \} \ | + | ::: <math>\Omega_1 = \{ \lnot \},\!</math> |

− | ::: <math>\Omega_2 \subseteq \{ \land, \lor, \rightarrow, \leftrightarrow \} \ | + | ::: <math>\Omega_2 \subseteq \{ \land, \lor, \rightarrow, \leftrightarrow \}.\!</math> |

− | : A frequently adopted option treats the constant | + | : A frequently adopted option treats the constant logical values as operators of arity zero, thus: |

− | ::: <math>\Omega_0 = \{0, | + | ::: <math>\Omega_0 = \{ 0, 1 \}.\!</math> |

− | : Some writers use the | + | : Some writers use the tilde (~) instead of (¬) and some use the ampersand (&) instead of (∧). Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, {0, 1}, and {<math>\bot</math>, <math>\top</math>} all being seen in various contexts. |

* Depending on the precise formal grammar or the grammar formalism that is being used, syntactic auxiliaries like the left parenthesis, "(", and the right parentheses, ")", may be necessary to complete the construction of formulas. | * Depending on the precise formal grammar or the grammar formalism that is being used, syntactic auxiliaries like the left parenthesis, "(", and the right parentheses, ")", may be necessary to complete the construction of formulas. | ||

− | The ''language'' of <math>\mathcal{L}</math>, also known as its set of ''formulas'', '' | + | The ''language'' of <math>\mathcal{L}</math>, also known as its set of ''formulas'', ''well-formed formulas'' or ''wffs'', is inductively or recursively defined by the following rules: |

# Base. Any element of the alpha set <math>\Alpha\!</math> is a formula of <math>\mathcal{L}</math>. | # Base. Any element of the alpha set <math>\Alpha\!</math> is a formula of <math>\mathcal{L}</math>. | ||

Line 59: | Line 59: | ||

# By rule 3, (¬''p'' ∨ ''q'') is a formula. | # By rule 3, (¬''p'' ∨ ''q'') is a formula. | ||

− | * The ''zeta set'' <math>\Zeta\!</math> is a finite set of ''transformation rules'' that are called '' | + | * The ''zeta set'' <math>\Zeta\!</math> is a finite set of ''transformation rules'' that are called ''inference rules'' when they acquire logical applications. |

− | * The ''iota set'' <math>\Iota\!</math> is a finite set of ''initial points'' that are called '' | + | * The ''iota set'' <math>\Iota\!</math> is a finite set of ''initial points'' that are called ''axioms'' when they receive logical interpretations. |

− | ==Example 1. | + | ==Example 1. Simple axiom system== |

Let <math>\mathcal{L}_1 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, where <math>\Alpha,\ \Omega,\ \Zeta,\ \Iota</math> are defined as follows: | Let <math>\mathcal{L}_1 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, where <math>\Alpha,\ \Omega,\ \Zeta,\ \Iota</math> are defined as follows: | ||

Line 77: | Line 77: | ||

::: <math>\Omega_1 = \{ \lnot \} \,,</math> | ::: <math>\Omega_1 = \{ \lnot \} \,,</math> | ||

− | ::: <math>\Omega_2 = \{ \ | + | ::: <math>\Omega_2 = \{ \Rightarrow \} \,.</math> |

An axiom system discovered by [[Jan Łukasiewicz|Jan Lukasiewicz]] formulates a propositional calculus in this language as follows: | An axiom system discovered by [[Jan Łukasiewicz|Jan Lukasiewicz]] formulates a propositional calculus in this language as follows: | ||

− | ::* <math>p \ | + | ::* <math>p \Rightarrow (q \Rightarrow p)</math> |

− | ::* <math>(p \ | + | ::* <math>(p \Rightarrow (q \Rightarrow r)) \Rightarrow ((p \Rightarrow q) \Rightarrow (p \Rightarrow r))</math> |

− | ::* <math>(\neg p \ | + | ::* <math>(\neg p \Rightarrow \neg q) \Rightarrow (q \Rightarrow p)</math> |

− | The inference rule is [[modus ponens]] | + | The inference rule is ''[[modus ponens]]'': |

− | ==Example 2. | + | ::* From ''p'', (''p'' ⇒ ''q''), infer ''q''. |

+ | |||

+ | Then we have the following definitions: | ||

+ | |||

+ | ::* ''p'' ∨ ''q'' is defined as ¬''p'' ⇒ ''q''. | ||

+ | |||

+ | ::* ''p'' ∧ ''q'' is defined as ¬(''p'' ⇒ ¬''q''). | ||

+ | |||

+ | ==Example 2. Natural deduction system== | ||

Let <math>\mathcal{L}_2 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, where <math>\Alpha,\ \Omega,\ \Zeta,\ \Iota</math> are defined as follows: | Let <math>\mathcal{L}_2 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, where <math>\Alpha,\ \Omega,\ \Zeta,\ \Iota</math> are defined as follows: | ||

Line 109: | Line 117: | ||

* The set of transformation rules, <math>\Zeta ,\!</math>, is described as follows: | * The set of transformation rules, <math>\Zeta ,\!</math>, is described as follows: | ||

− | + | ==Graphical calculi== | |

− | + | : ''Main article'' : [[Logical graph]] | |

− | |||

− | |||

− | |||

− | : | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | 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 well-suited 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. | |

− | + | ==References== | |

− | |||

− | |||

− | |||

− | === | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | * Brown, Frank Markham (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003. | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | * Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, 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. | |

− | + | ==Resources== | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | * Klement, Kevin C. (2006), “Propositional Logic”, in James Fieser and Bradley Dowden (eds.), ''Internet Encyclopedia of Philosophy''. [http://www.iep.utm.edu/p/prop-log.htm Online]. | |

− | : | ||

− | |||

− | |||

− | |||

− | + | * Magnus, P.D. ''[http://www.fecundity.com/logic/ Forall x : An Introduction to Formal Logic]''. | |

− | + | * [http://www.visualstatistics.net/Scaling/Propositional%20Calculus/Elements%20of%20Propositional%20Calculus.htm Elements of Propositional Calculus]. | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | * [http://www.ltn.lv/~podnieks/mlog/ml2.htm Introduction to Mathematical Logic]. | |

− | |||

− | + | ==Syllabus== | |

− | + | ===Focal nodes=== | |

− | + | * [[Inquiry Live]] | |

+ | * [[Logic Live]] | ||

− | + | ===Peer nodes=== | |

− | + | * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_calculus Propositional Calculus @ InterSciWiki] | |

+ | * [http://mywikibiz.com/Propositional_calculus Propositional Calculus @ MyWikiBiz] | ||

+ | * [http://ref.subwiki.org/wiki/Propositional_calculus Propositional Calculus @ Subject Wikis] | ||

+ | * [http://en.wikiversity.org/wiki/Propositional_calculus Propositional Calculus @ Wikiversity] | ||

+ | * [http://beta.wikiversity.org/wiki/Propositional_calculus Propositional Calculus @ Wikiversity Beta] | ||

− | + | ===Logical operators=== | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

{{col-begin}} | {{col-begin}} | ||

{{col-break}} | {{col-break}} | ||

Line 532: | Line 176: | ||

* [[Logical NAND]] | * [[Logical NAND]] | ||

* [[Logical NNOR]] | * [[Logical NNOR]] | ||

− | * [[Negation]] | + | * [[Logical negation|Negation]] |

{{col-end}} | {{col-end}} | ||

===Related topics=== | ===Related topics=== | ||

+ | |||

{{col-begin}} | {{col-begin}} | ||

{{col-break}} | {{col-break}} | ||

* [[Ampheck]] | * [[Ampheck]] | ||

− | |||

− | |||

* [[Boolean domain]] | * [[Boolean domain]] | ||

* [[Boolean function]] | * [[Boolean function]] | ||

− | |||

* [[Boolean-valued function]] | * [[Boolean-valued function]] | ||

− | * [[ | + | * [[Differential logic]] |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

{{col-break}} | {{col-break}} | ||

* [[Logical graph]] | * [[Logical graph]] | ||

− | |||

* [[Minimal negation operator]] | * [[Minimal negation operator]] | ||

* [[Multigrade operator]] | * [[Multigrade operator]] | ||

− | |||

* [[Parametric operator]] | * [[Parametric operator]] | ||

* [[Peirce's law]] | * [[Peirce's law]] | ||

− | * [[ | + | {{col-break}} |

+ | * [[Propositional calculus]] | ||

+ | * [[Sole sufficient operator]] | ||

* [[Truth table]] | * [[Truth table]] | ||

+ | * [[Universe of discourse]] | ||

+ | * [[Zeroth order logic]] | ||

{{col-end}} | {{col-end}} | ||

− | === | + | ===Relational concepts=== |

− | * [[ | + | {{col-begin}} |

+ | {{col-break}} | ||

+ | * [[Continuous predicate]] | ||

+ | * [[Hypostatic abstraction]] | ||

+ | * [[Logic of relatives]] | ||

+ | * [[Logical matrix]] | ||

+ | {{col-break}} | ||

+ | * [[Relation (mathematics)|Relation]] | ||

+ | * [[Relation composition]] | ||

+ | * [[Relation construction]] | ||

+ | * [[Relation reduction]] | ||

+ | {{col-break}} | ||

+ | * [[Relation theory]] | ||

+ | * [[Relative term]] | ||

+ | * [[Sign relation]] | ||

+ | * [[Triadic relation]] | ||

+ | {{col-end}} | ||

− | == | + | ===Information, Inquiry=== |

− | * [[ | + | {{col-begin}} |

+ | {{col-break}} | ||

+ | * [[Inquiry]] | ||

+ | * [[Dynamics of inquiry]] | ||

+ | {{col-break}} | ||

+ | * [[Semeiotic]] | ||

+ | * [[Logic of information]] | ||

+ | {{col-break}} | ||

+ | * [[Descriptive science]] | ||

+ | * [[Normative science]] | ||

+ | {{col-break}} | ||

+ | * [[Pragmatic maxim]] | ||

+ | * [[Truth theory]] | ||

+ | {{col-end}} | ||

− | + | ===Related articles=== | |

− | * [http:// | + | {{col-begin}} |

− | + | {{col-break}} | |

− | * | + | * [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language] |

+ | * [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs] | ||

+ | * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] | ||

+ | {{col-break}} | ||

+ | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] | ||

+ | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] | ||

+ | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] | ||

+ | {{col-break}} | ||

+ | * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] | ||

+ | * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] | ||

+ | * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] | ||

+ | {{col-end}} | ||

==Document history== | ==Document history== | ||

− | + | Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders. | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | * [http://intersci.ss.uci.edu/wiki/index.php/Propositional_calculus Propositional Calculus], [http://intersci.ss.uci.edu/wiki/index.php/Main_Page InterSciWiki] | |

+ | * [http://mywikibiz.com/Propositional_calculus Propositional Calculus], [http://mywikibiz.com/ MyWikiBiz] | ||

+ | * [http://planetmath.org/PropositionalCalculus Propositional Calculus], [http://planetmath.org/ PlanetMath] | ||

+ | * [http://wikinfo.org/w/index.php/Propositional_calculus Propositional Calculus], [http://wikinfo.org/w/ Wikinfo] | ||

+ | * [http://en.wikiversity.org/wiki/Propositional_calculus Propositional Calculus], [http://en.wikiversity.org/ Wikiversity] | ||

+ | * [http://beta.wikiversity.org/wiki/Propositional_calculus Propositional Calculus], [http://beta.wikiversity.org/ Wikiversity Beta] | ||

+ | * [http://en.wikipedia.org/w/index.php?title=Propositional_calculus&oldid=77110794 Propositional Calculus], [http://en.wikipedia.org/ Wikipedia] | ||

+ | [[Category:Charles Sanders Peirce]] | ||

[[Category:Computer Science]] | [[Category:Computer Science]] | ||

+ | [[Category:Formal Grammars]] | ||

[[Category:Formal Languages]] | [[Category:Formal Languages]] | ||

[[Category:Formal Sciences]] | [[Category:Formal Sciences]] | ||

[[Category:Formal Systems]] | [[Category:Formal Systems]] | ||

+ | [[Category:Inquiry]] | ||

+ | [[Category:Linguistics]] | ||

[[Category:Logic]] | [[Category:Logic]] | ||

[[Category:Mathematics]] | [[Category:Mathematics]] | ||

+ | [[Category:Model Theory]] | ||

[[Category:Normative Sciences]] | [[Category:Normative Sciences]] | ||

+ | [[Category:Pragmatics]] | ||

+ | [[Category:Proof Theory]] | ||

+ | [[Category:Semantics]] | ||

+ | [[Category:Semiotics]] | ||

+ | [[Category:Syntax]] |

## Latest revision as of 15:56, 7 November 2015

☞ This page belongs to resource collections on Logic and Inquiry.

A **propositional calculus** (or a **sentential calculus**) is a formal system that represents the materials and the principles of *propositional logic* (or *sentential logic*). Propositional logic is a domain of formal subject matter that is, up to somorphism, constituted by the structural relationships of mathematical objects called *propositions*.

In general terms, a calculus is a formal system that consists of a set of syntactic expressions (*well-formed formulas* or *wffs*), a distinguished subset of these expressions, plus a set of transformation rules that define a binary relation on the space of expressions.

When the expressions are interpreted for mathematical purposes, the transformation rules are typically intended to preserve some type of semantic equivalence relation among the expressions. In particular, when the expressions are interpreted as a logical system, the semantic equivalence is typically intended to be logical equivalence. In this setting, the transformation rules can be used to derive logically equivalent expressions from any given expression. These derivations include as special cases (1) the problem of *simplifying* expressions and (2) the problem of deciding whether a given expression is equivalent to an expression in the distinguished subset, typically interpreted as the subset of logical *axioms*.

The set of axioms may be empty, a nonempty finite set, a countably infinite set, or given by axiom schemata. A formal grammar recursively defines the expressions and well-formed formulas (wffs) of the language. In addition a semantics is given which defines truth and valuations (or interpretations). It allows us to determine which wffs are valid, that is, are theorems.

The language of a propositional calculus consists of (1) a set of primitive symbols, variously referred to as *atomic formulas*, *placeholders*, *proposition letters*, or *variables*, and (2) a set of operator symbols, variously interpreted as *logical operators* or *logical connectives*. A *well-formed formula* (*wff*) is any atomic formula or any formula that can be built up from atomic formulas by means of operator symbols.

The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their language, that is, the particular collection of primitive symbols and operator symbols, (2) the set of axioms, or distingushed formulas, and (3) the set of transformation rules that are available.

## Abstraction and application

Although it is possible to construct an abstract formal calculus that has no immediate practical use and next to nothing in the way of obvious applications, the very name *calculus* indicates that this species of formal system owes its origin to the utility of its prototypical members in practical calculation. Generally speaking, any mathematical calculus is designed with the intention of representing a given domain of formal objects, and typically with the aim of facilitating the computations and inferences that need to be carried out in this representation. Thus some idea of the intended denotation, the formal objects that the formulas of the calculus are intended to denote, is given in advance of developing the calculus itself.

Viewed over the course of its historical development, a formal calculus for any given subject matter normally arises through a process of gradual abstraction, stepwise refinement, and trial-and-error synthesis from the array of informal notational systems that inform prior use, each of which covers the object domain only in part or from a particular angle.

## Generic description of a propositional calculus

A **propositional calculus** is a formal system \(\mathcal{L} = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)\), whose formulas are constructed in the following manner:

- The
*alpha set*\(\Alpha\!\) is a finite set of elements called*proposition symbols*or*propositional variables*. Syntactically speaking, these are the most basic elements of the formal language \(\mathcal{L},\) otherwise referred to as*atomic formulas*or*terminal elements*. In the examples to follow, the elements of \(\Alpha\!\) are typically the letters*p*,*q*,*r*, and so on.

- The
*omega set*\(\Omega\!\) is a finite set of elements called*operator symbols*or*logical connectives*. The set \(\Omega\!\) is partitioned into disjoint subsets as follows:

- \[\Omega = \Omega_0 \cup \Omega_1 \cup \ldots \cup \Omega_j \cup \ldots \cup \Omega_m.\!\]

- In this partition, \(\Omega_j\!\) is the set of operator symbols of
*arity*\(j.\!\)

- In the more familiar propositional calculi, \(\Omega\!\) is typically partitioned as follows:

- \[\Omega_1 = \{ \lnot \},\!\]

- \[\Omega_2 \subseteq \{ \land, \lor, \rightarrow, \leftrightarrow \}.\!\]

- A frequently adopted option treats the constant logical values as operators of arity zero, thus:

- \[\Omega_0 = \{ 0, 1 \}.\!\]

- Some writers use the tilde (~) instead of (¬) and some use the ampersand (&) instead of (∧). Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, {0, 1}, and {\(\bot\), \(\top\)} all being seen in various contexts.

- Depending on the precise formal grammar or the grammar formalism that is being used, syntactic auxiliaries like the left parenthesis, "(", and the right parentheses, ")", may be necessary to complete the construction of formulas.

The *language* of \(\mathcal{L}\), also known as its set of *formulas*, *well-formed formulas* or *wffs*, is inductively or recursively defined by the following rules:

- Base. Any element of the alpha set \(\Alpha\!\) is a formula of \(\mathcal{L}\).
- Step (a). If
*p*is a formula, then ¬*p*is a formula. - Step (b). If
*p*and*q*are formulas, then (*p*∧*q*), (*p*∨*q*), (*p*→*q*), and (*p*↔*q*) are formulas. - Close. Nothing else is a formula of \(\mathcal{L}\).

Repeated applications of these rules permits the construction of complex formulas. For example:

- By rule 1,
*p*is a formula. - By rule 2, ¬
*p*is a formula. - By rule 1,
*q*is a formula. - By rule 3, (¬
*p*∨*q*) is a formula.

- The
*zeta set*\(\Zeta\!\) is a finite set of*transformation rules*that are called*inference rules*when they acquire logical applications.

- The
*iota set*\(\Iota\!\) is a finite set of*initial points*that are called*axioms*when they receive logical interpretations.

## Example 1. Simple axiom system

Let \(\mathcal{L}_1 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)\), where \(\Alpha,\ \Omega,\ \Zeta,\ \Iota\) are defined as follows:

- The alpha set \(\Alpha \!\) is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

- \[\Alpha = \{p, q, r, s, t, u \} \,.\]

Of the three connectives for conjunction, disjunction, and implication (∧, ∨, and →), one can be taken as primitive and the other two can be defined in terms of it and negation (¬). Indeed, all of the logical connectives can be defined in terms of a sole sufficient operator. The biconditional (↔) can of course be defined in terms of conjunction and implication, with a ↔ b defined as (a → b) ∧ (b → a).

Adopting negation and implication as the two primitive operations of a propositional calculus is tantamount to having the omega set \(\Omega = \Omega_1 \cup \Omega_2\) partition as follows:

- \[\Omega_1 = \{ \lnot \} \,,\]

- \[\Omega_2 = \{ \Rightarrow \} \,.\]

An axiom system discovered by Jan Lukasiewicz formulates a propositional calculus in this language as follows:

- \(p \Rightarrow (q \Rightarrow p)\)

- \((p \Rightarrow (q \Rightarrow r)) \Rightarrow ((p \Rightarrow q) \Rightarrow (p \Rightarrow r))\)

- \((\neg p \Rightarrow \neg q) \Rightarrow (q \Rightarrow p)\)

The inference rule is *modus ponens*:

- From
*p*, (*p*⇒*q*), infer*q*.

- From

Then we have the following definitions:

*p*∨*q*is defined as ¬*p*⇒*q*.

*p*∧*q*is defined as ¬(*p*⇒ ¬*q*).

## Example 2. Natural deduction system

Let \(\mathcal{L}_2 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)\), where \(\Alpha,\ \Omega,\ \Zeta,\ \Iota\) are defined as follows:

- The alpha set \(\Alpha \!\) is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

- \[\Alpha = \{p, q, r, s, t, u \} \,.\]

- The omega set \(\Omega = \Omega_1 \cup \Omega_2\) partitions as follows:

- \[\Omega_1 = \{ \lnot \} \,,\]

- \[\Omega_2 = \{ \land, \lor, \rightarrow, \leftrightarrow \} \,.\]

In the following example of a propositional calculus,
the transformation rules are intended to be interpreted as the inference rules of a so-called *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, \(\Iota = \varnothing \,.\)

- The set of transformation rules, \(\Zeta ,\!\), is described as follows:

## Graphical calculi

*Main article*: Logical graph

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 well-suited 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.

## References

- Brown, Frank Markham (2003),
*Boolean Reasoning: The Logic of Boolean Equations*, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

- Chang, C.C., and Keisler, H.J. (1973),
*Model Theory*, North-Holland, 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.

## Resources

- Klement, Kevin C. (2006), “Propositional Logic”, in James Fieser and Bradley Dowden (eds.),
*Internet Encyclopedia of Philosophy*. Online.

- Magnus, P.D.
*Forall x : An Introduction to Formal Logic*.

## Syllabus

### Focal nodes

### Peer nodes

- Propositional Calculus @ InterSciWiki
- Propositional Calculus @ MyWikiBiz
- Propositional Calculus @ Subject Wikis
- Propositional Calculus @ Wikiversity
- Propositional Calculus @ Wikiversity Beta

### Logical operators

Template:Col-breakTemplate:Col-breakTemplate:Col-end### Related topics

- Propositional calculus
- Sole sufficient operator
- Truth table
- Universe of discourse
- Zeroth order logic

### Relational concepts

### Information, Inquiry

### Related articles

- Differential Logic : Introduction
- Differential Propositional Calculus
- Differential Logic and Dynamic Systems

- Prospects for Inquiry Driven Systems
- Introduction to Inquiry Driven Systems
- Inquiry Driven Systems : Inquiry Into Inquiry

## Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

- Propositional Calculus, InterSciWiki
- Propositional Calculus, MyWikiBiz
- Propositional Calculus, PlanetMath
- Propositional Calculus, Wikinfo
- Propositional Calculus, Wikiversity
- Propositional Calculus, Wikiversity Beta
- Propositional Calculus, Wikipedia