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74-469: Any problem that can be expressed as a set of well-formed formulas (WFFs) or Horn clauses , and that constitutes a directed graph with one or more sources (that is, hypotheses ) and sinks (that is, desired conclusions), can be solved, in principle, by GPS. Proofs in the predicate logic and Euclidean geometry problem spaces are prime examples of the domain of applicability of GPS. It was based on Simon and Newell's theoretical work on logic machines. GPS

148-735: A {\displaystyle a} is assigned T and b {\displaystyle b} is assigned F , or a {\displaystyle a} is assigned F and b {\displaystyle b} is assigned T . Since L {\displaystyle {\mathcal {L}}} has ℵ 0 {\displaystyle \aleph _{0}} , that is, denumerably many propositional symbols, there are 2 ℵ 0 = c {\displaystyle 2^{\aleph _{0}}={\mathfrak {c}}} , and therefore uncountably many distinct possible interpretations of L {\displaystyle {\mathcal {L}}} as

222-575: A {\displaystyle a} , for example, there are 2 1 = 2 {\displaystyle 2^{1}=2} possible interpretations: either a {\displaystyle a} is assigned T , or a {\displaystyle a} is assigned F . And for the pair a {\displaystyle a} , b {\displaystyle b} there are 2 2 = 4 {\displaystyle 2^{2}=4} possible interpretations: either both are assigned T , or both are assigned F , or

296-547: A well-formed formula , abbreviated WFF or wff , often simply formula , is a finite sequence of symbols from a given alphabet that is part of a formal language . The abbreviation wff is pronounced "woof", or sometimes "wiff", "weff", or "whiff". A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation . Two key uses of formulas are in propositional logic and predicate logic. A key use of formulas

370-429: A broader category that includes logical connectives. Sentential connectives are any linguistic particles that bind sentences to create a new compound sentence, or that inflect a single sentence to create a new sentence. A logical connective , or propositional connective , is a kind of sentential connective with the characteristic feature that, when the original sentences it operates on are (or express) propositions ,

444-429: A counterexample , where a counterexample is defined as a case I {\displaystyle {\mathcal {I}}} in which the argument's premises { φ 1 , φ 2 , φ 3 , . . . , φ n } {\displaystyle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\}} are all true but

518-608: A formula comes in several parts. First, the set of terms is defined recursively. Terms, informally, are expressions that represent objects from the domain of discourse . The next step is to define the atomic formulas . Finally, the set of formulas is defined to be the smallest set containing the set of atomic formulas such that the following holds: If a formula has no occurrences of ∃ x {\displaystyle \exists x} or ∀ x {\displaystyle \forall x} , for any variable x {\displaystyle x} , then it

592-450: A formula of propositional logic is defined recursively by these definitions: Writing the result of applying c n m {\displaystyle c_{n}^{m}} to ⟨ {\displaystyle \langle } A, B, C, … ⟩ {\displaystyle \rangle } in functional notation, as c n m {\displaystyle c_{n}^{m}} (A, B, C, …), we have

666-402: A formula. The formulas of propositional calculus , also called propositional formulas , are expressions such as ( A ∧ ( B ∨ C ) ) {\displaystyle (A\land (B\lor C))} . Their definition begins with the arbitrary choice of a set V of propositional variables . The alphabet consists of the letters in V along with the symbols for

740-473: A line, called the inference line , separated by a comma , which indicates combination of premises. The conclusion is written below the inference line. The inference line represents syntactic consequence , sometimes called deductive consequence , which is also symbolized with ⊢. So the above can also be written in one line as P → Q , P ⊢ Q {\displaystyle P\to Q,P\vdash Q} . Syntactic consequence

814-438: A new era in logic's history; however, advances in propositional logic were still made after Frege, including natural deduction , truth trees and truth tables . Natural deduction was invented by Gerhard Gentzen and Stanisław Jaśkowski . Truth trees were invented by Evert Willem Beth . The invention of truth tables, however, is of uncertain attribution. Within works by Frege and Bertrand Russell , are ideas influential to

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888-628: A particularly brief one, for the common set of five connectives, is this single clause: This clause, due to its self-referential nature (since ϕ {\displaystyle \phi } is in some branches of the definition of ϕ {\displaystyle \phi } ), also acts as a recursive definition , and therefore specifies the entire language. To expand it to add modal operators , one need only add …  |   ◻ ϕ   |   ◊ ϕ {\displaystyle |~\Box \phi ~|~\Diamond \phi } to

962-742: A set of propositional connectives c 1 1 {\displaystyle c_{1}^{1}} , c 2 1 {\displaystyle c_{2}^{1}} , c 3 1 {\displaystyle c_{3}^{1}} , ..., c 1 2 {\displaystyle c_{1}^{2}} , c 2 2 {\displaystyle c_{2}^{2}} , c 3 2 {\displaystyle c_{3}^{2}} , ..., c 1 3 {\displaystyle c_{1}^{3}} , c 2 3 {\displaystyle c_{2}^{3}} , c 3 3 {\displaystyle c_{3}^{3}} , ...,

1036-449: A value ( excluded middle ), are distinctive features of classical logic. To learn about nonclassical logics with more than two truth-values, and their unique semantics, one may consult the articles on " Many-valued logic ", " Three-valued logic ", " Finite-valued logic ", and " Infinite-valued logic ". For a given language L {\displaystyle {\mathcal {L}}} , an interpretation , valuation , or case ,

1110-904: A whole. Where I {\displaystyle {\mathcal {I}}} is an interpretation and φ {\displaystyle \varphi } and ψ {\displaystyle \psi } represent formulas, the definition of an argument , given in § Arguments , may then be stated as a pair ⟨ { φ 1 , φ 2 , φ 3 , . . . , φ n } , ψ ⟩ {\displaystyle \langle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\},\psi \rangle } , where { φ 1 , φ 2 , φ 3 , . . . , φ n } {\displaystyle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\}}

1184-444: Is Misplaced Pages?", and imperative statements, such as "Please add citations to support the claims in this article.". Such non-declarative sentences have no truth value , and are only dealt with in nonclassical logics , called erotetic and imperative logics . In propositional logic, a statement can contain one or more other statements as parts. Compound sentences are formed from simpler sentences and express relationships among

1258-412: Is a classically valid form. So, in classical logic, the argument is valid , although it may or may not be sound , depending on the meteorological facts in a given context. This example argument will be reused when explaining § Formalization . An argument is valid if, and only if, it is necessary that, if all its premises are true, its conclusion is true. Alternatively, an argument

1332-463: Is a free online encyclopedia that anyone can edit" evaluates to True , while "Misplaced Pages is a paper encyclopedia " evaluates to False . In other respects, the following formal semantics can apply to the language of any propositional logic, but the assumptions that there are only two semantic values ( bivalence ), that only one of the two is assigned to each formula in the language ( noncontradiction ), and that every formula gets assigned

1406-407: Is a logical consequence of its premises, which, when this is understood as semantic consequence , means that there is no case in which the premises are true but the conclusion is not true – see § Semantics below. Propositional logic is typically studied through a formal system in which formulas of a formal language are interpreted to represent propositions . This formal language

1480-436: Is a branch of logic . It is also called (first-order) propositional logic , statement logic , sentential calculus , sentential logic , or sometimes zeroth-order logic . It deals with propositions (which can be true or false ) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing

1554-435: Is a formula in which there are no free occurrences of any variable . If A is a formula of a first-order language in which the variables v 1 , …, v n have free occurrences, then A preceded by ∀ v 1 ⋯ ∀ v n is a universal closure of A . In earlier works on mathematical logic (e.g. by Church ), formulas referred to any strings of symbols and among these strings, well-formed formulas were

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1628-454: Is a formula, because it is grammatically correct. The sequence of symbols is not a formula, because it does not conform to the grammar. A complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules (akin to the standard mathematical order of operations ) are assumed among the operators, making some operators more binding than others. For example, assuming

1702-439: Is an assignment of semantic values to each formula of L {\displaystyle {\mathcal {L}}} . For a formal language of classical logic, a case is defined as an assignment , to each formula of L {\displaystyle {\mathcal {L}}} , of one or the other, but not both, of the truth values , namely truth ( T , or 1) and falsity ( F , or 0). An interpretation that follows

1776-440: Is called quantifier-free . An existential formula is a formula starting with a sequence of existential quantification followed by a quantifier-free formula. An atomic formula is a formula that contains no logical connectives nor quantifiers , or equivalently a formula that has no strict subformulas. The precise form of atomic formulas depends on the formal system under consideration; for propositional logic , for example,

1850-418: Is common to represent propositional constants by A , B , and C , propositional variables by P , Q , and R , and schematic letters are often Greek letters, most often φ , ψ , and χ . However, some authors recognize only two "propositional constants" in their formal system: the special symbol ⊤ {\displaystyle \top } , called "truth", which always evaluates to True , and

1924-414: Is contrasted with semantic consequence , which is symbolized with ⊧. In this case, the conclusion follows syntactically because the natural deduction inference rule of modus ponens has been assumed. For more on inference rules, see the sections on proof systems below. The language (commonly called L {\displaystyle {\mathcal {L}}} ) of a propositional calculus

1998-404: Is defined as a pair of things, namely a set of sentences, called the premises , and a sentence, called the conclusion . The conclusion is claimed to follow from the premises, and the premises are claimed to support the conclusion. The following is an example of an argument within the scope of propositional logic: The logical form of this argument is known as modus ponens , which

2072-473: Is defined in terms of: A well-formed formula is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar. The language L {\displaystyle {\mathcal {L}}} , then, is defined either as being identical to its set of well-formed formulas, or as containing that set (together with, for instance, its set of connectives and variables). Usually

2146-656: Is far from clear that any one person should be given the title of 'inventor' of truth-tables". Propositional logic, as currently studied in universities, is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered in evaluating the conditions for the truth of a sentence, or whether a sentence logically follows from some other sentence or group of sentences. Propositional logic deals with statements , which are defined as declarative sentences having truth value. Examples of statements might include: Declarative sentences are contrasted with questions , such as "What

2220-403: Is in propositional logic and predicate logic such as first-order logic . In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven. Although

2294-625: Is interpreted as "It's raining" and Q as "it's cloudy" these symbolic expressions correspond exactly with the original expression in natural language. Not only that, but they will also correspond with any other inference with the same logical form . When a formal system is used to represent formal logic, only statement letters (usually capital roman letters such as P {\displaystyle P} , Q {\displaystyle Q} and R {\displaystyle R} ) are represented directly. The natural language propositions that arise when they're interpreted are outside

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2368-439: Is not concerned with the structure of propositions beyond the point where they cannot be decomposed any more by logical connectives, it is typically studied by replacing such atomic (indivisible) statements with letters of the alphabet, which are interpreted as variables representing statements ( propositional variables ). With propositional variables, the § Example argument would then be symbolized as follows: When P

2442-519: Is not specifically required by the other definitions in the syntax. In particular, it excludes infinitely long formulas from being well-formed . An alternative to the syntax definitions given above is to write a context-free (CF) grammar for the language L {\displaystyle {\mathcal {L}}} in Backus-Naur form (BNF). This is more common in computer science than in philosophy . It can be done in many ways, of which

2516-423: Is the basis for proof systems , which allow a conclusion to be derived from premises if, and only if, it is a logical consequence of them. This section will show how this works by formalizing the § Example argument . The formal language for a propositional calculus will be fully specified in § Language , and an overview of proof systems will be given in § Proof systems . Since propositional logic

2590-437: Is the foundation of first-order logic and higher-order logic. Propositional logic is typically studied with a formal language , in which propositions are represented by letters, which are called propositional variables . These are then used, together with symbols for connectives, to make compound propositions. Because of this, the propositional variables are called atomic formulas of a formal zeroth-order language. While

2664-436: Is the interpretation function for M {\displaystyle {\mathfrak {M}}} . Some of these connectives may be defined in terms of others: for instance, implication, p → q, may be defined in terms of disjunction and negation, as ¬p ∨ q; and disjunction may be defined in terms of negation and conjunction, as ¬(¬p ∧ ¬q). In fact, a truth-functionally complete system, in

2738-456: Is the interpretation of φ {\displaystyle \varphi } , the five connectives are defined as: Instead of I ( φ ) {\displaystyle {\mathcal {I}}(\varphi )} , the interpretation of φ {\displaystyle \varphi } may be written out as | φ | {\displaystyle |\varphi |} , or, for definitions such as

2812-496: Is the set of premises and ψ {\displaystyle \psi } is the conclusion. The definition of an argument's validity , i.e. its property that { φ 1 , φ 2 , φ 3 , . . . , φ n } ⊨ ψ {\displaystyle \{\varphi _{1},\varphi _{2},\varphi _{3},...,\varphi _{n}\}\models \psi } , can then be stated as its absence of

2886-405: Is valid if, and only if, it is impossible for all the premises to be true while the conclusion is false. Validity is contrasted with soundness . An argument is sound if, and only if, it is valid and all its premises are true. Otherwise, it is unsound . Logic, in general, aims to precisely specify valid arguments. This is done by defining a valid argument as one in which its conclusion

2960-471: The propositional connectives and parentheses "(" and ")", all of which are assumed to not be in V . The formulas will be certain expressions (that is, strings of symbols) over this alphabet. The formulas are inductively defined as follows: This definition can also be written as a formal grammar in Backus–Naur form , provided the set of variables is finite: Using this grammar, the sequence of symbols

3034-443: The truth functions of conjunction , disjunction , implication , biconditional , and negation . Some sources include other connectives, as in the table below. Unlike first-order logic , propositional logic does not deal with non-logical objects, predicates about them, or quantifiers . However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic

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3108-480: The truth values that they take when the propositional variables that they're applied to take either of the two possible truth values, the semantic definition of the connectives is usually represented as a truth table for each of the connectives, as seen below: This table covers each of the main five logical connectives : conjunction (here notated p ∧ q), disjunction (p ∨ q), implication (p → q), biconditional (p ↔ q) and negation , (¬p, or ¬q, as

3182-511: The 3rd century BC and expanded by his successor Stoics . The logic was focused on propositions . This was different from the traditional syllogistic logic , which focused on terms . However, most of the original writings were lost and, at some time between the 3rd and 6th century CE, Stoic logic faded into oblivion, to be resurrected only in the 20th century, in the wake of the (re)-discovery of propositional logic. Symbolic logic , which would come to be important to refine propositional logic,

3256-522: The above, I ( φ ) = T {\displaystyle {\mathcal {I}}(\varphi )={\mathsf {T}}} may be written simply as the English sentence " φ {\displaystyle \varphi } is given the value T {\displaystyle {\mathsf {T}}} ". Yet other authors may prefer to speak of a Tarskian model M {\displaystyle {\mathfrak {M}}} for

3330-450: The atomic formulas are the propositional variables . For predicate logic , the atoms are predicate symbols together with their arguments, each argument being a term . According to some terminology, an open formula is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers. This is not to be confused with a formula which is not closed. A closed formula , also ground formula or sentence ,

3404-464: The atomic propositions are typically represented by letters of the alphabet , there is a variety of notations to represent the logical connectives. The following table shows the main notational variants for each of the connectives in propositional logic. The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic , in which formulas are interpreted as having precisely one of two possible truth values ,

3478-401: The atoms as ultimate building blocks. Composite formulas (all formulas besides atoms) are called molecules , or molecular sentences . (This is an imperfect analogy with chemistry , since a chemical molecule may sometimes have only one atom, as in monatomic gases .) The definition that "nothing else is a formula", given above as Definition 3 , excludes any formula from the language which

3552-405: The biconditional. (As to equivalence, Howson calls it "truth-functional equivalence", while Cunningham calls it "logical equivalence".) Equivalence is symbolized with ⇔ and is a metalanguage symbol, while a biconditional is symbolized with ↔ and is a logical connective in the object language L {\displaystyle {\mathcal {L}}} . Regardless, an equivalence or biconditional

3626-451: The case may be). It is sufficient for determining the semantics of each of these operators. For more truth tables for more different kinds of connectives, see the article " Truth table ". Some authors (viz., all the authors cited in this subsection) write out the connective semantics using a list of statements instead of a table. In this format, where I ( φ ) {\displaystyle {\mathcal {I}}(\varphi )}

3700-422: The conclusion ψ {\displaystyle \psi } is not true. As will be seen in § Semantic truth, validity, consequence , this is the same as to say that the conclusion is a semantic consequence of the premises. An interpretation assigns semantic values to atomic formulas directly. Molecular formulas are assigned a function of the value of their constituent atoms, according to

3774-457: The concrete string representation of formulas (using this or that symbol for connectives and quantifiers, using this or that parenthesizing convention , using Polish or infix notation, etc.) as a mere notational problem. The expression "well-formed formulas" (WFF) also crept into popular culture. WFF is part of an esoteric pun used in the name of the academic game " WFF 'N PROOF : The Game of Modern Logic", by Layman Allen, developed while he

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3848-465: The connective used; the connectives are defined in such a way that the truth-value of a sentence formed from atoms with connectives depends on the truth-values of the atoms that they're applied to, and only on those. This assumption is referred to by Colin Howson as the assumption of the truth-functionality of the connectives . Since logical connectives are defined semantically only in terms of

3922-796: The constituent sentences. This is done by combining them with logical connectives : the main types of compound sentences are negations , conjunctions , disjunctions , implications , and biconditionals , which are formed by using the corresponding connectives to connect propositions. In English , these connectives are expressed by the words "and" ( conjunction ), "or" ( disjunction ), "not" ( negation ), "if" ( material conditional ), and "if and only if" ( biconditional ). Examples of such compound sentences might include: If sentences lack any logical connectives, they are called simple sentences , or atomic sentences ; if they contain one or more logical connectives, they are called compound sentences , or molecular sentences . Sentential connectives are

3996-409: The end of the clause. 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, or schematic letters , however, range over all formulas. (Schematic letters are also called metavariables .) It

4070-484: The following as examples of well-formed formulas: What was given as Definition 2 above, which is responsible for the composition of formulas, is referred to by Colin Howson as the principle of composition . It is this recursion in the definition of a language's syntax which justifies the use of the word "atomic" to refer to propositional variables, since all formulas in the language L {\displaystyle {\mathcal {L}}} are built up from

4144-573: The invention of truth tables. The actual tabular structure (being formatted as a table), itself, is generally credited to either Ludwig Wittgenstein or Emil Post (or both, independently). Besides Frege and Russell, others credited with having ideas preceding truth tables include Philo, Boole, Charles Sanders Peirce , and Ernst Schröder . Others credited with the tabular structure include Jan Łukasiewicz , Alfred North Whitehead , William Stanley Jevons , John Venn , and Clarence Irving Lewis . Ultimately, some have concluded, like John Shosky, that "It

4218-393: The language, so that instead they'll use the notation M ⊨ φ {\displaystyle {\mathfrak {M}}\models \varphi } , which is equivalent to saying I ( φ ) = T {\displaystyle {\mathcal {I}}(\varphi )={\mathsf {T}}} , where I {\displaystyle {\mathcal {I}}}

4292-398: The new sentence that results from its application also is (or expresses) a proposition . Philosophers disagree about what exactly a proposition is, as well as about which sentential connectives in natural languages should be counted as logical connectives. Sentential connectives are also called sentence-functors , and logical connectives are also called truth-functors . An argument

4366-516: The objects, and GPS generated heuristics by means–ends analysis in order to solve problems. It focused on the available operations, finding what inputs were acceptable and what outputs were generated. It then created subgoals to get closer and closer to the goal. The GPS paradigm eventually evolved into the Soar architecture for artificial intelligence . Well-formed formula In mathematical logic , propositional logic and predicate logic ,

4440-413: The precedence (from most binding to least binding) 1. ¬   2. →  3. ∧  4. ∨. Then the formula may be abbreviated as This is, however, only a convention used to simplify the written representation of a formula. If the precedence was assumed, for example, to be left-right associative, in following order: 1. ¬   2. ∧  3. ∨  4. →, then

4514-896: The rules of classical logic is sometimes called a Boolean valuation . An interpretation of a formal language for classical logic is often expressed in terms of truth tables . Since each formula is only assigned a single truth-value, an interpretation may be viewed as a function , whose domain is L {\displaystyle {\mathcal {L}}} , and whose range is its set of semantic values V = { T , F } {\displaystyle {\mathcal {V}}=\{{\mathsf {T}},{\mathsf {F}}\}} , or V = { 1 , 0 } {\displaystyle {\mathcal {V}}=\{1,0\}} . For n {\displaystyle n} distinct propositional symbols there are 2 n {\displaystyle 2^{n}} distinct possible interpretations. For any particular symbol

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4588-440: The same formula above (without parentheses) would be rewritten as The definition of a formula in first-order logic Q S {\displaystyle {\mathcal {QS}}} is relative to the signature of the theory at hand. This signature specifies the constant symbols, predicate symbols, and function symbols of the theory at hand, along with the arities of the function and predicate symbols. The definition of

4662-510: The same formula may be written more than once, and a formula might in principle be so long that it cannot be written at all within the physical universe. Formulas themselves are syntactic objects. They are given meanings by interpretations. For example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not be interpreted, however, to be considered solely as

4736-425: The scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself. If we assume that the validity of modus ponens has been accepted as an axiom , then the same § Example argument can also be depicted like this: This method of displaying it is Gentzen 's notation for natural deduction and sequent calculus . The premises are shown above

4810-518: The search was easily lost in the combinatorial explosion . Put another way, the number of "walks" through the inferential digraph became computationally untenable. (In practice, even a straightforward state space search such as the Towers of Hanoi can become computationally infeasible, albeit judicious prunings of the state space can be achieved by such elementary AI techniques as A* and IDA* ). The user defined objects and operations that could be done on

4884-663: The sense that all and only the classical propositional tautologies are theorems, may be derived using only disjunction and negation (as Russell , Whitehead , and Hilbert did), or using only implication and negation (as Frege did), or using only conjunction and negation, or even using only a single connective for "not and" (the Sheffer stroke ), as Jean Nicod did. A joint denial connective ( logical NOR ) will also suffice, by itself, to define all other connectives, but no other connectives have this property. Some authors, namely Howson and Cunningham, distinguish equivalence from

4958-546: The special symbol ⊥ {\displaystyle \bot } , called "falsity", which always evaluates to False . Other authors also include these symbols, with the same meaning, but consider them to be "zero-place truth-functors", or equivalently, " nullary connectives". To serve as a model of the logic of a given natural language , a formal language must be semantically interpreted. In classical logic , all propositions evaluate to exactly one of two truth-values : True or False . For example, " Misplaced Pages

5032-407: The strings that followed the formation rules of (correct) formulas. Several authors simply say formula. Modern usages (especially in the context of computer science with mathematical software such as model checkers , automated theorem provers , interactive theorem provers ) tend to retain of the notion of formula only the algebraic concept and to leave the question of well-formedness , i.e. of

5106-542: The syntax of L {\displaystyle {\mathcal {L}}} is defined recursively by just a few definitions, as seen next; some authors explicitly include parentheses as punctuation marks when defining their language's syntax, while others use them without comment. Given a set of atomic propositional variables p 1 {\displaystyle p_{1}} , p 2 {\displaystyle p_{2}} , p 3 {\displaystyle p_{3}} , ..., and

5180-483: The term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence of symbols being expressed, with the marks being a token instance of formula. This distinction between the vague notion of "property" and the inductively-defined notion of well-formed formula has roots in Weyl's 1910 paper "Uber die Definitionen der mathematischen Grundbegriffe". Thus

5254-433: The truth value of true or the truth value of false . The principle of bivalence and the law of excluded middle are upheld. By comparison with first-order logic , truth-functional propositional logic is considered to be zeroth-order logic . Although propositional logic (also called propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic ( Stoic logic ) by Chrysippus in

5328-580: Was at Yale Law School (he was later a professor at the University of Michigan ). The suite of games is designed to teach the principles of symbolic logic to children (in Polish notation ). Its name is an echo of whiffenpoof , a nonsense word used as a cheer at Yale University made popular in The Whiffenpoof Song and The Whiffenpoofs . Propositional calculus The propositional calculus

5402-551: Was first developed by the 17th/18th-century mathematician Gottfried Leibniz , whose calculus ratiocinator was, however, unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan , completely independent of Leibniz. Gottlob Frege's predicate logic builds upon propositional logic, and has been described as combining "the distinctive features of syllogistic logic and propositional logic." Consequently, predicate logic ushered in

5476-450: Was the first computer program that separated its knowledge of problems (rules represented as input data) from its strategy of how to solve problems (a generic solver engine ). GPS was implemented in the third-order programming language, IPL . While GPS solved simple problems such as the Towers of Hanoi that could be sufficiently formalized, it could not solve any real-world problems because

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