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ell others credited with having ideas preceding truth tables include philo boole charles sanders peirce 29 and ernst schröder others credited with the tabular structure include jan łukasiewicz alfred north whitehead william stanley jevons john venn and clarence irving lewis 28 ultimately some have concluded like john shosky that it is far from clear that any one person should be given the title of inventor of truth tables 28 sentences edit main article proposition propositional logic as currently studied in universities is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered in evaluating the conditions for the truth of a sentence or whether a sentence logically follows from some other sentence or group of sentences 2 declarative sentences edit propositional logic deals with statements which are defined as declarative sentences having truth value 30 1 examples of statements might include wikipedia is a free online encyclopedia that anyone can edit london is the capital of england all wikipedia editors speak at least three languages declarative sentences are contrasted with questions such as what is wikipedia and imperative statements such as please add citations to support the claims in this article 31 32 such non declarative sentences have no truth value 33 and are only dealt with in nonclassical logics called erotetic and imperative logics compounding sentences with connectives edit see also atomic formula and atomic sentence in propositional logic a statement can contain one or more other statements as parts 1 compound sentences are formed from simpler sentences and express relationships among the constituent sentences 34 this is done by combining them with logical connectives 34 35 the main types of compound sentences are negations conjunctions disjunctions implications and biconditionals 34 which are formed by using the corresponding connectives to connect propositions 36 37 in english these connectives are expressed by the words and conjunction or disjunction not negation if material conditional and if and only if biconditional 1 14 examples of such compound sentences might include wikipedia is a free online encyclopedia that anyone can edit and millions already have conjunction it is not true that all wikipedia editors speak at least three languages negation either london is the capital of england or london is the capital of the united kingdom or both disjunction f if sentences lack any logical connectives they are called simple sentences 1 or atomic sentences 35 if they contain one or more logical connectives they are called compound sentences 34 or molecular sentences 35 sentential connectives are a broader category that includes logical connectives 2 35 sentential connectives are any linguistic particles that bind sentences to create a new compound sentence 2 35 or that inflect a single sentence to create a new sentence 2 a logical connective or propositional connective is a kind of sentential connective with the characteristic feature that when the original sentences it operates on are or express propositions the new sentence that results from its application also is or expresses a proposition 2 philosophers disagree about what exactly a proposition is 11 2 as well as about which sentential connectives in natural languages should be counted as logical connectives 35 2 sentential connectives are also called sentence functors 38 and logical connectives are also called truth functors 38 arguments edit main article argument an argument is defined as a pair of things namely a set of sentences called the premises g and a sentence called the conclusion 39 35 38 the conclusion is claimed to follow from the premises 38 and the premises are claimed to support the conclusion 35 example argument edit the following is an example of an argument within the scope of propositional logic premise 1 if it s raining then it s cloudy premise 2 it s raining conclusion it s cloudy the logical form of this argument is known as modus ponens 40 which is a classically valid form 41 so in classical logic the argument is valid although it may or may not be sound depending on the meteorological facts in a given context this example argument will be reused when explaining formalization validity and soundness edit main articles validity logic and soundness an argument is valid if and only if it is necessary that if all its premises are true its conclusion is true 39 42 43 alternatively an argument is valid if and only if it is impossible for all the premises to be true while the conclusion is false 43 39 validity is contrasted with soundness 43 an argument is sound if and only if it is valid and all its premises are true 39 43 otherwise it is unsound 43 logic in general aims to precisely specify valid arguments 35 this is done by defining a valid argument as one in which its conclusion is a logical consequence of its premises 35 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 35 see semantics below formalization edit propositional logic is typically studied through a formal system in which formulas of a formal language are interpreted to represent propositions this formal language is the basis for proof systems which allow a conclusion to be derived from premises if and only if it is a logical consequence of them this section will show how this works by formalizing the example argument the formal language for a propositional calculus will be fully specified in language and an overview of proof systems will be given in proof systems propositional variables edit main article propositional variable since propositional logic is not concerned with the structure of propositions beyond the point where they cannot be decomposed any more by logical connectives 40 1 it is typically studied by replacing such atomic indivisible statements with letters of the alphabet which are interpreted as variables representing statements propositional variables 1 with propositional variables the example argument would then be symbolized as follows premise 1 p q displaystyle p to q premise 2 p displaystyle p conclusion q displaystyle q when p is interpreted as it s raining and q as it s cloudy these symbolic expressions correspond exactly with the original expression in natural language not only that but they will also correspond with any other inference with the same logical form when a formal system is used to represent formal logic only statement letters usually capital roman letters such as p displaystyle p q displaystyle q and r displaystyle r are represented directly the natural language propositions that arise when they re interpreted are outside the scope of the system and the relation between the formal system and its interpretation is likewise outside the formal system itself gentzen notation edit if we assume that the validity of modus ponens has been accepted as an axiom then the same example argument can also be depicted like this p q p q displaystyle frac p to q p q this method of displaying it is gentzen s notation for natural deduction and sequent calculus 44 the premises are shown above a line called the inference line 16 separated by a comma which indicates combination of premises 45 the conclusion is written below the inference line 16 the inference line represents syntactic consequence 16 sometimes called deductive consequence 46 which is also symbolized with 47 46 so the above can also be written in one line as p q p q displaystyle p to q p vdash q h syntactic consequence is contrasted with semantic consequence 48 which is symbolized with 47 46 in this case the conclusion follows syntactically because the natural deduction inference rule of modus ponens has been assumed for more on inference rules see the sections on proof systems below language edit part of a series on formal languages key concepts formal system alphabet syntax formal semantics semantics programming languages formal grammar formation rule well formed formula automata theory regular expression production ground expression atomic formula applications formal methods propositional calculus predicate logic mathematical notation natural language processing programming language theory mathematical linguistics computational linguistics syntax analysis formal verification automated theorem proving v t e the language commonly called l displaystyle mathcal l 46 49 35 of a propositional calculus is defined in terms of 2 15 a set of primitive symbols called atomic formulas atomic sentences 40 35 atoms 50 placeholders prime formulas 50 proposition letters sentence letters 40 or variables and a set of operator symbols called connectives 19 1 51 logical connectives 1 logical operators 1 truth functional connectives 1 truth functors 38 or propositional connectives 2 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 49 or as containing that set together with for instance its set of connectives and variables 15 35 usually the syntax of l displaystyle mathcal l is defined recursively by just a few definitions as seen next some authors explicitly include parentheses as punctuation marks when defining their language s syntax 35 52 while others use them without comment 2 15 syntax edit given a set of atomic propositional variables p 1 displaystyle p_ 1 p 2 displaystyle p_ 2 p 3 displaystyle p_ 3 and a set of propositional connectives c 1 1 displaystyle c_ 1 1 c 2 1 displaystyle c_ 2 1 c 3 1 displaystyle c_ 3 1 c 1 2 displaystyle c_ 1 2 c 2 2 displaystyle c_ 2 2 c 3 2 displaystyle c_ 3 2 c 1 3 displaystyle c_ 1 3 c 2 3 displaystyle c_ 2 3 c 3 3 displaystyle c_ 3 3 a formula of propositional logic is defined recursively by these definitions 2 15 51 i definition 1 atomic propositional variables are formulas definition 2 if c n m displaystyle c_ n m is a propositional connective and displaystyle langle a b c displaystyle rangle is a sequence of m possibly but not necessarily atomic possibly but not necessarily distinct formulas then the result of applying c n m displaystyle c_ n m to displaystyle langle a b c displaystyle rangle is a formula definition 3 nothing else is a formula writing the result of applying c n m displaystyle c_ n m to displaystyle langle a b c displaystyle rangle in functional notation as c n m displaystyle c_ n m a b c we have the following as examples of well formed formulas p 5 displaystyle p_ 5 c 3 2 p 2 p 9 displaystyle c_ 3 2 p_ 2 p_ 9 c 3 2 p 1 c 2 1 p 3 displaystyle c_ 3 2 p_ 1 c_ 2 1 p_ 3 c 1 3 p 4 p 6 c 2 2 p 1 p 2 displaystyle c_ 1 3 p_ 4 p_ 6 c_ 2 2 p_ 1 p_ 2 c 4 2 c 1 1 p 7 c 3 1 p 8 displaystyle c_ 4 2 c_ 1 1 p_ 7 c_ 3 1 p_ 8 c 2 3 c 1 2 p 3 p 4 c 2 1 p 5 c 3 2 p 6 p 7 displaystyle c_ 2 3 c_ 1 2 p_ 3 p_ 4 c_ 2 1 p_ 5 c_ 3 2 p_ 6 p_ 7 c 3 1 c 1 3 p 2 p 3 c 2 2 p 4 p 5 displaystyle c_ 3 1 c_ 1 3 p_ 2 p_ 3 c_ 2 2 p_ 4 p_ 5 what was given as definition 2 above which is responsible for the composition of formulas is referred to by colin howson as the principle of composition 40 j it is this recursion in the definition of a language s syntax which justifies the use of the word atomic to refer to propositional variables since all formulas in the language l displaystyle mathcal l are built up from the atoms as ultimate building blocks 2 composite formulas all formulas besides atoms are called molecules 50 or molecular sentences 35 this is an imperfect analogy with chemistry since a chemical molecule may sometimes have only one atom as in monatomic gases 50 the definition that nothing else is a formula given above as definition 3 excludes any formula from the language which is not specifically required by the other definitions in the syntax 38 in particular it excludes infinitely long formulas from being well formed 38 it is sometimes called the closure clause 54 cf grammar in bnf edit 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 55 56 this is more common in computer science than in philosophy 56 it can be done in many ways 55 of which a particularly brief one for the common set of five connectives is this single clause 56 57 ϕ a 1 a 2 ϕ ϕ ψ ϕ ψ ϕ ψ ϕ ψ displaystyle phi a_ 1 a_ 2 ldots neg phi phi psi phi vee psi phi rightarrow psi phi leftrightarrow psi this clause due to its self referential nature since ϕ displaystyle phi is in some branches of the definition of ϕ displaystyle phi also acts as a recursive definition and therefore specifies the entire language to expand it to add modal operators one need only add ϕ ϕ displaystyle box phi diamond phi to the end of the clause 56 constants and schemata edit mathematicians sometimes distinguish between propositional constants propositional variables and schemata propositional constants represent some particular proposition 58 while propositional variables range over the set of all atomic propositions 58 schemata or schematic letters however range over all formulas 38 1 schematic letters are also called metavariables 39 it is common to represent propositional constants by a b and c propositional variables by p q and r and schematic letters are often greek letters most often φ ψ and χ 38 1 however some authors recognize only two propositional constants in their formal system the special symbol displaystyle top called truth which always evaluates to true and the special symbol displaystyle bot called falsity which always evaluates to false 59 60 61 other authors also include these symbols with the same meaning but consider them to be zero place truth functors 38 or equivalently nullary connectives 51 semantics edit main articles semantics of logic and model theory to serve as a model of the logic of a given natural language a formal language must be semantically interpreted 35 in classical logic all propositions evaluate to exactly one of two truth values true or false 1 62 for example wikipedia is a free online encyclopedia that anyone can edit evaluates to true 63 while wikipedia is a paper encyclopedia evaluates to false 64 in other respects the following formal semantics can apply to the language of any propositional logic but the assumptions that there are only two semantic values bivalence that only one of the two is assigned to each formula in the language noncontradiction and that every formula gets assigned a value excluded middle are distinctive features of classical logic 62 65 38 to learn about nonclassical logics with more than two truth values and their unique semantics one may consult the articles on many valued logic three valued logic finite valued logic and infinite valued logic interpretation case and arg...
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