In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are "theorems" of the formal system. Logic, from Classical Greek λόγος logos (the word), is the study of patterns found in reasoning. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... A boolean-valued function, in some usages a predicate or a proposition, is a function of the type f : X → B, where X is an arbitrary set and where B is a boolean domain. ... In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ...

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 formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions. In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... In logic, statements p and q are logically equivalent if they have the same logical content. ...

When the formal system is intended to be a logical system, the expressions are meant to be interpreted as mathematical statements, and the rules, known as inference rules, are typically intended to be truth-preserving. In this setting, the rules (which may include axioms) can then be used to derive ("infer") formulas representing true statements from given formulas representing true statements. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... For the algebra software named Axiom, see Axiom computer algebra system. ...

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 well-formed formulas (wffs) of the language. In addition a semantics may be given which defines truth and valuations (or interpretations). In computer science and linguistics, a formal grammar, or sometimes simply grammar, is a precise description of a formal language — that is, of a set of strings. ... Semantics (Greek semantikos, giving signs, significant, symptomatic, from sema, sign) refers to the aspects of meaning that are expressed in a language, code, or other form of representation. ... Valuation can mean: Look up valuation in Wiktionary, the free dictionary. ...

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 according to the rules of the grammar. In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

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

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 , whose formulas are constructed in the following manner: In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ...

• The alpha set is a finite set of elements called proposition symbols or propositional variables. Syntactically speaking, these are the most basic elements of the formal language , otherwise referred to as atomic formulas or terminal elements. In the examples to follow, the elements of are typically the letters p, q, r, and so on.

• The omega set is a finite set of elements called operator symbols or logical connectives. The set is partitioned into disjoint subsets as follows:

In this partition, is the set of operator symbols of arity .

In the more familiar propositional calculi, is typically partitioned as follows:

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

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 {, } 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 , also known as its set of formulas, well-formed formulas or wffs, is inductively or recursively defined by the following rules: In mathematical logic, a propositional variable (also called a sentential variable) is a variable which can either be true or false. ... In mathematical logic, an atomic formula, or atom, is a formula that has no subformulas. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ... Look up partition in Wiktionary, the free dictionary. ... In mathematics and computer programming the arity of a function or an operator is the number of arguments or operands it takes (arity is sometimes referred to as valency, although that actually refers to another meaning of valency in mathematics). ... In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... A tilde. ... This article or section does not cite its references or sources. ... In logic, WFF is an abbreviation for well-formed formula. ... In logic, WFF is an abbreviation for well-formed formula. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page — a list of pages that otherwise might share the same title. ...

1. Base. Any element of the alpha set is a formula of .
2. Step (a). If p is a formula, then ¬p is a formula.
3. Step (b). If p and q are formulas, then (p q), (p ∨ q), (p → q), and (p ↔ q) are formulas.
4. Close. Nothing else is a formula of .

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

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

• The zeta set is a finite set of transformation rules that are called inference rules when they acquire logical applications.

• The iota set is a finite set of initial points that are called axioms when they receive logical interpretations.

In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... This article does not cite its references or sources. ...

Example 1. Simple axiom system

Let , where are defined as follows:

• The alpha set is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

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). Sole sufficient operator, or sole sufficient connective, is a term used to describe an operator that is sufficient by itself to generate all of the operators in a specified class of operators. ...

Adopting negation and implication as the two primitive operations of a propositional calculus is tantamount to having the omega set partition as follows:

An axiom system discovered by Jan Lukasiewicz formulates a propositional calculus in this language as follows. The axioms are all substitution instances of: // Jan Łukasiewicz (21 December 1878 - 13 February 1956) was a Polish mathematician born in Lemberg, Galicia, Austria-Hungary (now Lviv, Ukraine). ...

The rule of inference is modus ponens (i.e. from p and (p → q), infer q). Then a ∨ b is defined as ¬a → b, and a ∧ b is defined as ¬(a → ¬b). In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P → Q P ⊢ Q where ⊢ represents the logical assertion. ...

Example 2. Natural deduction system

Let , where are defined as follows:

• The alpha set is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

• The omega set partitions as follows:

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. In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it naturally occurs. ... A theorem (IPA pronunciation: , from vulgar Latin theōrēma, Greek θεώρημα spectacle, speculation, theory) is a proposition that has been or is to be proved on the basis of explicit assumptions. ...

• The set of initial points is empty, that is,

• The set of transformation rules, , 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 premiss 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 non-hypothetical rules, and the last one as a hypothetical rule.

Reductio ad absurdum (negation introduction) 

From (p→q), (p→ ¬q), infer ¬p.

Double negative elimination 

From ¬¬p, infer p.

Conjunction introduction 

From p and q, infer (p ∧ q).

Conjunction elimination 

From (p ∧ q), infer p

From (p ∧ q), infer q.

Disjunction introduction 

From p, infer (p ∨ q)

From p, infer (q ∨ p).

Disjunction elimination 

From (p ∨ q), (p → r), (q → r), infer r.

Biconditional introduction 

From (p → q), (q → p), infer (p ↔ q).

Biconditional elimination 

From (p ↔ q), infer (p → q);

From (p ↔ q), infer (q → p).

Modus ponens (conditional elimination) 

From p, (p → q), infer q.

Conditional proof (conditional introduction) 

If accepting p allows a proof of q, infer (p → q).

This article or section does not cite its references or sources. ... In logic and the propositional calculus, double negative elimination is a rule that states that double negatives can be removed from a proposition without changing its meaning: means the same as: Formally: ¬ ¬ A ∴ A The rule of double negative introduction states the converse, that double negatives can be added without... Conjunction introduction is the inference that, if p is true, and q is true, then the conjunction p and q is true. ... In logic, conjunction elimination is the inference that, if the conjunction A and B is true, then A is true, and B is true. ... Disjunction introduction is the logic principle that, if A is true, then its true that either A or B is true. ... In propositional calculus disjunction elimination is the inference that, if A or B is true, and A entails C, and B entails C, then we may justifiably infer C. The reasoning is simple: since at least one of the statements A and B is true, and since either of them... Biconditional introduction is the inference that, if B follows from A, and A follows from B, then A if and only if B. For example: if Im breathing, then Im alive; also, if Im alive, then Im breathing. ... Biconditional elimination allows one to infer a conditional from a biconditional: if ( A ↔ B ) is true, then one may infer one direction of the biconditional, either ( A → B ) or ( B → A ). For example, if its true that Im breathing if and only if Im... In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P → Q P ⊢ Q where ⊢ represents the logical assertion. ... Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ...

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 formulas. 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 sequences of numbered lines, with each line consisting of a single formula followed by a reason or justification for introducing that formula. Each premiss of the argument, that is, an assumption introduced as a hypothesis of the argument, is listed at the beginning of the sequence and is marked as a "premiss" in lieu of other justification. The conclusion is listed on the last line. A proof is complete if every line follows from previous ones by the correct application of a transformation rule. (For a contrasting approach, see proof-trees).

Example of a proof

• To be shown that A → A.

• One possible proof of this may be arranged as follows:

Interpret A ├ A as "Assuming A, infer A". Read ├ A → 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. (This article discusses the soundess notion of informal logic. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ...

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. Partial plot of a function f. ... State of affairs has some technical usages in philosophy, as well as being a phrase in everyday speech in English. ... In philosophy and logic, the concept of possible worlds is used to express modal claims. ...

We define when such a truth assignment A satisfies a certain wff with the following rules: In logic, WFF is an abbreviation for well-formed formula. ...

• A satisfies the propositional variable P if and only if A(P) = true
• A satisfies ¬φ if and only if A does not satisfy φ
• A satisfies (φ ∧ ψ) if and only if A satisfies both φ and ψ
• A satisfies (φ ∨ ψ) if and only if A satisfies at least one of either φ or ψ
• A satisfies (φ → ψ) if and only if it is not the case that A satisfies φ but not ψ
• A satisfies (φ ↔ ψ) if and only if A satisfies both φ and ψ or satisfies neither one of them

With this definition we can now formalize what it means for a formula φ to be implied by a certain set S of formulae. Informally this is true if in all worlds that are possible given the set of formulae S the formula φ also holds. This leads to the following formal definition: We say that a set S of wffs semantically entails (or implies) a certain wff φ if all truth assignments that satisfy all the formulae in S also satisfy φ. It has been suggested that this article or section be merged with Logical biconditional. ...

Finally we define syntactical entailment such that φ is syntactically entailed by S 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 syntactically entails wff φ then S semantically entails φ

Completeness 

If the set of wffs S semantically entails wff φ then S syntactically entails φ

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) Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...

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.

• I. Basis. Show: If A is a member of G, then G implies A.
• II. Basis. Show: If A is an axiom, then G implies A.
• III. Inductive step (induction on n, the length of the proof):

(a) Assume for arbitrary G and A that if G proves A in n or fewer steps, then G implies A.

(b) 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. In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it naturally occurs. ...

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 case-by-case analysis of all the rules of inference, showing that each "preserves" semantic implication. Case analysis is one of the most general and applicable methods of analytical thinking, depending only on the division of a problem, decision or situation into a sufficient number of separate cases. ...

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. In traditional logic, contraposition is a form of immediate inference in which from a given categorical proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality (affirmation or negation). ...

• I. G does not prove A. (Assumption)
• II. 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.
  • (a)Place an "ordering" on all the sentences in the language. (e.g., alphabetical ordering), and number them E₁, E₂, ...
  • (b)Define a series G_n of sets (G₀, G₁ ... ) inductively, as follows. (i)G₀ = G. (ii) If {G_k, E_(k+1)} proves A, then G_(k+1) = G_k. (iii) If {G_k, E_(k+1)} does not prove A, then G_(k+1) = {G_k, E_(k+1)}
  • (c)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).
  • (d) It can be easily shown that (i) G* contains (is a superset of) G (by (b.i)); (ii) 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 (iii) 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)
• III. If G* is a Maximal Set (wrt A), then it is "truth-like". 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.
• IV. If G* is truth-like 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.
• V. A G*-canonical valuation will make our original set G all true, and make A false.
• VI. If there is a valuation on which G are true and A is false, then G does not (semantically) imply A.

QED Q.E.D. is an abbreviation of the Latin phrase (literally, which was to be demonstrated). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven. ...

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. Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

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 φ, χ and ψ stand for well-formed formulas. (The wffs themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are as follows:

Axiom THEN-2 may be considered to be a "distributive property of implication with respect to implication."
Axioms AND-1 and AND-2 correspond to "conjunction elimination". The relation between AND-1 and AND-2 reflects the commutativity of the conjunction operator.
Axiom AND-3 corresponds to "conjunction introduction."
Axioms OR-1 and OR-2 correspond to "disjunction introduction." The relation between OR-1 and OR-2 reflects the commutativity of the disjunction operator.
Axiom NOT-1 corresponds to "reductio ad absurdum."
Axiom NOT-2 says that "anything can be deduced from a contradiction."
Axiom NOT-3 is called "tertium non datur" (Latin: "a third is not given") and reflects the semantic valuation of propositional formulae: a formula can have a truth-value of either true or false. There is no third truth-value, at least not in classical logic. Intuitionistic logicians do not accept the axiom NOT-3.
To meet Wikipedias quality standards, this article or section may require cleanup. ... Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...

Inference rule

The inference rule is modus ponens: In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P → Q P ⊢ Q where ⊢ represents the logical assertion. ...

• .

Meta-inference 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: In mathematical logic, the deduction theorem states that if a formula F is deducible from E then the implication E → F is demonstrable (i. ...

If the sequence

has been demonstrated, then it is also possible to demonstrate the sequence

.

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 meta-theorem, 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. Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ...

The converse of DT is also valid:

If the sequence

has been demonstrated, then it is also possible to demonstrate the sequence

in fact, the validity of the converse of DT is almost trivial compared to that of DT:

If

then

1:

2:

and from (1) and (2) can be deduced

3:

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 AND-1,

can be transformed by means of the converse of the deduction theorem into the inference rule

which is conjunction elimination, one of the ten inference rules used in the first version (in this article) of the propositional calculus. In logic, conjunction elimination is the inference that, if the conjunction A and B is true, then A is true, and B is true. ...

Example of a proof

The following is an example of a (syntactical) demonstration, involving only axioms THEN-1 and THEN-2:
Prove: A → A (Reflexivity of implication).
Proof:

1. (A → ((B → A) → A)) → ((A → (B → A)) → (A → A))

Axiom THEN-2 with φ = A, χ = B → A, ψ = A

2. A → ((B → A) → A)

Axiom THEN-1 with φ = A, χ = B → A

3. (A → (B → A)) → (A → A)

From (1) and (2) by modus ponens.

4. A → (B → A)

Axiom THEN-1 with φ = A, χ = B

5. A → A

From (3) and (4) by modus ponens.

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. A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for 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. A parse tree or concrete syntax tree is a tree that represents the syntactic structure of a string according to some formal grammar. ... In computer science, a pointer is a programming language datatype whose value refers directly to (or “points to”) another value stored elsewhere in the computer memory using its address. ... This article or section is in need of attention from an expert on the subject. ... In computer science, tree traversal is the process of visiting each node in a tree data structure exactly one time. ...

Other logical calculi

Propositional calculus is about the simplest kind of logical calculus in any current use. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler — but in other ways more complex — than propositional calculus.) It can be extended in several ways.

The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used. When the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, they yield first-order logic, or first-order predicate logic, which keeps all the rules of propositional logic and adds some new ones. (For example, from "All dogs are mammals" we may infer "If Rover is a dog then Rover is a mammal".) There is no really adequate definition of singular term. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... In linguistics and logic, a predicate is an expression that can be true of something. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...

With the tools of first-order 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. Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Mereology is a collection of axiomatic formal systems dealing with parts and their respective wholes. ...

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". In philosophical logic, a modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ...

Many-valued 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. Multi-valued logics are logical calculi in which there are more than two possible truth values. ...

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.
• 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.

Joachim Lambek (1922-) is the Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD in 1950. ...

See also

Logical levels

• Zeroth-order logic
• First-order logic
• Second-order logic
• Higher-order logic

Zeroth-order logic is a term in popular use among practitioners for the subject matter otherwise known as boolean functions, monadic predicate logic, propositional calculus, or sentential calculus. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ... In mathematics, higher-order logic is distinguished from first-order logic in a number of ways. ...

Building blocks

Exclusive disjunction, also known as exclusive or and symbolized by XOR or EOR, is a logical operation on two operands that results in a logical value of true if and only if one of the operands, but not both, has a value of true. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... OR logic gate. ... XNOR Logic Gate Symbol Logical equality is a logical operator that corresponds to equality in boolean algebra and to the logical biconditional in propositional calculus. ... In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ... NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ... Negation (i. ...

Related topics

Ampheck, from Greek double-edged, is a term coined by Charles Sanders Peirce for either one of the pair of logically dual operators, variously referred to as Peirce arrows, Sheffer strokes, or NAND and NNOR. Either of these logical operators is a sole sufficient operator for deriving or generating all... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... Algebra of sets Ampheck Boole, George Boolean algebra Boolean domain Boolean function Boolean logic Boolean implicant Boolean prime ideal theorem Boolean-valued function Boolean-valued model Boolean satisfiability problem Booles syllogistic Canonical form (Boolean algebra) Characteristic function Compactness theorem Complete Boolean algebra De Morgan, Augustus De Morgans laws... A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. ... In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ... Boolean logic is a complete system for logical operations. ... A boolean-valued function is a function of the type , where is an arbitrary set, where is a generic 2-element set, typically , and where the latter is frequently interpreted for logical applications as . ... Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science. ... This article is not about combinatory logic, a topic in mathematical logic. ... Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. ... Elsie the cat is sitting on a mat John F. Sowas conceptual graphs (CGs) are a system of logic based on the existential graphs of Charles Sanders Peirce and the semantic networks of artificial intelligence. ... A disjunctive syllogism, also known as modus tollendo ponens (literally: mode which, by denying, affirms) is a valid, simple argument form: P or Q Not P Therefore, Q In logical operator notation: ¬ where represents the logical assertion. ... An entitative graph is an element of the graphical syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned. ... An existential graph is a type of diagrammatic or visual notation for logical expressions, proposed by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ... In mathematical logic Freges propositional calculus was the first axiomatization of propositional calculus. ... In mathematical logic, the implicational propositional calculus is a version of classical (two-valued) propositional calculus which uses only one connective, called implication or conditional. ... Intuitionistic logic, or constructivist logic, is the symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwers programme of intuitionism. ... The book Laws of Form (hereinafter abbreviated LoF), by G. Spencer-Brown, describes three distinct logical systems: The primary arithmetic (described in Chapter 4), which can be interpreted as Boolean arithmetic; The primary algebra (Chapter 6), which can be interpreted via two-element Boolean algebra (hereinafter abbreviated 2), Boolean logic... A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. ... In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... In logic and mathematics, the minimal negation operator is a multigrade operator where each is a k-ary boolean function defined in such a way that if and only if exactly one of the arguments is 0. ... In logic and mathematics, a multigrade operator is a parametric operator with parameter k in the set N of non-negative integers. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In logic and mathematics, a parametric operator with parameter in the parametric set is a indexed family of operators with index in the index set . ... Peirces law in logic is named after the philosopher and logician Charles Sanders Peirce. ... In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ... It has been suggested that this article or section be merged with fallacy. ... The argument from fallacy, also known as argumentum ad logicam or fallacy fallacy, is a logical fallacy which assumes that if an argument is fallacious, its conclusion must be false. ... In philosophical logic, a modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... I know who X is. ... The appeal to probability is a logical fallacy, often used in conjunction with other fallacies. ... The logical fallacy of affirming a disjunct occurs in a disjunctive syllogism when an argument takes the form: Either A or B (this is the disjunct) A (Affirming the middle term) Therefore, not B The fallacy lies in concluding that B must be false because A is true; in fact... Affirming the consequent is a logical fallacy in the form of a hypothetical proposition. ... Denying the antecedent (also known as vacuous implication) is a type of logical fallacy. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... The existential fallacy is a logical fallacy committed in a categorical syllogism that is invalid because it has two universal premises and a particular conclusion. ... Syllogistic fallacies are logical fallacies that occur in syllogisms. ... Affirmative conclusion from a negative premise is a logical fallacy that is committed when a categorical syllogism has a positive conclusion, but one or two negative premises. ... The fallacy of exclusive premises is a formal fallacy committed in a categorical syllogism that is invalid because both of its premises are negative. ... A fallacy of necessity is a fallacy in the logic of a syllogism whereby a degree of unwarranted necessity is placed in the conclusion. ... The fallacy of four terms (Latin: quaternio terminorum) is a logical fallacy that occurs when a three-part syllogism has four terms. ... This article may be too technical for most readers to understand. ... Illicit minor is a logical fallacy committed in a categorical syllogism that is invalid because its minor term is undistributed in the minor premise but distributed in the conclusion. ... The fallacy of the undistributed middle is a logical fallacy that is committed when the middle term in a categorical syllogism isnt distributed. ...

Related works

• Hofstadter, D.R., Gödel, Escher, Bach

Douglas Richard Hofstadter (born February 15, 1945) is an American academic. ... GEB cover Gödel, Escher, Bach: an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ...

External links

• Klement, Kevin C. (2006), "Propositional Logic", in James Fieser and Bradley Dowden (eds.), Internet Encyclopedia of Philosophy, Eprint.
• Introduction to Mathematical Logic
• Elements of Propositional Calculus
• forall x: an introduction to formal logic, by P.D. Magnus, covers formal semantics and proof theory for sentential logic.