# Category: Propositional calculus

Functional completeness
In logic, a functionally complete set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expressio
DiVincenzo's criteria
The DiVincenzo criteria are conditions necessary for constructing a quantum computer, conditions proposed in 2000 by the theoretical physicist David P. DiVincenzo, as being those necessary to construc
Resolution (logic)
In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order l
Principle of distributivity
The principle of distributivity states that the algebraic distributive law is valid, where both logical conjunction and logical disjunction are distributive over each other so that for any proposition
Predicate (mathematical logic)
In logic, a predicate is a symbol which represents a property or a relation. For instance, in the first order formula , the symbol is a predicate which applies to the individual constant . Similarly,
Nicod's axiom
In logic, Nicod's axiom (named after the French logician and philosopher Jean Nicod) is a formula that can be used as the sole axiom of a semantically complete system of propositional calculus. The on
Substitution (logic)
Substitution is a fundamental concept in logic.A substitution is a syntactic transformation on formal expressions.To apply a substitution to an expression means to consistently replace its variable, o
Tautology (logic)
In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, o
Clause (logic)
In logic, a clause is a propositional formula formed from a finite collection of literals (atoms or their negations) and logical connectives. A clause is true either whenever at least one of the liter
Propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions
Frege's propositional calculus
In mathematical logic, Frege's propositional calculus was the first axiomatization of propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus, in 1879 as part of
Logical consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statem
Deductive closure
In mathematical logic, a set of logical formulae is deductively closed if it contains every formula that can be logically deduced from , formally: if always implies . If is a set of formulae, the dedu
Unsatisfiable core
In mathematical logic, given an unsatisfiable Boolean propositional formula in conjunctive normal form, a subset of clauses whose conjunction is still unsatisfiable is called an unsatisfiable core of
Propositional formula
In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it deter
System L
System L is a natural deductive logic developed by E.J. Lemmon. Derived from Suppes' method, it represents natural deduction proofs as sequences of justified steps. Both methods are derived from Gentz
Negation introduction
Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus. Negation introduction states that if a given antecedent implies both the consequent and it
Literal (mathematical logic)
In mathematical logic, a literal is an atomic formula (also known as an atom or prime formula) or its negation. The definition mostly appears in proof theory (of classical logic), e.g. in conjunctive
Frege system
In proof complexity, a Frege system is a propositional proof system whose proofs are sequences of formulas derived using a finite set of sound and implicationally complete inference rules. Frege syste
Second-order propositional logic
A second-order propositional logic is a propositional logic extended with quantification over propositions. A special case are the logics that allow second-order Boolean propositions, where quantifier
Negation normal form
In mathematical logic, a formula is in negation normal form (NNF) if the negation operator is only applied to variables and the only other allowed Boolean operators are conjunction and disjunction . N
Propositional proof system
In propositional calculus and proof complexity a propositional proof system (pps), also called a Cook–Reckhow propositional proof system, is a system for proving classical propositional tautologies.
Truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expre
Formation rule
In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. These rules only add
Proof by contrapositive
In logic, the contrapositive of a conditional statement is formed by negating both terms and reversing the direction of inference. More specifically, the contrapositive of the statement "if A, then B"
Proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same me
Zeroth-order logic
Zeroth-order logic is first-order logic without variables or quantifiers. Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus, but an alternative definition ex
Propositional variable
In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional vari
Implicational propositional calculus
In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional. In formulas, this bi
Intermediate logic
In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent supe
Rule of replacement
In logic, a rule of replacement is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of
Rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion
Minimal axioms for Boolean algebra
In mathematical logic, minimal axioms for Boolean algebra are assumptions which are equivalent to the axioms of Boolean algebra (or propositional calculus), chosen to be as short as possible. For exam
Stoic logic
Stoic logic is the system of propositional logic developed by the Stoic philosophers in ancient Greece. It was one of the two great systems of logic in the classical world. It was largely built and sh
List of Hilbert systems
This article contains a list of sample Hilbert-style deductive systems for propositional logics.