# Propositional calculus

**propositional logicpropositionalsentential logicclassical propositional logicsentential calculuspropositionslogicalpropositionpropositional or sententialsentential**

Propositional calculus is a branch of logic.wikipedia

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### Logic

**logicianlogicallogics**

Propositional calculus is a branch of logic.

Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is often divided into two main branches: propositional logic and predicate logic.

### Stoic logic

**LogicThe Stoic logician**

Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic (Stoic logic) by Chrysippus in the 3rd century BC and expanded by his successor Stoics.

Stoic logic is the system of propositional logic developed by the Stoic philosophers in ancient Greece.

### Interpretation (logic)

**interpretationinterpretationsinterpreted**

Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions.

The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation.

### Theorem

**theoremspropositionconverse**

These derived formulas are called theorems and may be interpreted to be true propositions.

Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English.

### Chrysippus

**Chrysippus the Stoic**

Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic (Stoic logic) by Chrysippus in the 3rd century BC and expanded by his successor Stoics.

He created an original system of propositional logic in order to better understand the workings of the universe and role of humanity within it. He adhered to a deterministic view of fate, but nevertheless sought a role for personal freedom in thought and action.

### First-order logic

**predicate logicfirst-orderpredicate calculus**

Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers.

This distinguishes it from propositional logic, which does not use quantifiers or relations.

### Method of analytic tableaux

**analytic tableauanalytic tableauxsemantic tableau**

Consequently, predicate logic ushered in 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.

In proof theory, the semantic tableau (plural: tableaux, also called 'truth tree') is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic.

### Rule of inference

**inference rulerules of inferenceinference rules**

The premises are taken for granted and then with the application of modus ponens (an inference rule) the conclusion follows. A system of inference rules and axioms allows certain formulas to be derived.

Popular rules of inference in propositional logic include modus ponens, modus tollens, and contraposition.

### Stoicism

**StoicStoicsStoic philosopher**

Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic (Stoic logic) by Chrysippus in the 3rd century BC and expanded by his successor Stoics.

Diodorus Cronus, who was one of Zeno's teachers, is considered the philosopher who first introduced and developed an approach to logic now known as propositional logic, which is based on statements or propositions, rather than terms, making it very different from Aristotle's term logic.

### Jan Łukasiewicz

**ŁukasiewiczJ. LukasiewiczJan '''Ł'''ukasiewicz**

Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz.

He thought innovatively about traditional propositional logic, the principle of non-contradiction and the law of excluded middle.

### Law of excluded middle

**law of the excluded middleexcluded middletertium non datur**

The principle of bivalence and the law of excluded middle are upheld.

The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

### Syllogism

**syllogisticcategorical syllogismsyllogisms**

This advancement was different from the traditional syllogistic logic which was focused on terms.

This led to the rapid development of sentential logic and first-order predicate logic, subsuming syllogistic reasoning, which was, therefore, after 2000 years, suddenly considered obsolete by many.

### Principle of bivalence

**bivalenttwo-valued logicbinary logic**

The principle of bivalence and the law of excluded middle are upheld.

In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra, "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element.

### Logical connective

**logical operatorconnectivesconnective**

Compound propositions are formed by connecting propositions by logical connectives.

Logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic.

### Axiom

**axiomspostulateaxiomatic**

A system of inference rules and axioms allows certain formulas to be derived.

A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions.

### Double negation

**double negation eliminationnegation of negation**

Double negation elimination: From \neg \neg p, infer.

In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true."

### Zeroth-order logic

It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and argument flow.

Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus, but an alternative definition extends propositional logic by adding constants, operations, and relations on non-Boolean values.

### Atomic formula

**atomatomicatomic expressions**

1) a set of primitive symbols, variously referred to as atomic formulas, placeholders, proposition letters, or variables, and

The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, the atomic formulas are the propositional variables.

### Conjunction elimination

**SimplificationConjunctive elimination**

Conjunction elimination: From (p \land q), infer.

In propositional logic, conjunction elimination (also called and elimination, ∧ elimination, or simplification) is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction A and B is true, then A is true, and B is true.

### Modus ponens

**implicationfollowsif all As are Bs and all Bs are Cs, then all As are Cs**

The premises are taken for granted and then with the application of modus ponens (an inference rule) the conclusion follows.

In propositional logic, modus ponens (MP; also modus ponendo ponens (Latin for "mode that affirms by affirming") or implication elimination) is a rule of inference.

### Mathematical logic

**formal logicsymbolic logiclogic**

Propositional logic was eventually refined using symbolic logic.

The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties.

### Modus tollens

**denying the consequentMTconsidered unlikely versus**

In propositional logic, modus tollens (MT; also modus tollendo tollens (Latin for "mode that denies by denying") or denying the consequent) is a valid argument form and a rule of inference.

### Disjunction elimination

**Case analysis**

Disjunction elimination: From (p \lor q) and (p \to r) and (q \to r), infer.

In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof.

### Biconditional introduction

Biconditional introduction: From (p \to q) and (q \to p), infer.

In propositional logic, biconditional introduction is a valid rule of inference.

### De Morgan's laws

**De Morgan's TheoremDe Morgan's lawDeMorgan's Laws**

In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference.