In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

Frege's Begriffsschrift ( 1879 ) introduced both a complete propositional calculus and what is essentially modern predicate logic.

More ambitious was the Logic Theory Machine, a deduction system for the propositional logic of the Principia Mathematica, developed by Allen Newell, Herbert A. Simon and J. C. Shaw.

For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.

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

or as the statement of a truth-functional tautology or theorem of propositional logic:

Category: Theorems in propositional logic

Biconditional elimination is the name of two valid rules of inference of propositional logic.

or as the statement of a truth-functional tautology or theorem of propositional logic:

Category: Theorems in propositional logic

A formula of propositional logic is said to be satisfiable if logical values can be assigned to its variables in a way that makes the formula true.

Conjunction introduction ( often abbreviated simply as conjunction ) is a valid rule of inference of propositional logic.

In propositional logic, disjunctive syllogism ( also known as disjunction elimination and or elimination, or abbreviated ∨ E ), is a valid rule of inference.

and expressed as a truth-functional tautology or theorem of propositional logic:

Disjunction introduction or addition is a simple valid argument form, an immediate inference and a rule of inference of propositional logic.

and expressed as a truth-functional tautology or theorem of propositional logic:

Category: Theorems in propositional logic

: For the theorem of propositional logic which expresses Disjunction elimination, see Case analysis.

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

and expressed as a truth-functional tautology or theorem of propositional logic:

He includes here syllogistic classic logic, propositional logic and that of predicates.

This software showed a radical innovation: it used propositional logic (" Zeroth order logic ") to execute expert systems, reasoning on a knowledge base written with everyday language rules, producing explanations and detecting logic contradictions between the facts.

Also running on a JOHANNIAC, the Logic Theory Machine constructed proofs from a small set of propositional axioms and three deduction rules: modus ponens, ( propositional ) variable substitution, and the replacement of formulas by their definition.

Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

A propositional argument using modus ponens is said to be deductive.

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

In propositional logic, the inference rule is modus ponens

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

This is just the modus ponens rule of propositional logic.

In his doctoral thesis, Post proved, among other things, that the propositional calculus of Principia Mathematica was complete: all tautologies are theorems, given the Principia axioms and the rules of substitution and modus ponens.

Modus ponendo tollens ( Latin: " mode that by affirming, denies ") is a valid rule of inference for propositional logic, sometimes abbreviated MPT.

* Modus ponens, a principle in propositional logic

The propositional formulas could then be checked for unsatisfiability using a number of methods.

Except for propositional logic, all are complex and can only be understood by mathematicians, logicians or computer scientists.

For Avicenna ( Ibn Sina ), for example, the a tabula rasa is a pure potentiality that is actualized through education, and knowledge is attained through " empirical familiarity with objects in this world from which one abstracts universal concepts " developed through a " syllogistic method of reasoning in which observations lead to propositional statements which when compounded lead to further abstract concepts.

In propositional logic these are treated as two unrelated propositions, denoted for example by p and q.

The reason for that is the completeness of propositional logic, with the existential quantifiers playing no role.

We may represent any given proposition with a letter which we call a propositional constant, analogous to representing a number by a letter in mathematics, for instance,.

A simple way to generate this is by truth-tables, in which one writes,, …, for any list of propositional constants — that is to say, any list of propositional constants with entries.

Below one fills in one-quarter of the rows with T, then one-quarter with F, then one-quarter with T and the last quarter with F. The next column alternates between true and false for each eighth of the rows, then sixteenths, and so on, until the last propositional constant varies between T and F for each row.

This will give a complete listing of cases or truth-value assignments possible for those propositional constants.

A very strong propositional hand-one that is more likely to win with a straight or a flush-is one of the hands that can be played for effect with an aggressive style.

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.

An attempt to defend a system based on propositional meaning for semantic underspecification can be found in the generative lexicon model of James Pustejovsky, who extends contextual operations ( based on type shifting ) into the lexicon.

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

First, his language is artificially impoverished, and second, the rules for the propositional modal logic must be weakened.

Rather, complementarity means that the composition of physical properties for S ( such as position and momentum both having values within certain ranges ), using propositional connectives, does not obey the rules of classical propositional logic ( see also Quantum logic ).

Quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables.