Propositional calculus in the context of "Satisfiable theory"

In propositional logic, modus ponens (/ˈmoʊdəs ˈpoʊnɛnz/; MP), also known as modus ponendo ponens (from Latin 'mode that by affirming affirms'), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q. P is true. Therefore, Q must also be true."

Modus ponens is a mixed hypothetical syllogism and is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent. Constructive dilemma is the disjunctive version of modus ponens.

In propositional logic, modus tollens (/ˈmoʊdəs ˈtɒlɛnz/) (MT), also known as modus tollendo tollens (Latin for "mode that by denying denies") and denying the consequent, is a deductive argument form and a rule of inference. Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.

The history of the inference rule modus tollens goes back to antiquity. The first to explicitly describe the argument form modus tollens was Theophrastus.

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory ${\displaystyle T}$ is consistent if there is no formula ${\displaystyle \varphi }$ such that both ${\displaystyle \varphi }$ and its negation ${\displaystyle \lnot \varphi }$ are elements of the set of consequences of ${\displaystyle T}$. Let ${\displaystyle A}$ be a set of closed sentences (informally "axioms") and ${\displaystyle \langle A\rangle }$ the set of closed sentences provable from ${\displaystyle A}$ under some (specified, possibly implicitly) formal deductive system. The set of axioms ${\displaystyle A}$ is consistent when there is no formula ${\displaystyle \varphi }$ such that ${\displaystyle \varphi \in \langle A\rangle }$ and ${\displaystyle \lnot \varphi \in \langle A\rangle }$. A trivial theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what consistent meant in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.

In a sound formal system, every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete. The completeness of the propositional calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of (first order) predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.

Constructive dilemma is a valid rule of inference of propositional logic. It is the inference that, if P implies Q and R implies S and either P or R is true, then either Q or S has to be true. In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too. Constructive dilemma is the disjunctive version of modus ponens, whereas destructive dilemma is the disjunctive version of modus tollens. The constructive dilemma rule can be stated:

where the rule is that whenever instances of "${\displaystyle P\to Q}$", "${\displaystyle R\to S}$", and "${\displaystyle P\lor R}$" appear on lines of a proof, "${\displaystyle Q\lor S}$" can be placed on a subsequent line.

In propositional logic, affirming the consequent (also known as converse error, fallacy of the converse, or confusion of necessity and sufficiency) is a formal fallacy (or an invalid form of argument) that is committed when, in the context of an indicative conditional statement, it is stated that because the consequent is true, therefore the antecedent is true. It takes on the following form:

In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value. The typical example is in propositional logic, wherein a compound statement is constructed using individual statements connected by logical connectives; if the truth value of the compound statement is entirely determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and any logical connectives used are said to be truth functional.

Classical propositional logic is a truth-functional logic, in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function. On the other hand, modal logic is non-truth-functional.

Fitch notation, also known as Fitch diagrams (named after Frederic Fitch), is a method of presenting natural deduction proofs in propositional calculus and first-order logics using a structured, line-by-line format that explicitly shows assumptions, inferences, and their scope. It was invented by Frederic Brenton Fitch in the 1930s and later popularized through his textbook Symbolic Logic (1952). Fitch notation is notable for its use of indentation or boxes to indicate the scope of subordinate assumptions, making it one of the most pedagogically accessible systems for teaching formal logic.

In logic and theoretical computer science, and specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse the computational resources that are required to prove or refute statements. Research in proof complexity is predominantly concerned with proving proof-length lower and upper bounds in various propositional proof systems. For example, among the major challenges of proof complexity is showing that the Frege system, the usual propositional calculus, does not admit polynomial-size proofs of all tautologies. Here the size of the proof is simply the number of symbols in it, and a proof is said to be of polynomial size if it is polynomial in the size of the tautology it proves.

Systematic study of proof complexity began with the work of Stephen Cook and Robert Reckhow (1979) who provided the basic definition of a propositional proof system from the perspective of computational complexity. Specifically Cook and Reckhow observed that proving proof size lower bounds on stronger and stronger propositional proof systems can be viewed as a step towards separating NP from coNP (and thus P from NP), since the existence of a propositional proof system that admits polynomial size proofs for all tautologies is equivalent to NP=coNP.