Well-formed formula in the context of "Logical assertion"

In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formulas (also called wffs or simply formulas).

The validity of an argument can be tested, proved or disproved, and depends on its logical form.

Modal logic is a kind of logic used to represent statements about necessity and possibility. In philosophy and related fieldsit is used as a tool for understanding concepts such as knowledge, obligation, and causation. For instance, in epistemic modal logic, the formula ${\displaystyle \Box P}$ can be used to represent the statement that ${\displaystyle P}$ is known. In deontic modal logic, that same formula can represent that ${\displaystyle P}$ is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula ${\displaystyle \Box P\rightarrow P}$ as a tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false.

Modal logics are formal systems that include unary operators such as ${\displaystyle \Diamond }$ and ${\displaystyle \Box }$, representing possibility and necessity respectively. For instance the modal formula ${\displaystyle \Diamond P}$ can be read as "possibly ${\displaystyle P}$" while ${\displaystyle \Box P}$ can be read as "necessarily ${\displaystyle P}$". In the standard relational semantics for modal logic, formulas are assigned truth values relative to a possible world. A formula's truth value at one possible world can depend on the truth values of other formulas at other accessible possible worlds. In particular, ${\displaystyle \Diamond P}$ is true at a world if ${\displaystyle P}$ is true at some accessible possible world, while ${\displaystyle \Box P}$ is true at a world if ${\displaystyle P}$ is true at every accessible possible world. A variety of proof systems exist which are sound and complete with respect to the semantics one gets by restricting the accessibility relation. For instance, the deontic modal logic D is sound and complete if one requires the accessibility relation to be serial.

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.

Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.

In logic and deductive reasoning, an argument is sound if it is both valid in form and has no false premises. Soundness has a related meaning in mathematical logic, wherein a formal system of logic is sound if and only if every well-formed formula that can be proven in the system is logically valid with respect to the logical semantics of the system.

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence, according to the rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but there may be no method by which we can reliably find proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof.

The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus (of some formal system) to the previous well-formed formulas in the proof sequence.

In mathematical logic, a tautology (from Ancient Greek: ταυτολογία) is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. It is a logical truth. For example, a formula that states "the ball is green or the ball is not green" is always true, regardless of what a ball is and regardless of its colour. Tautology is usually, though not always, used to refer to valid formulas of propositional logic.

The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false.

In mathematics, a law is a formula that is always true within a given context. Laws describe a relationship, between two or more expressions or terms (which may contain variables), usually using equality or inequality, or between formulas themselves, for instance, in mathematical logic. For example, the formula ${\displaystyle a^{2}\geq 0}$ is true for all real numbers a, and is therefore a law. Laws over an equality are called identities. For example, ${\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}$ and ${\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}$ are identities. Mathematical laws are distinguished from scientific laws which are based on observations, and try to describe or predict a range of natural phenomena. The more significant laws are often called theorems.

Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields.

In natural or formal languages, self-reference occurs when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding.

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 inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a whole logical expression, a rule of replacement may be applied to only a particular segment. Within the context of a logical proof, logically equivalent expressions may replace each other. Rules of replacement are used in propositional logic to manipulate propositions.

Common rules of replacement include de Morgan's laws, commutation, association, distribution, double negation, transposition, material implication, logical equivalence, exportation, and tautology.