Well-formed formula in the context of "Metaphysical contingency"

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.

In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete.The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantic validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true.

In logic, syntax is an arrangement of well-structured entities in the formal languages or formal systems that express something. Syntax is concerned with the rules used for constructing or transforming the symbols and words of a language, as contrasted with the semantics of a language, which is concerned with its meaning.

The symbols, formulas, systems, theorems and proofs expressed in formal languages are syntactic entities whose properties may be studied without regard to any meaning they may be given, and, in fact, need not be given any.

In logic, a strict conditional (symbol: ${\displaystyle \Box }$, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic. For any two propositions p and q, the formula p → q says that p materially implies q while ${\displaystyle \Box (p\rightarrow q)}$ says that p strictly implies q. Strict conditionals are the result of Clarence Irving Lewis's attempt to find a conditional for logic that can adequately express indicative conditionals in natural language. They have also been used in studying Molinist theology.

In mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula ${\displaystyle x+3=y}$ is satisfiable because it is true when ${\displaystyle x=3}$ and ${\displaystyle y=6}$, while the formula ${\displaystyle x+1=x}$ is not satisfiable over the integers. The dual concept to satisfiability is validity; a formula is valid if every assignment of values to its variables makes the formula true. For example, ${\displaystyle x+3=3+x}$ is valid over the integers, but ${\displaystyle x+3=y}$ is not.

Formally, satisfiability is studied with respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being syntactic, however, satisfiability is a semantic property because it relates to the meaning of the symbols, for example, the meaning of ${\displaystyle +}$ in a formula such as ${\displaystyle x+1=x}$. Formally, we define an interpretation (or model) to be an assignment of values to the variables and an assignment of meaning to all other non-logical symbols, and a formula is said to be satisfiable if there is some interpretation which makes it true. While this allows non-standard interpretations of symbols such as ${\displaystyle +}$, one can restrict their meaning by providing additional axioms. The satisfiability modulo theories problem considers satisfiability of a formula with respect to a formal theory, which is a (finite or infinite) set of axioms.

In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory.

Judgments are used in formalizing deduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequence of judgments, and their conclusion is a judgment as well (thus, hypotheses and conclusions of proofs are judgments). A characteristic feature of the variants of Hilbert-style deduction systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only in the derivability of tautologies, not hypothetical judgments, then we can formalize the Hilbert-style deduction system in such a way that its rules of inference contain only judgments of a rather simple form. The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want to use them just for proving derivability of tautologies.

In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.

First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, quantifies over relations. For example, the second-order sentence ${\displaystyle \forall P\,\forall x(Px\lor \neg Px)}$ says that for every formula P, and every individual x, either Px is true or not(Px) is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified.
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