First-order logic in the context of "Hilbert system"

Logic translation is the process of representing a text in the formal language of a logical system. If the original text is formulated in ordinary language then the term natural language formalization is often used. An example is the translation of the English sentence "some men are bald" into first-order logic as ${\displaystyle \exists x(M(x)\land B(x))}$. The purpose is to reveal the logical structure of arguments. This makes it possible to use the precise rules of formal logic to assess whether these arguments are correct. It can also guide reasoning by arriving at new conclusions.

Many of the difficulties of the process are caused by vague or ambiguous expressions in natural language. For example, the English word "is" can mean that something exists, that it is identical to something else, or that it has a certain property. This contrasts with the precise nature of formal logic, which avoids such ambiguities. Natural language formalization is relevant to various fields in the sciences and humanities. It may play a key role for logic in general since it is needed to establish a link between many forms of reasoning and abstract logical systems. The use of informal logic is an alternative to formalization since it analyzes the cogency of ordinary language arguments in their original form. Natural language formalization is distinguished from logic translations that convert formulas from one logical system into another, for example, from modal logic to first-order logic. This form of logic translation is specifically relevant for logic programming and metalogic.

In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, giving rise to a formal system that combines the language with deduction rules. An element ${\displaystyle \phi \in T}$ of a deductively closed theory ${\displaystyle T}$ is then called a theorem of the theory. In many deductive systems there is usually a subset ${\displaystyle \Sigma \subseteq T}$ that is called "the set of axioms" of the theory ${\displaystyle T}$, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.

In predicate logic, existential generalization (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition. In first-order logic, it is often used as a rule for the existential quantifier (${\displaystyle \exists }$) in formal proofs.

Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail."

Propositional logic is a branch of logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called first-order propositional logic to contrast it with System F, but it should not be confused with first-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication, biconditional, and negation. Some sources include other connectives, as in the table below.

Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

In logic and computer science, the Davis–Putnam algorithm was developed by Martin Davis and Hilary Putnam for checking the validity of a first-order logic formula using a resolution-based decision procedure for propositional logic. Since the set of valid first-order formulas is recursively enumerable but not recursive, there exists no general algorithm to solve this problem. Therefore, the Davis–Putnam algorithm only terminates on valid formulas. Today, the term "Davis–Putnam algorithm" is often used synonymously with the resolution-based propositional decision procedure (Davis–Putnam procedure) that is actually only one of the steps of the original algorithm.

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.

Willard Van Orman Quine (/kwaɪn/ KWYNE; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". He was the Edgar Pierce Chair of Philosophy at Harvard University from 1956 to 1978.

Quine was a teacher of logic and set theory. He was famous for his position that first-order logic is the only kind worthy of the name, and developed his own system of mathematics and set theory, known as New Foundations. In the philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the Quine–Putnam indispensability argument, an argument for the reality of mathematical entities. He was the main proponent of the view that philosophy is not conceptual analysis, but continuous with science; it is the abstract branch of the empirical sciences. This led to his famous quip that "philosophy of science is philosophy enough". He led a "systematic attempt to understand science from within the resources of science itself" and developed an influential naturalized epistemology that tried to provide "an improved scientific explanation of how we have developed elaborate scientific theories on the basis of meager sensory input". He also advocated holism in science, known as the Duhem–Quine thesis.

In mathematical logic, a term is an arrangement of dependent/bound symbols that denotes a mathematical object within an expression/formula. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact.

A first-order term is recursively constructed from constant symbols, variable symbols, and function symbols.An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation.For example, ⁠${\displaystyle (x+1)*(x+1)}$⁠ is a term built from the constant 1, the variable x, and the binary function symbols ⁠${\displaystyle +}$⁠ and ⁠${\displaystyle *}$⁠; it is part of the atomic formula ⁠${\displaystyle (x+1)*(x+1)\geq 0}$⁠ which evaluates to true for each real-numbered value of x.

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.