Formal language in the context of Substitution (logic)

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.

Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like ${\displaystyle \land }$ (and) or ${\displaystyle \to }$ (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts.

A natural language or ordinary language is any spoken language or signed language used organically in a human community, first emerging without conscious premeditation and subject to: replication across generations of people in the community, regional expansion or contraction, and gradual internal and structural changes. The vast majority of languages in the world are natural languages. As a category, natural language includes both standard dialects (ones with high social prestige) as well as nonstandard or vernacular dialects. Even an official language with a regulating academy such as Standard French, overseen by the Académie Française, is still classified as a natural language (e.g. in the field of natural language processing), as its prescriptive aspects do not make it regulated enough to be considered a constructed or controlled natural language. Linguists broadly consider writing to be a static visual representation of a particular natural language, though, in many cases in highly literate modern societies, writing itself can also be considered natural language.

Excluded from the definition of natural language are: artificial and constructed languages, such as those developed for works of fiction; languages of formal logic, such as those in computer programming; and non-human communication systems in nature, such as whale vocalizations or honey bees' waggle dance. The academic consensus is that particular key features prevent animal communication systems from being classified as languages at all. Certain human communication or linguistic systems with no native speakers, as sometimes used in cross-cultural contexts, are also not natural languages.

In philosophy and linguistics, the denotation of a word or expression is its strictly literal meaning. For instance, the English word "warm" denotes the property of having high temperature. Denotation is contrasted with other aspects of meaning, in particular connotation. For instance, the word "warm" may evoke calmness, coziness, or kindness (as in the warmth of someone's personality) but these associations are not part of the word's denotation. Similarly, an expression's denotation is separate from pragmatic inferences it may trigger. For instance, describing something as "warm" often implicates that it is not hot, but this is once again not part of the word's denotation.

Denotation plays a major role in several fields. Within semantics and philosophy of language, denotation is studied as an important aspect of meaning. In mathematics and computer science, assignments of denotations are assigned to expressions are a crucial step in defining interpreted formal languages. The main task of formal semantics is to reverse engineer the computational system which assigns denotations to expressions of natural languages.

Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.

Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e., without regard to any personal interpretations of the sentences) the sentence must be true if every sentence in the set is true.

In logic, the logical form of a statement is a precisely specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.

The logical form of an argument is called the argument form of the argument.

An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.

The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of an object language. For example, an interpretation function could take the predicate symbol ${\displaystyle T}$ and assign it the extension ${\displaystyle \{(\mathrm {a} )\}}$. All our interpretation does is assign the extension ${\displaystyle \{(\mathrm {a} )\}}$ to the non-logical symbol ${\displaystyle T}$, and does not make a claim about whether ${\displaystyle T}$ is to stand for tall and ${\displaystyle \mathrm {a} }$ for Abraham Lincoln. On the other hand, an interpretation does not have anything to say about logical symbols, e.g. logical connectives "${\displaystyle \mathrm {and} }$", "${\displaystyle \mathrm {or} }$" and "${\displaystyle \mathrm {not} }$". Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.

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.

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 mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.

The abbreviation wff is pronounced "woof", or sometimes "wiff", "weff", or "whiff".

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 mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other.As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954.Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.

Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit to classical mathematics.This has prompted the comment that "if proof theory is about the sacred, then model theory is about the profane".The applications of model theory to algebraic and Diophantine geometry reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

In logic, a logical constant or constant symbol of a language ${\displaystyle {\mathcal {L}}}$ is a symbol that has the same semantic value under every interpretation of ${\displaystyle {\mathcal {L}}}$. Two important types of logical constants are logical connectives and quantifiers. The equality predicate (usually written '=') is also treated as a logical constant in many systems of logic.

One of the fundamental questions in the philosophy of logic is "What is a logical constant?"; that is, what special feature of certain constants makes them logical in nature?