Symbol (formal) in the context of "Universally quantified"

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 predicate logic, an existential quantification is a type of quantifier which asserts the existence of an object with a given property. It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)"), read as "there exists", "there is at least one", or "for some". Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain. Some sources use the term existentialization to refer to existential quantification.

Quantification in general is covered in the article on quantification (logic). The existential quantifier is encoded as U+2203 ∃ THERE EXISTS in Unicode, and as \exists in LaTeX and related formula editors.

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".

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?

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 mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.

It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("∀x", "∀(x)", or sometimes by "(x)" alone). Universal quantification is distinct from existential quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain.

In formal language theory, an alphabet, often called a vocabulary in the context of terminal and nonterminal symbols, is a non-empty set of indivisible symbols/characters/glyphs, typically thought of as representing letters, characters, digits, phonemes, or even words. The definition is used in a diverse range of fields including logic, mathematics, computer science, and linguistics. An alphabet may have any cardinality ("size") and, depending on its purpose, may be finite (e.g., the alphabet of letters "a" through "z"), countable (e.g., ${\displaystyle \{v_{1},v_{2},\ldots \}}$), or even uncountable (e.g., ${\displaystyle \{v_{x}:x\in \mathbb {R} \}}$).

Strings, also known as "words" or "sentences", over an alphabet are defined as a sequence of the symbols from the alphabet set. For example, the alphabet of lowercase letters "a" through "z" can be used to form English words like "iceberg" while the alphabet of both upper and lower case letters can also be used to form proper names like "Wikipedia". A common alphabet is {0,1}, the binary alphabet, and "00101111" is an example of a binary string. Infinite sequences of symbols may be considered as well (see Omega language).