Formulation is a term used in various senses in various applications, both the material and the abstract or formal . Its fundamental meaning is the putting together of components in appropriate relationships or structures, according to a formula . Etymologically formula is the diminutive of the Latin forma , meaning shape. In that sense a formulation is created according to the standard for the product.
28-455: Disciplines in which one might use the word formulation in the abstract sense include logic , mathematics , linguistics , legal theory , and computer science . For details, see the related articles. In more material senses the concept of formulation appears in the physical sciences , such as physics , chemistry , and biology . It also is ubiquitous in industry , engineering and medicine , especially pharmaceutics . In pharmacy ,
56-547: 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". A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation . Two key uses of formulas are in propositional logic and predicate logic. A key use of formulas
84-608: A formula comes in several parts. First, the set of terms is defined recursively. Terms, informally, are expressions that represent objects from the domain of discourse . The next step is to define the atomic formulas . Finally, the set of formulas is defined to be the smallest set containing the set of atomic formulas such that the following holds: If a formula has no occurrences of ∃ x {\displaystyle \exists x} or ∀ x {\displaystyle \forall x} , for any variable x {\displaystyle x} , then it
112-502: A formula, create a formulation . Some components impart specific properties to the formulation when it is put into use. For example, certain components (polymers) are used in paint formulations to achieve deforming or levelling properties. Some components of a formulation may only be active in particular applications. A formulation may be created for any of the following purposes: Formula (mathematical logic) In mathematical logic , propositional logic and predicate logic ,
140-402: A formula. The formulas of propositional calculus , also called propositional formulas , are expressions such as ( A ∧ ( B ∨ C ) ) {\displaystyle (A\land (B\lor C))} . Their definition begins with the arbitrary choice of a set V of propositional variables . The alphabet consists of the letters in V along with the symbols for
168-440: A formulation is a mixture or a structure such as a capsule , tablet , or an emulsion , prepared according to a specific procedure (called a "formula"). Formulations are a very important aspect of creating medicines, since they are essential to ensuring that the active part of the drug is delivered to the correct part of the body, in the right concentration, and at the right rate (not too fast and not too slowly). A good example
196-406: A universe of discourse of character-string theory, or concatenation theory . Peirce's original words are the following. A common mode of estimating the amount of matter in a ... printed book is to count the number of words. There will ordinarily be about twenty 'thes' on a page, and, of course, they count as twenty words. In another sense of the word 'word,' however, there is but one word 'the' in
224-758: Is a drug delivery system that exploits supersaturation . They also need to have an acceptable taste (in the case of pills, tablets or syrups), last long enough in storage still to be safe and effective when used, and be sufficiently stable both physically and chemically to be transported from where they are manufactured to the eventual consumer. Competently designed formulations for particular applications are safer, more effective, and more economical than any of their components used singly. Formulations are commercially produced for drugs , cosmetics , coatings , dyes , alloys , cleaning agents , foods , lubricants , fuels , fertilisers , pesticides and many others. Components (also called ingredients), when mixed according to
252-435: Is a formula in which there are no free occurrences of any variable . If A is a formula of a first-order language in which the variables v 1 , …, v n have free occurrences, then A preceded by ∀ v 1 ⋯ ∀ v n is a universal closure of A . In earlier works on mathematical logic (e.g. by Church ), formulas referred to any strings of symbols and among these strings, well-formed formulas were
280-454: Is a formula, because it is grammatically correct. The sequence of symbols is not a formula, because it does not conform to the grammar. A complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules (akin to the standard mathematical order of operations ) are assumed among the operators, making some operators more binding than others. For example, assuming
308-440: Is called quantifier-free . An existential formula is a formula starting with a sequence of existential quantification followed by a quantifier-free formula. An atomic formula is a formula that contains no logical connectives nor quantifiers , or equivalently a formula that has no strict subformulas. The precise form of atomic formulas depends on the formal system under consideration; for propositional logic , for example,
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#1732851902508336-403: Is in propositional logic and predicate logic such as first-order logic . In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven. Although
364-399: Is the difference between a class (type) of objects and the individual instances (tokens) of that class. Since each type may be instantiated by multiple tokens, there are generally more tokens than types of an object. For example, the sentence " A rose is a rose is a rose " contains three word types: three word tokens of the type a , two word tokens of the type is, and three word tokens of
392-471: The propositional connectives and parentheses "(" and ")", all of which are assumed to not be in V . The formulas will be certain expressions (that is, strings of symbols) over this alphabet. The formulas are inductively defined as follows: This definition can also be written as a formal grammar in Backus–Naur form , provided the set of variables is finite: Using this grammar, the sequence of symbols
420-451: The English language; and it is impossible that this word should lie visibly on a page, or be heard in any voice .... Such a ... Form, I propose to term a Type. A Single ... Object ... such as this or that word on a single line of a single page of a single copy of a book, I will venture to call a Token. .... In order that a Type may be used, it has to be embodied in a Token which shall be a sign of
448-450: The atomic formulas are the propositional variables . For predicate logic , the atoms are predicate symbols together with their arguments, each argument being a term . According to some terminology, an open formula is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers. This is not to be confused with a formula which is not closed. A closed formula , also ground formula or sentence ,
476-457: The concrete string representation of formulas (using this or that symbol for connectives and quantifiers, using this or that parenthesizing convention , using Polish or infix notation, etc.) as a mere notational problem. The expression "well-formed formulas" (WFF) also crept into popular culture. WFF is part of an esoteric pun used in the name of the academic game " WFF 'N PROOF : The Game of Modern Logic", by Layman Allen, developed while he
504-413: The precedence (from most binding to least binding) 1. ¬ 2. → 3. ∧ 4. ∨. Then the formula may be abbreviated as This is, however, only a convention used to simplify the written representation of a formula. If the precedence was assumed, for example, to be left-right associative, in following order: 1. ¬ 2. ∧ 3. ∨ 4. →, then
532-440: The same formula above (without parentheses) would be rewritten as The definition of a formula in first-order logic Q S {\displaystyle {\mathcal {QS}}} is relative to the signature of the theory at hand. This signature specifies the constant symbols, predicate symbols, and function symbols of the theory at hand, along with the arities of the function and predicate symbols. The definition of
560-510: The same formula may be written more than once, and a formula might in principle be so long that it cannot be written at all within the physical universe. Formulas themselves are syntactic objects. They are given meanings by interpretations. For example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not be interpreted, however, to be considered solely as
588-422: The same while the class of its tokens is continually gaining new members and losing old members. In typography , the type–token distinction is used to determine the presence of a text printed by movable type : The defining criteria which a typographic print has to fulfill is that of the type identity of the various letter forms which make up the printed text. In other words: each letter form which appears in
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#1732851902508616-433: The sentence " the bicycle is becoming more popular " the word bicycle represents the abstract concept of bicycles and this abstract concept is a type, whereas in the sentence " the bicycle is in the garage ", it represents a particular object and this particular object is a token. Similarly, the word type 'letter' uses only four letter types: L , E , T and R . Nevertheless, it uses both E and T twice. One can say that
644-407: The strings that followed the formation rules of (correct) formulas. Several authors simply say formula. Modern usages (especially in the context of computer science with mathematical software such as model checkers , automated theorem provers , interactive theorem provers ) tend to retain of the notion of formula only the algebraic concept and to leave the question of well-formedness , i.e. of
672-483: The term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence of symbols being expressed, with the marks being a token instance of formula. This distinction between the vague notion of "property" and the inductively-defined notion of well-formed formula has roots in Weyl's 1910 paper "Uber die Definitionen der mathematischen Grundbegriffe". Thus
700-421: The text has to be shown as a particular instance ("token") of one and the same type which contains a reverse image of the printed letter . The distinctions between using words as types or tokens were first made by American logician and philosopher Charles Sanders Peirce in 1906 using terminology that he established. Peirce's type–token distinction applies to words, sentences, paragraphs and so on: to anything in
728-438: The type rose . The distinction is important in disciplines such as logic , linguistics , metalogic , typography , and computer programming . Beware, in what follows, the type "rose" may refer to appearances of the word in language, or instances of physical roses in material reality. The type–token distinction separates types (abstract descriptive concepts) from tokens (objects that instantiate concepts). For example, in
756-426: The word type 'letter' has six letter tokens, with two tokens each of the letter types E and T . Whenever a word type is inscribed, the number of letter tokens created equals the number of letter occurrences in the word type. Some logicians consider a word type to be the class of its tokens. Other logicians counter that the word type has a permanence and constancy not found in the class of its tokens. The type remains
784-579: Was at Yale Law School (he was later a professor at the University of Michigan ). The suite of games is designed to teach the principles of symbolic logic to children (in Polish notation ). Its name is an echo of whiffenpoof , a nonsense word used as a cheer at Yale University made popular in The Whiffenpoof Song and The Whiffenpoofs . Type-token distinction The type–token distinction
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