55 Sentences With "domain of discourse" | Random Sentence Generator

The word quantifier in the introduction refers to a variable used in a domain of discourse, a collection of objects under discussion. In daily life, the domain of discourse could be 'apples', or 'persons', or even everything. In a more technical arena, the domain of discourse could be 'integers', say. The quantifier variable x, say, in the given domain of discourse can take on the 'value' or designate any object in the domain.

The size of the resulting Markov network thus depends strongly (exponentially) on the number of constants in the domain of discourse.

The domain of discourse is the set of considered objects. For example, one can take D to be the set of integer numbers. The interpretation of a function symbol is a function. For example, if the domain of discourse consists of integers, a function symbol f of arity 2 can be interpreted as the function that gives the sum of its arguments.

A formula \exists x \phi(x) is true according to M if there is some d in the domain of discourse such that \phi(c_d) holds. Here \phi(c_d) is the result of substituting cd for every free occurrence of x in φ. # Universal quantifiers (alternate). A formula \forall x \phi(x) is true according to M if, for every d in the domain of discourse, \phi(c_d) is true according to M. This alternate approach gives exactly the same truth values to all sentences as the approach via variable assignments.

An interpretation (or model) of a first-order formula specifies what each predicate means, and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in an interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.

A formula evaluates to true or false given an interpretation, and a variable assignment μ that associates an element of the domain of discourse with each variable. The reason that a variable assignment is required is to give meanings to formulas with free variables, such as y = x. The truth value of this formula changes depending on whether x and y denote the same individual. First, the variable assignment μ can be extended to all terms of the language, with the result that each term maps to a single element of the domain of discourse.

This indicates the importance of the domain of discourse, which specifies which values n can take.Further information on using domains of discourse with quantified statements can be found in the Quantification (logic) article. In particular, note that if the domain of discourse is restricted to consist only of those objects that satisfy a certain predicate, then for universal quantification this requires a logical conditional. For example, > For all composite numbers n, 2·n > 2 + n is logically equivalent to > For all natural numbers n, if n is composite, then 2·n > 2 + n.

Fictionalism consists in at least the following three theses: # Claims made within the domain of discourse are taken to be truth-apt; that is, true or false # The domain of discourse is to be interpreted at face value—not reduced to meaning something else # The aim of discourse in any given domain is not truth, but some other virtue(s) (e.g., simplicity, explanatory scope). Two important strands of fictionalism are modal fictionalism developed by Gideon Rosen, which states that possible worlds, regardless of whether they exist or not, may be a part of a useful discourse, and mathematical fictionalism advocated by Hartry Field.Field, Hartry, Science Without Numbers, Blackwell, 1980.

In other words, the symbol f is associated with the function I(f) which, in this interpretation, is addition. The interpretation of a constant symbol is a function from the one-element set D0 to D, which can be simply identified with an object in D. For example, an interpretation may assign the value I(c)=10 to the constant symbol c. The interpretation of an n-ary predicate symbol is a set of n-tuples of elements of the domain of discourse. This means that, given an interpretation, a predicate symbol, and n elements of the domain of discourse, one can tell whether the predicate is true of those elements according to the given interpretation.

A formula t_1 = t_2 is assigned true if t_1 and t_2 evaluate to the same object of the domain of discourse (see the section on equality below). # Logical connectives. A formula in the form eg \phi, \phi \rightarrow \psi, etc. is evaluated according to the truth table for the connective in question, as in propositional logic.

According to Lee (1999) an information model is a representation of concepts, relationships, constraints, rules, and operations to specify data semantics for a chosen domain of discourse. It can provide sharable, stable, and organized structure of information requirements for the domain context.Y. Tina Lee (1999). "Information modeling from design to implementation" National Institute of Standards and Technology.

"For all instances x of 'winged creatures', 'x is a mammal'" has a truth value of "falsity"; "Flying insects are mammals" is false); : However over the broad domain of discourse "all winged creatures" (e.g. "birds" + "flying insects" + "flying squirrels" + "bats") we can assert ∃xf(x) (read: "There exists at least one winged creature that is a mammal'"; it yields a truth value of "truth" because the objects x can come from the category "bats" and perhaps "flying squirrels" (depending on how we define "winged"). But the formula yields "falsity" when the domain of discourse is restricted to "flying insects" or "birds" or both "insects" and "birds". Kleene remarks that "the predicate calculus (without or with equality) fully accomplishes (for first order theories) what has been conceived to be the role of logic" (Kleene 1967:322).

Peirce notated predicates using intuitive English phrases; the standard notation of contemporary logic, capital Latin letters, may also be employed. A dot asserts the existence of some individual in the domain of discourse. Multiple instances of the same object are linked by a line, called the "line of identity". There are no literal variables or quantifiers in the sense of first-order logic.

In classical logic there are theorems that clearly presuppose that there is something in the domain of discourse. Consider the following classically valid theorems. :1. \forall xA \Rightarrow \exists xA :2. \forall x \forall rA(x) \Rightarrow \forall rA(r) :3. \forall rA(r) \Rightarrow \exists xA(x) A valid scheme in the theory of equality which exhibits the same feature is :4.

The following rules are used to make this assignment: # Variables. Each variable x evaluates to μ(x) # Functions. Given terms t_1, \ldots, t_n that have been evaluated to elements d_1, \ldots, d_n of the domain of discourse, and a n-ary function symbol f, the term f(t_1, \ldots, t_n) evaluates to (I(f))(d_1,\ldots,d_n). Next, each formula is assigned a truth value.

An alt= An information model in software engineering is a representation of concepts and the relationships, constraints, rules, and operations to specify data semantics for a chosen domain of discourse. Typically it specifies relations between kinds of things, but may also include relations with individual things. It can provide sharable, stable, and organized structure of information requirements or knowledge for the domain context.Y. Tina Lee (1999).

Any grouping together of mathematical, abstract, or concrete objects, however formed, is a collection, a synonym for what other set theories refer to as a class. The things that make up a collection are called elements or members. A common instance of a collection is the domain of discourse of a first-order theory. All sets are collections, but there are collections that are not sets.

This, however, takes an entirely different scope from (3). As in (3a) a necessarily distinct set of cats is everywhere in the domain of discourse, whereas in (3) the set at each place is not necessarily unique. Finally (4) seems to completely resist traditional quantificational analysis, acting on the entire set of objects, not on any individual member. Several theories have been put forth to explain this discrepancy.

Philosophers can classify ontologies in various ways, using criteria such as the degree of abstraction and field of application: #Upper ontology: concepts supporting development of an ontology, meta-ontology. #Domain ontology: concepts relevant to a particular topic, domain of discourse, or area of interest, for example, to information technology or to computer. languages, or to particular branches of science. #Interface ontology: concepts relevant to the juncture of two disciplines.

The smallest possible domains have those variables that can only have two values, also called binary (or dichotomous) variables. Bigger domains have non-dichotomous variables and the ones with a higher level of measurement. (See also domain of discourse.) Semantically, greater precision can be obtained when considering an object's characteristics by distinguishing 'attributes' (characteristics that are attributed to an object) from 'traits' (characteristics that are inherent to the object).

While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification. A predicate takes an entity or entities in the domain of discourse as input while outputs are either True or False. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated, and might be denoted, for example, by variables such as p and q.

Ordinary first-order interpretations have a single domain of discourse over which all quantifiers range. Many-sorted first-order logic allows variables to have different sorts, which have different domains. This is also called typed first-order logic, and the sorts called types (as in data type), but it is not the same as first-order type theory. Many-sorted first-order logic is often used in the study of second-order arithmetic.

This contrasts with kinds possessing an attribute determined by the predicate. So "cats are everywhere" holds true if and only if a stage exists of individuals of the kind "cat" who are everywhere relevant in the domain of discourse. Crucially, it is not the case that "cats" as a kind possess the property of being "everywhere". This creates an ontological distinction between the two predicate types, i(ndividual)-level and s(tage)-level.

All of the above "systems of logic" are considered to be "classical" meaning propositions and predicate expressions are two-valued, with either the truth value "truth" or "falsity" but not both(Kleene 1967:8 and 83). While intuitionistic logic falls into the "classical" category, it objects to extending the "for all" operator to the Law of Excluded Middle; it allows instances of the "Law", but not its generalization to an infinite domain of discourse.

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic.Shapiro (1991) and Hinman (2005) give complete introductions to the subject, with full definitions. 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, also quantifies over relations.

An interpretation of a first-order language assigns a denotation to each non-logical symbol in that language. It also determines a domain of discourse that specifies the range of the quantifiers. The result is that each term is assigned an object that it represents, each predicate is assigned a property of objects, and each sentence is assigned a truth value. In this way, an interpretation provides semantic meaning to the terms, the predicates, and formulas of the language.

In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a propositional function 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.

All of these types of variables can be quantified. There are two kinds of interpretations commonly employed for higher-order logic. Full semantics require that, once the domain of discourse is satisfied, the higher-order variables range over all possible elements of the correct type (all subsets of the domain, all functions from the domain to itself, etc.). Thus the specification of a full interpretation is the same as the specification of a first-order interpretation.

The following axioms are known as the basic axioms, or sometimes the Robinson axioms. The resulting first-order theory, known as Robinson arithmetic, is essentially Peano arithmetic without induction. The domain of discourse for the quantified variables is the natural numbers, collectively denoted by N, and including the distinguished member \ 0, called "zero." The primitive functions are the unary successor function, denoted by prefix S, and two binary operations, addition and multiplication, denoted by infix "+" and " \cdot", respectively.

First-order logic is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse. Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality.

In the study of set theory and category theory, it is sometimes useful to consider structures in which the domain of discourse is a proper class instead of a set. These structures are sometimes called class models to distinguish them from the "set models" discussed above. When the domain is a proper class, each function and relation symbol may also be represented by a proper class. In Bertrand Russell's Principia Mathematica, structures were also allowed to have a proper class as their domain.

Here we informally survey some of those operations. For simplicity we assume that the domain of discourse is always the set of all natural numbers: {0,1,2,...}. The operation ¬ of negation ("not") switches the roles of the two players, turning moves and wins by the machine into those by the environment, and vice versa. For instance, if Chess is the game of chess (but with ties ruled out) from the white player's perspective, then ¬Chess is the same game from the black player's perspective.

In the metavariable sense, a predicate variable can be used to define an axiom schema. Predicate variables should be distinguished from predicate constants, which could be represented either with a different (exclusive) set of predicate letters, or by their own symbols which really do have their own specific meaning in their domain of discourse: e.g. =, \ \in , \ \le,\ <, \ \sub,... . If letters are used for predicate constants as well as for predicate variables, then there has to be a way of distinguishing between them.

This assignment can be uniquely extended to an assignment of truth values to all propositional formulas. In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols. Formulas are built out of atomic formulas using logical connectives and quantifiers. A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language.

Propositional variables can be considered nullary predicates in first order logic, because there are no object variables such as x and y attached to predicate letters such as Px and xRy. The internal structure of propositional variables contains predicate letters such as P and Q, in association with individual variables (e.g., x, y), individual constants such as a and b (singular terms from a domain of discourse D), ultimately taking a form such as Pa, aRb.(or with parenthesis, P(11) and R(1, 3)).

However, the study of methicillin as its name derived from penicillin historically might best be described, according to Elk, as orismologic. Elk's use of the words orismology and terminology echo the historical notions of Kirby and Spence rather than modern disciplinary senses of these words. Orismology does not entail etymology, but nomenography may well delve into etymological analysis to construct such neologisms as are needed to satisfy logical requirements for terms within a domain of discourse. Orismology and nomenography are studies overlapping both terminology and specialized lexicography.

There is a second common approach to defining truth values that does not rely on variable assignment functions. Instead, given an interpretation M, one first adds to the signature a collection of constant symbols, one for each element of the domain of discourse in M; say that for each d in the domain the constant symbol cd is fixed. The interpretation is extended so that each new constant symbol is assigned to its corresponding element of the domain. One now defines truth for quantified formulas syntactically, as follows: # Existential quantifiers (alternate).

There are several different conventions for using equality (or identity) in first-order logic. The most common convention, known as first-order logic with equality, includes the equality symbol as a primitive logical symbol which is always interpreted as the real equality relation between members of the domain of discourse, such that the "two" given members are the same member. This approach also adds certain axioms about equality to the deductive system employed. These equality axioms are:Fitting, M., First-Order Logic and Automated Theorem Proving (Berlin/Heidelberg: Springer, 1990), pp. 198–200.

This particular predicate calculus is "restricted to the first order". To the propositional calculus it adds two special symbols that symbolize the generalizations "for all" and "there exists (at least one)" that extend over the domain of discourse. The calculus requires only the first notion "for all", but typically includes both: (1) the notion "for all x" or "for every x" is symbolized in the literature as variously as (x), ∀x, ∏x etc., and the (2) notion of "there exists (at least one x)" variously symbolized as Ex, ∃x.

For example, the second-order sentence \forall P\,\forall x (Px \lor eg Px) says that for every formula P, and every individual x, either Px is true or not(Px) is true (this is the principle of bivalence). Second-order logic also includes quantification over sets, functions, and other variables as explained in the section Syntax and fragments. 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.

If a sentence φ evaluates to True under a given interpretation M, one says that M satisfies φ; this is denoted M \vDash \varphi. A sentence is satisfiable if there is some interpretation under which it is true. Satisfiability of formulas with free variables is more complicated, because an interpretation on its own does not determine the truth value of such a formula. The most common convention is that a formula with free variables is said to be satisfied by an interpretation if the formula remains true regardless which individuals from the domain of discourse are assigned to its free variables.

In another example, the famous theory of "six degrees of separation" between people tacitly presumes that the domain of discourse is the set of people alive at any one time. The number of degrees of separation between Albert Einstein and Alexander the Great is almost certainly greater than 30Einstein and Alexander the Great lived 2202 years apart. Assuming an age difference of 70 years between any two connected people in the chain that connects the two, this would necessitate at least 32 connections between Einstein and Alexander the Great. and this network does not have small-world properties.

All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, in addition to the axioms given below (although he notes that he does so only "for emphasis").. Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC is typically formalized, the domain of discourse must be nonempty. Hence, it is a logical theorem of first-order logic that something exists — usually expressed as the assertion that something is identical to itself, \exists x ( x = x ).

In predicate logic, what is described in layman's terms as "something" can more specifically be regarded as existential quantification, that is, the predication of a property or relation to at least one member of the domain. It is a type of quantifier, a logical constant which is interpreted as "there exists," "there is at least one," or "for some." It expresses that a propositional function can be satisfied by at least one member of a domain of discourse. In other terms, it is the predication of a property or relation to at least one member of the domain.

One introduces into the single- sorted theory a unary predicate symbol for each sort in the many-sorted theory, and adds an axiom saying that these unary predicates partition the domain of discourse. For example, if there are two sorts, one adds predicate symbols P_1(x) and P_2(x) and the axiom :\forall x ( P_1(x) \lor P_2(x)) \land \lnot \exists x (P_1(x) \land P_2(x)). Then the elements satisfying P_1 are thought of as elements of the first sort, and elements satisfying P_2 as elements of the second sort. One can quantify over each sort by using the corresponding predicate symbol to limit the range of quantification.

For example, (4) is true if and only if the kind "cats" possess the attribute "common" in the domain of discourse. The perceived difference in what level of generic quantification applies is then a pragmatic property of the predicate, determined by what is perceived to be necessary for the statement to hold true. In order for the existential readings in sentences like (3) to hold, another semantic object is defined called "stages". These represent locations in time and space, and are created to reconcile the fact that sentences like (3) only hold true if there exists a specific spatio-temporal place in which the predicate applies.

Feminist Rhetoric is the study of effectively persuading others to learn about the movement of women who fight for social, political and economic equality of genders. The Another main idea in feminist rhetoric is making the point of view, and the rhetorical canon, more expansive than what lies within the United States' domain of discourse. Feminist rhetoric scholars ask the questions of how rhetoric, writing studies, and social change intersect, or may be influenced by politics, the economy, religions, cultures, and education. A key term used in this field is "transnationality", defined as the culture of one nation moving through borders to another nation.

As stated above, a first-order interpretation is usually required to specify a nonempty set as the domain of discourse. The reason for this requirement is to guarantee that equivalences such as ::(\phi \lor \exists x \psi) \leftrightarrow \exists x (\phi \lor \psi), where x is not a free variable of φ, are logically valid. This equivalence holds in every interpretation with a nonempty domain, but does not always hold when empty domains are permitted. For example, the equivalence :[\forall y (y = y) \lor \exists x ( x = x)] \equiv \exists x [ \forall y ( y = y) \lor x = x] fails in any structure with an empty domain.

The study of the interpretations of formal languages is called formal semantics. What follows is a description of the standard or Tarskian semantics for first-order logic. (It is also possible to define game semantics for first-order logic, but aside from requiring the axiom of choice, game semantics agree with Tarskian semantics for first-order logic, so game semantics will not be elaborated herein.) The domain of discourse D is a nonempty set of "objects" of some kind. Intuitively, a first-order formula is a statement about these objects; for example, \exists x P(x) states the existence of an object x such that the predicate P is true where referred to it.

The Löwenheim–Skolem theorem shows that if a first-order theory of cardinality λ has an infinite model, then it has models of every infinite cardinality greater than or equal to λ. One of the earliest results in model theory, it implies that it is not possible to characterize countability or uncountability in a first-order language with a countable signature. That is, there is no first-order formula φ(x) such that an arbitrary structure M satisfies φ if and only if the domain of discourse of M is countable (or, in the second case, uncountable). The Löwenheim–Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic.

That is, the equality relation may now be interpreted by an arbitrary equivalence relation on the domain of discourse that is congruent with respect to the functions and relations of the interpretation. When this second convention is followed, the term normal model is used to refer to an interpretation where no distinct individuals a and b satisfy a = b. In first-order logic with equality, only normal models are considered, and so there is no term for a model other than a normal model. When first-order logic without equality is studied, it is necessary to amend the statements of results such as the Löwenheim–Skolem theorem so that only normal models are considered.

Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion. Deductive reasoning goes in the same direction as that of the conditionals, and links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. Deductive reasoning ("top-down logic") contrasts with inductive reasoning ("bottom-up logic") in the following way; in deductive reasoning, a conclusion is reached reductively by applying general rules which hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion(s) is left (there is no epistemic uncertainty; i.e.

In philosophy, a proposition is the meaning of a declarative sentence, where "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the non- linguistic bearer of truth or falsity which makes any sentence that expresses it either true or false. In mathematics the term proposition refers to a statement that may or may not be true, whilst the term axiom refers to a statement that is taken to be true within a domain of discourse. While the term "proposition" may sometimes be used in everyday language to refer to a linguistic statement which can be either true or false, the technical philosophical term, which differs from the mathematical usage, refers exclusively to the non-linguistic meaning behind the statement.

Unlike deductive reasoning, it does not rely on universals holding over a closed domain of discourse to draw conclusions, so it can be applicable even in cases of epistemic uncertainty (technical issues with this may arise however; for example, the second axiom of probability is a closed-world assumption). Another crucial difference between these two types of argument is that deductive certainty is impossible in non-axiomatic systems such as reality, leaving inductive reasoning as the primary route to (probabilistic) knowledge of such systems. Given that "if A is true then that would cause B, C, and D to be true", an example of deduction would be "A is true therefore we can deduce that B, C, and D are true". An example of induction would be "B, C, and D are observed to be true therefore A might be true".

The restriction is that the generalization "for all" applies only to the variables (objects x, y, z etc. drawn from the domain of discourse) and not to functions, in other words the calculus will permit ∀xf(x) ("for all creatures x, x is a bird") but not ∀f∀x(f(x)) [but if "equality" is added to the calculus it will permit ∀f:f(x); see below under Tarski]. Example: : Let the predicate "function" f(x) be "x is a mammal", and the subject-domain (or universe of discourse) (cf Kleene 1967:84) be the category "bats": : The formula ∀xf(x) yields the truth value "truth" (read: "For all instances x of objects 'bats', 'x is a mammal'" is a truth, i.e. "All bats are mammals"); : But if the instances of x are drawn from a domain "winged creatures" then ∀xf(x) yields the truth value "false" (i.e.