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Note that the last one (carry the piano together) can be made non-collective by removing the word together. Quantifiers differ with respect to whether or not they can be the subject of a collective predicate. For example, quantifiers formed with all the can, while ones formed with every or each cannot. All the students formed a line.
As an example, the only difference in the definition of uniform continuity and (ordinary) continuity is the order of quantifications. First order quantifiers approximate the meanings of some natural language quantifiers such as "some" and "all".
Example requires a quantifier over predicates, which cannot be implemented in single-sorted first-order logic: Zj → ∃X(Xj∧Xp). Quantification over properties Santa Claus has all the attributes of a sadist. Example requires quantifiers over predicates, which cannot be implemented in single-sorted first-order logic: ∀X(∀x(Sx → Xx) → ...
Sentences are then built up out of atomic sentences by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory.
In these examples, the predicate is tall and the QNPs are a girl, many girls, every girl and no girl. The logical meaning of these sentences indicates that the property of being tall is attributed to some form of the QNP referring to girl. Along with the QNP and the predicate, there is also an inference of truth value.
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". 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 ...
The truth value of an arbitrary sentence is then defined inductively using the T-schema, which is a definition of first-order semantics developed by Alfred Tarski. The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example, φ ∧ ψ is satisfied if and only if both φ and ψ are satisfied.
In linguistics and grammar, a sentence is a linguistic expression, such as the English example "The quick brown fox jumps over the lazy dog." In traditional grammar , it is typically defined as a string of words that expresses a complete thought, or as a unit consisting of a subject and predicate .