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The following example in first-order logic (=) is a sentence. This sentence means that for every y, there is an x such that =. This sentence is true for positive real numbers, false for real numbers, and true for complex numbers. However, the formula
But when taken in its assembled form, the sentence a TRUTH. This is an example of the paradoxes that result from an impredicative definition—that is, when an object m has a property P, but the object m is defined in terms of property P. [22] The best advice for a rhetorician or one involved in deductive analysis is avoid impredicative ...
In mathematics education, a number sentence is an equation or inequality expressed using numbers and mathematical symbols. The term is used in primary level mathematics teaching in the US, [ 1 ] Canada, UK, [ 2 ] Australia, New Zealand [ 3 ] and South Africa.
In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x) that is not defined or specified (thus being a free variable), which leaves the statement undetermined.
Propositional logic, as currently studied in universities, is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered in evaluating the conditions for the truth of a sentence, or whether a sentence logically follows from some other sentence or group of sentences.
A closed formula, also ground formula or sentence, 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.
For example, consider the sentence "There exists x such that x is a philosopher." This sentence is seen as being true in an interpretation such that the domain of discourse consists of all human beings, and that the predicate "is a philosopher" is understood as "was the author of the Republic." It is true, as witnessed by Plato in that text.
are two different sentences that make the same statement. In either case, a statement is viewed as a truth bearer. Examples of sentences that are (or make) true statements: "Socrates is a man." "A triangle has three sides." "Madrid is the capital of Spain." Examples of sentences that are also statements, even though they aren't true: