Ambiguity is a type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty. It is thus an attribute of any idea or statement whose intended meaning cannot be definitively resolved according to a rule or process with a finite number of steps.
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase, or another symbol. In the case of a word, the word's definition often implies an intension. For instance, the intensions of the word plant include properties such as "being composed of cellulose", "alive", and "organism", among others. A comprehension is the collection of all such intensions.
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
Referential transparency and referential opacity are properties of parts of computer programs. An expression is called referentially transparent if it can be replaced with its corresponding value without changing the program's behavior. This requires that the expression be pure, that is to say the expression value must be the same for the same inputs and its evaluation must have no side effects. An expression that is not referentially transparent is called referentially opaque.
The semicolon or semi-colon; is a symbol commonly used as orthographic punctuation. In the English language, a semicolon can be used between two closely related independent clauses, provided they are not already joined by a coordinating conjunction. Semicolons can also be used in place of commas to separate the items in a list, particularly when the elements of that list contain commas. The semicolon is likely the least understood of the standard marks, and so it is not used by many English speakers.
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.
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
In analytic philosophy, a definite description is a denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun. The definite description is proper if X applies to a unique individual or object. For example: "the first person in space" and "the 42nd President of the United States of America", are proper. The definite descriptions "the person in space" and "the Senator from Ohio" are improper because the noun phrase X applies to more than one thing, and the definite descriptions "the first man on Mars" and "the Senator from some Country" are improper because X applies to nothing. Improper descriptions raise some difficult questions about the law of excluded middle, denotation, modality, and mental content.
The open-question argument is a philosophical argument put forward by British philosopher G. E. Moore in §13 of Principia Ethica (1903), to refute the equating of the property of goodness with some non-moral property, X, whether natural or supernatural. That is, Moore's argument attempts to show that no moral property is identical to a natural property. The argument takes the form of a syllogism modus tollens:
In philosophy and mathematics, a logical form of a syntactic expression is a precisely-specified semantic version of that expression in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.
In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.
In the philosophy of language, the descriptivist theory of proper names is the view that the meaning or semantic content of a proper name is identical to the descriptions associated with it by speakers, while their referents are determined to be the objects that satisfy these descriptions. Bertrand Russell and Gottlob Frege have both been associated with the descriptivist theory, which is sometimes called the Frege–Russell view.
In philosophical logic, a slingshot argument is one of a group of arguments claiming to show that all true sentences stand for the same thing.
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.
Avalency refers to the property of a predicate, often a verb, taking no arguments. Valency refers to how many and what kinds of arguments a predicate licenses—i.e. what arguments the predicate selects grammatically. Avalent verbs are verbs which have no valency, meaning that they have no logical arguments, such as subject or object. Languages known as pro-drop or null-subject languages do not require clauses to have an overt subject when the subject is easily inferred, meaning that a verb can appear alone. However, non-null-subject languages such as English require a pronounced subject in order for a sentence to be grammatical. This means that the avalency of a verb is not readily apparent, because, despite the fact that avalent verbs lack arguments, the verb nevertheless has a subject. According to some, avalent verbs may have an inserted subject, which is syntactically required, yet semantically meaningless, making no reference to anything that exists in the real world. An inserted subject is referred to as a pleonastic, or expletive it. Because it is semantically meaningless, pleonastic it is not considered a true argument, meaning that a verb with this it as the subject is truly avalent. However, others believe that it represents a quasi-argument, having no real-world referent, but retaining certain syntactic abilities. Still others consider it to be a true argument, meaning that it is referential, and not merely a syntactic placeholder. There is no general consensus on how it should be analyzed under such circumstances, but determining the status of it as a non-argument, a quasi-argument, or a true argument, will help linguists to understand what verbs, if any, are truly avalent. A common example of such verbs in many languages is the set of verbs describing weather. In providing examples for the avalent verbs below, this article must assume the analysis of pleonastic it, but will delve into the other two analyses following the examples.
Logical consequence is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.
Logic is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments.
