In logic Logic is the study of reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic and philosophy Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. It is distinguished from other ways of addressing fundamental questions by its critical, generally systematic approach and its reliance on rational argument. The word "philosophy" comes from the, the term proposition (from the word "proposal") refers to both (a) the "content" or "meaning" The nature of meaning, its definition, elements, and types, was discussed by Aristotle, Augustine, and Aquinas . According to them 'meaning is a relationship between two sorts of things: signs and the kinds of things they mean (intend, express or signify)'.[citation needed] One term in the relation of meaning necessarily causes something else to of a meaningful declarative sentence In the field of linguistics, a sentence is an expression in natural language, often defined to indicate a grammatical and lexical unit consisting of one or more words that represent distinct concepts. A sentence can include words grouped meaningfully to express a statement, question, exclamation, request or command or (b) the pattern of symbols A symbol is an idea, abstraction or concept, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in, marks, or sounds that make up a meaningful declarative sentence. The meaning of a proposition includes that it has the quality or property of being either true Truth can have a variety of meanings, from the state of being the case, being in accord with a particular fact or reality, being in accord with the body of real things, events, actuality, or fidelity to an original or to a standard, truth "behind" everything, the ontological truth. In archaic usage it could be fidelity, constancy or or false Falsity or falsehood is a perversion of truth originating in the deceitfulness of one party, and culminating in the damage of another party. Falsity is also a measure of the quality or extent of the falseness of something, while a falsehood may also mean simply an incorrect (false) statement, independent of any intention to deceive, and as such propositions are called truthbearers Truthbearer is a term used to designate entities that are either true or false and nothing else. The acceptance that some things are true while others are false raises the question of the nature of such things. Since there is no agreement on the matter, the term truthbearer is used to be neutral among the various theories. Candidates truthbearers.

The existence of propositions in the abstract sense, as well as the existence of "meanings", is disputed by some philosophers. Where the concept of a "meaning" is admitted, its nature is controversial. In earlier texts writers have not always made it sufficiently clear whether they are using the term proposition in sense of the words or the "meaning" expressed by the words.^[1] To avoid the controversies and ontological Ontology (from the Greek ὄν, genitive ὄντος: of being and -λογία, -logia: science, study, theory) is the philosophical study of the nature of being, existence or reality in general, as well as the basic categories of being and their relations. Traditionally listed as a part of the major branch of philosophy known as metaphysics, implications, the term sentence is often now used instead of proposition to refer to just those strings of symbols that are truthbearers, being either true or false under an interpretation. Strawson advocated the use of the term "statement" In logic a statement is a declarative sentence that is either true or false. A statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement. The term "statement" may to refer to a sentence or the idea expressed by a sentence, and this is the current usage in mathematical logic.

Historical usage

Usage in Aristotle

Aristotelian logic The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic. The works are Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics and Sophistical Refutations identifies a proposition as a sentence which affirms or denies a predicate Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common. The notation P is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x) written {x | P(x)}, is just a of a subject In philosophy, a subject is a being that has subjective experiences, subjective consciousness or a relationship with another entity . A subject is an observer and an object is a thing observed. This concept is especially important in continental philosophy, where 'the Subject' is a central term in debates over human autonomy and the nature of the. An Aristotelian proposition may take the form "All men are mortal" or "Socrates is a man." In the first example the subject is "men" and the predicate "are mortal". In the second example the subject is "Socrates" and the predicate is "is a man".

Usage by the logical positivists

Often propositions are related to closed sentences In mathematical logic, a sentence of a predicate logic is a well formed formula with no free variables. A sentence is viewed by some as expressing a proposition. It makes an assertion, potentially concerning any structure of L. This assertion has a fixed truth value with respect to the structure. In contrast, the truth value of a formula may be to distinguish them from what is expressed by an open sentence. In this sense, propositions are "statements" that are truth bearers Truthbearer is a term used to designate entities that are either true or false and nothing else. The acceptance that some things are true while others are false raises the question of the nature of such things. Since there is no agreement on the matter, the term truthbearer is used to be neutral among the various theories. Candidates truthbearers. This conception of a proposition was supported by the philosophical school of logical positivism Logical positivism is a school of philosophy that combines empiricism – the idea that observational evidence is indispensable for knowledge of the world – with a version of rationalism incorporating mathematical and logico-linguistic constructs and deductions in epistemology.

Some philosophers argue that some (or all) kinds of speech or actions besides the declarative ones also have propositional content. For example, yes-no questions A question may be either a linguistic expression used to make a request for information, or else the request itself made by such an expression. This information is provided with an answer present propositions, being inquiries into the truth value In logic and mathematics, a logical value, also called a truth value, is a value indicating the relation of a proposition to truth of them. On the other hand, some signs In linguistics, semiotics, also called semiotic studies or semiology, is the study of sign processes , or signification and communication, signs and symbols. It is usually divided into the three following branches: can be declarative assertions of propositions without forming a sentence nor even being linguistic, e.g. traffic signs convey definite meaning which is either true or false.

Propositions are also spoken of as the content of beliefs Belief is the psychological state in which an individual holds a proposition or premise to be true and similar intentional attitudes A propositional attitude is a relational mental state connecting a person to a proposition. They are often assumed to be the simplest components of thought and can express meanings or content that can be true or false. In being a type of attitude they imply that a person can have different mental postures towards a proposition, for example, such as desires, preferences, and hopes. For example, "I desire that I have a new car," or "I wonder whether it will snow" (or, whether it is the case that "it will snow"). Desire, belief, and so on, are thus called propositional attitudes when they take this sort of content.

Usage by Russell

Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, atheist, socialist, pacifist, and social critic. Although he spent most of his life in England, he was born in Wales where he also died, aged 97 held that propositions were structured entities with objects and properties as constituents. Others have held that a proposition is the set of possible worlds/states of affairs in which it is true. One important difference between these views is that on the Russellian account, two propositions that are true in all the same states of affairs can still be differentiated. For instance, the proposition that two plus two equals four is distinct on a Russellian account from three plus three equals six. If propositions are sets of possible worlds, however, then all mathematical truths are the same set (the set of all possible worlds).

Relation to the mind

In relation to the mind, propositions are discussed primarily as they fit into propositional attitudes A propositional attitude is a relational mental state connecting a person to a proposition. They are often assumed to be the simplest components of thought and can express meanings or content that can be true or false. In being a type of attitude they imply that a person can have different mental postures towards a proposition, for example,. Propositional attitudes are simply attitudes characteristic of folk psychology Folk psychology is the set of assumptions, constructs, and convictions that makes up the everyday language in which people discuss human psychology. Folk psychology embraces everyday concepts like “beliefs”, "desires”, “fear”, and “hope" (belief, desire, etc.) that one can take toward a proposition (e.g. 'it is raining', 'snow is white', etc.). In English, propositions usually follow folk psychological attitudes by a "that clause" (e.g. "Jane believes that it is raining"). In philosophy of mind Philosophy of mind is a branch of modern analytic philosophy that studies the nature of the mind, mental events, mental functions, mental properties, consciousness and their relationship to the physical body, particularly the brain. The mind-body problem, i.e. the relationship of the mind to the body, is commonly seen as the central issue in and psychology Psychology is the scientific study of human or other animal mental functions and behaviors. In this field, a professional practitioner or researcher is called a psychologist. Psychologists are classified as social or behavioral scientists. Psychological research can be considered either basic or applied. Psychologists attempt to understand the, mental states are often taken to primarily consist in propositional attitudes. The propositions are usually said to be the "mental content" of the attitude. For example, if Jane has a mental state of believing that it is raining, her mental content is the proposition 'it is raining'. Furthermore, since such mental states are about something (namely propositions), they are said to be intentional The term intentionality was introduced by Jeremy Bentham as a principle of utility in his doctrine of consciousness for the purpose of distinguishing acts that are intentional and acts that are not . The term was later used by Edmund Husserl in his doctrine that consciousness is always intentional, a concept that he undertook in connection with mental states. Philosophical debates surrounding propositions as they relate to propositional attitudes have also recently centered on whether they are internal or external to the agent or whether they are mind-dependent or mind-independent entities (see the entry on internalism and externalism in philosophy of mind).

Treatment in logic

As noted above, in Aristotelian logic The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic. The works are Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics and Sophistical Refutations a proposition is a particular kind of sentence, one which affirms or denies a predicate Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common. The notation P is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x) written {x | P(x)}, is just a of a subject In philosophy, a subject is a being that has subjective experiences, subjective consciousness or a relationship with another entity . A subject is an observer and an object is a thing observed. This concept is especially important in continental philosophy, where 'the Subject' is a central term in debates over human autonomy and the nature of the. Aristotelian propositions take forms like "All men are mortal" and "Socrates is a man."

In mathematical logic Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the, propositions, also called "propositional formulas In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula" or "statement forms", are statements In logic a statement is a declarative sentence that is either true or false. A statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement. The term "statement" may to refer to a sentence or the idea expressed by a sentence that do not contain quantifiers Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method. They are composed of well-formed formulas consisting entirely of atomic formulas In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the, the five logical connectives In logic, a logical connective is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences, and symbols of grouping (parentheses etc.). Propositional logic In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true propositions. The series of formulas which is constructed is one of the few areas of mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions that is totally solved, in the sense that it has been proven internally consistent, every theorem is true, and every true statement can be proved.^[2] (From this fact, and Gödel's Theorem Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that, it is easy to see that propositional logic is not sufficient to construct the set of integers.) The most common extension of propositional logic In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true propositions. The series of formulas which is constructed is called predicate logic In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas contain variables which can be quantified. Two common quantifiers are the existential ∃ and universal, which adds variables A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use. The concepts of constants and variables are fundamental to all modern mathematics, science, engineering, and computer programming and quantifiers Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method.

Objections to propositions

Attempts to provide a workable definition of proposition include

Two meaningful declarative sentences express the same proposition if and only if they mean the same thing.

thus defining proposition in terms of synonymity. For example, "Snow is white" (in English) and "Schnee ist weiß" (in German) are different sentences, but they say the same thing, so they express the same proposition.

Two meaningful declarative sentence-tokens express the same proposition if and only if they mean the same thing.

Unfortunately, the above definition has the result that two sentences/sentence-tokens which have the same meaning and thus express the same proposition, could have different truth-values, e.g. "I am Spartacus" said by Spartacus and said by John Smith; and e.g. "It is Wednesday" said on a Wednesday and on a Thursday.

In mathematical logic, this problem is solved with quantifiers. Both sentences are predicates, not propositions, because "I" and "It" are variables, and predicates only have a truth value when they are quantified. "For all days, it is Wednesday." is false. "There exist a day, such that it is Wednesday." is true.^[3]

A number of philosophers and linguists claim that all definitions of a proposition are too vague to be useful. For them, it is just a misleading concept that should be removed from philosophy and semantics Semantics is the study of meaning, usually in language. The word "semantics" itself denotes a range of ideas, from the popular to the highly technical. It is often used in ordinary language to denote a problem of understanding that comes down to word selection or connotation. This problem of understanding has been the subject of many. W.V. Quine Willard Van Orman Quine (known to intimates as "Van") was an American philosopher and logician in the analytic tradition. From 1930 until his death 70 years later, Quine was continuously affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of mathematics, and maintained that the indeterminacy of translation prevented any meaningful discussion of propositions, and that they should be discarded in favor of sentences In mathematical logic, a sentence of a predicate logic is a well formed formula with no free variables. A sentence is viewed by some as expressing a proposition. It makes an assertion, potentially concerning any structure of L. This assertion has a fixed truth value with respect to the structure. In contrast, the truth value of a formula may be.^[4] Strawson advocated the use of the term "statement" In logic a statement is a declarative sentence that is either true or false. A statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement. The term "statement" may to refer to a sentence or the idea expressed by a sentence.

See also

• Main contention
• Premise In logic, an argument is a set of one or more declarative sentences known as the premises along with another declarative sentence (or "proposition") known as the conclusion. Aristotle held that any logical argument could be reduced to two premises and a conclusion. Premises are sometimes left unstated in which case they are called

References

1. ^ see eg http://plato.stanford.edu/entries/propositions/
2. ^ A. G. Hamilton, Logic for Mathematicians, Cambridge University Press, 1980, ISBN 0521292913
3. ^ A. G. Hamilton, Logic for Mathematicians, Cambridge University Press, 1978, ISBN 0521292913.
4. ^ Quine W.V. Philosophy of Logic, Prentice-Hall NJ USA: 1970, pp 1-14

External links

• Stanford Encyclopedia of Philosophy articles on:
  • Propositions, by Matthew McGrath
  • Singular Propositions, by Greg Fitch
  • Structured Propositions, by Jeffrey C. King

Categories: Semantic units | Logic | Philosophy of language | Statements | Logical syntax

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