Proposition
Encyclopedia
In logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

and philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

, the term proposition refers to either (a) the "content" or "meaning" of a meaningful declarative sentence or (b) the pattern of symbols
Symbol (formal)
For other uses see Symbol In logic, symbols build literal utility to illustrate ideas. A symbol is an abstraction, tokens of which may be marks or a configuration of marks which form a particular pattern...

, marks, or sounds that make up a meaningful declarative sentence. The meaning of a proposition includes having the quality or property of being either true
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...

or false
Falsity
Falsity or falsehood is a perversion of truth originating in the deceitfulness of one party, and culminating in the damage of another party...

, and as such propositions are claimed to be truthbearer
Truthbearer
Truth-bearer is a term used to designate entities that are either true or false and nothing else. The thesis that some things are true while others are false raises the question of the nature of these things. Since there is divergence of opinion on the matter, the term truthbearer is used to be...

s.

The existence of propositions in sense (a) above, 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. To avoid the controversies and ontological
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...

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"
Statement (logic)
In logic a statement is either a meaningful declarative sentence that is either true or false, or what is asserted or made by the use of a declarative sentence...

, and this is the current usage in mathematical logic.

### Usage in Aristotle

Aristotelian logic identifies a proposition as a sentence which affirms or denies a predicate of a subject
Subject (grammar)
The subject is one of the two main constituents of a clause, according to a tradition that can be tracked back to Aristotle and that is associated with phrase structure grammars; the other constituent is the predicate. According to another tradition, i.e...

. 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
Sentence (mathematical logic)
In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false...

to distinguish them from what is expressed by an open sentence
Open sentence
In mathematics, an open sentence is described as "open" in the sense that its truth value is meaningless until its variables are replaced with specific numbers, at which point the truth value can usually be determined...

. In this sense, propositions are "statements" that are truth bearers
Truthbearer
Truth-bearer is a term used to designate entities that are either true or false and nothing else. The thesis that some things are true while others are false raises the question of the nature of these things. Since there is divergence of opinion on the matter, the term truthbearer is used to be...

. This conception of a proposition was supported by the philosophical school of logical positivism
Logical positivism
Logical positivism is a philosophy that combines empiricism—the idea that observational evidence is indispensable for knowledge—with a version of rationalism incorporating mathematical and logico-linguistic constructs and deductions of epistemology.It may be considered as a type of analytic...

.

Some philosophers argue that some (or all) kinds of speech or actions besides the declarative ones also have propositional content. For example, yes-no question
Question
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 may be provided with an answer....

s present propositions, being inquiries into the truth value of them. On the other hand, some sign
Semiotics
Semiotics, also called semiotic studies or semiology, is the study of signs and sign processes , indication, designation, likeness, analogy, metaphor, symbolism, signification, and communication...

s 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 belief
Belief
Belief is the psychological state in which an individual holds a proposition or premise to be true.-Belief, knowledge and epistemology:The terms belief and knowledge are used differently in philosophy....

s and similar intentional attitudes
Propositional attitude
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...

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 Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

held that propositions were structured entities with objects and properties as constituents. Wittgenstein 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. Propositional attitudes are simply attitudes characteristic of folk psychology
Folk psychology
Folk psychology is the set of assumptions, constructs, and convictions that makes up the everyday language in which people discuss human psychology...

(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
Philosophy of mind is a branch of 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...

and psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...

, 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
Intentionality
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...

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 a proposition is a particular kind of sentence, one which affirms or denies a predicate of a subject
Subject (philosophy)
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...

. Aristotelian propositions take forms like "All men are mortal" and "Socrates is a man."

In mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...

, propositions, also called "propositional formula
Propositional formula
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...

s" or "statement forms", are statement
Statement (logic)
In logic a statement is either a meaningful declarative sentence that is either true or false, or what is asserted or made by the use of a declarative sentence...

s that do not contain quantifiers. They are composed of well-formed formulas consisting entirely of atomic formula
Atomic formula
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...

s, the five logical connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences 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.Each logical connective can be expressed as a...

s, and symbols of grouping (parentheses etc.). Propositional logic
Propositional calculus
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...

is one of the few areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

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. (From this fact, and Gödel's Theorem
Gödel's theorem
Gödel's theorem may refer to:*Gödel's incompleteness theorems*Gödel's completeness theorem*Gödel's speedup theorem...

, it is easy to see that propositional logic is not sufficient to construct the set of integers.) The most common extension of propositional logic
Propositional calculus
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...

is called predicate logic, which adds variables and quantifiers.

## 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.

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
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....

. W.V. Quine maintained that the indeterminacy of translation prevented any meaningful discussion of propositions, and that they should be discarded in favor of sentences
Sentence (mathematical logic)
In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false...

. Strawson advocated the use of the term "statement"
Statement (logic)
In logic a statement is either a meaningful declarative sentence that is either true or false, or what is asserted or made by the use of a declarative sentence...

.

## Related concepts

Fact
Fact
A fact is something that has really occurred or is actually the case. The usual test for a statement of fact is verifiability, that is whether it can be shown to correspond to experience. Standard reference works are often used to check facts...

s are verifiable information. Simple facts are often stated as propositions: "Apples are a type of fruit." The opposite statement—"Apples are not a type of fruit"—is still a properly formulated proposition, even though it is false (not a fact). Most statements of fact are compound facts: e.g., that apples exist, that fruit exists, that there are multiple types of fruit, etc.

A premise
Premise
Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...

is a proposition that is used as the foundation for drawing conclusions. For example:
• Premise: "Apples are a type of fruit."
• Premise: "All types of fruit are food."
• Conclusion: "Therefore, apples are food."

If the conclusion is false then either one or more of the premisses is false or the process of combining the premises is logically invalid
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

. If the premisses are true and the process is logically valid, then the conclusion must be true.