Propositions

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## Historical usage

### By Aristotle

### By the logical positivists

### By Russell

## Relation to the mind

## Treatment in logic

## Objections to propositions

## See also

## References

## External links

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Propositions

The term **proposition** has a broad use in contemporary analytic philosophy. The most basic meaning is a statement proposing an idea that can be true or false. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other "propositional attitudes" (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of declarative sentences. Propositions are the sharable objects of attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates for propositions, including thought- and utterance-tokens which are not sharable, and concrete events or facts, which cannot be false.^{[1]}

Aristotelian logic identifies a proposition as a sentence which affirms or denies a predicate of a subject with the help of a 'Copula'. An Aristotelian proposition may take the form of "All men are mortal" or "Socrates is a man." In the first example, the subject is "men", predicate is "mortal" and copula is "are", while in the second example, the subject is "Socrates", the predicate is "a man" and copula is "is".^{[2]}

Often, propositions are related to closed formulae to distinguish them from what is expressed by an open formula. In this sense, propositions are "statements" that are truth-bearers. This conception of a proposition was supported by the philosophical school of logical positivism.

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 present propositions, being inquiries into the truth value of them. On the other hand, some signs 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 and similar intentional attitudes, 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, doubt, and so on, are thus called propositional attitudes when they take this sort of content.^{[1]}

Bertrand Russell held that propositions were structured entities with objects and properties as constituents. One important difference between Ludwig Wittgenstein's view (according to which a proposition is the set of possible worlds/states of affairs in which it is true) 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 "two plus two equals four" is distinct on a Russellian account from the proposition "three plus three equals six". If propositions are sets of possible worlds, however, then all mathematical truths (and all other necessary truths) are the same set (the set of all possible worlds).^{[]}

In relation to the mind, propositions are discussed primarily as they fit into propositional attitudes. Propositional attitudes are simply attitudes characteristic of folk 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 and psychology, 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 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).

As noted above, in Aristotelian logic a proposition is a particular kind of sentence, one which affirms or denies a predicate of a subject with the help of a copula.^{[2]} Aristotelian propositions take forms like "All men are mortal" and "Socrates is a man."

Propositions show up in modern formal logic as objects of a formal language. A formal language begins with different types of symbols. These types can include variables, operators, function symbols, predicate (or relation) symbols, quantifiers, and propositional constants. (Grouping symbols are often added for convenience in using the language but do not play a logical role.) Symbols are concatenated together according to recursive rules, in order to construct strings to which truth-values will be assigned. The rules specify how the operators, function and predicate symbols, and quantifiers are to be concatenated with other strings. A proposition is then a string with a specific form. The form that a proposition takes depends on the type of logic.

The type of logic called propositional, sentential, or statement logic includes only operators and propositional constants as symbols in its language. The propositions in this language are propositional constants, which are considered atomic propositions, and composite (or compound) propositions,^{[3]} which are composed by recursively applying operators to propositions. *Application* here is simply a short way of saying that the corresponding concatenation rule has been applied.

The types of logics called predicate, quantificational, or *n*-order logic include variables, operators, predicate and function symbols, and quantifiers as symbols in their languages. The propositions in these logics are more complex. First, one typically starts by defining a term as follows:

- A variable, or
- A function symbol applied to the number of terms required by the function symbol's arity.

For example, if *+* is a binary function symbol and *x*, *y*, and *z* are variables, then *x+(y+z)* is a term, which might be written with the symbols in various orders. Once a term is defined, a proposition can then be defined as follows:

- A predicate symbol applied to the number of terms required by its arity, or
- An operator applied to the number of propositions required by its arity, or
- A quantifier applied to a proposition.

For example, if *=* is a binary predicate symbol and *?* is a quantifier, then *?x,y,z [(x = y) -> (x+z = y+z)]* is a proposition. This more complex structure of propositions allows these logics to make finer distinctions between inferences, i.e., to have greater expressive power.

In this context, propositions are also called sentences, statements, statement forms, formulas, and well-formed formulas, though these terms are usually not synonymous within a single text. This definition treats propositions as syntactic objects, as opposed to semantic or mental objects. That is, propositions in this sense are meaningless, formal, abstract objects. They are assigned meaning and truth-values by mappings called interpretations and valuations, respectively.

In mathematics, propositions are often constructed and interpreted in a way similar to that in predicate logic--albeit in a more informal way. However, the term can also be used to refer to a proven mathematical statement whose importance is generally neutral by nature.^{[4]} Other similar terms in this category include:

- Theorem (a proven mathematical statement of notable importance)
- Lemma (a proven mathematical statement whose importance is derived from the theorem it aims to prove)
- Corollary (a proven mathematical statement whose truth readily follows from a theorem).
^{[5]}

Propositions are called **structured propositions** if they have constituents, in some broad sense.^{[1]}^{[6]}

Assuming a structured view of propositions, one can distinguish between **singular propositions** (also **Russellian propositions**, named after Bertrand Russell) which are about a particular individual, **general propositions**, which are not about any particular individual, and **particularized propositions**, which are about a particular individual but do not contain that individual as a constituent.^{[7]}

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 definitions have the result that two identical sentences/sentence-tokens may appear to have the same meaning, and thus express the same proposition and yet 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. These examples reflect the problem of ambiguity in common language resulting in mistaken equivocation of the statements. "I am Spartacus" spoken by Spartacus is the declaration that the individual speaking is called Spartacus and it is true. When spoken by John Smith, it is a declaration about a different speaker and it is false. The term "I" means different things, so "I am Spartacus" means different things.

A related problem is when identical sentences have the same truth-value, yet express different propositions. The sentence "I am a philosopher" could have been spoken by both Socrates and Plato. In both instances, the statement is true, but means something different.

These problems are addressed in predicate logic by using a variable for the problematic term, so that "X is a philosopher" can have Socrates or Plato substituted for X, illustrating that "Socrates is a philosopher" and "Plato is a philosopher" are different propositions. Similarly, "I am Spartacus" becomes "X is Spartacus" where X is replaced with terms representing the individuals Spartacus and John Smith.

The example problems are therefore averted if sentences are formulated with sufficient precision that their terms have unambiguous meanings.

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. 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.^{[8]} Strawson advocated the use of the term "statement".

- ^
^{a}^{b}^{c}McGrath, Matthew; Frank, Devin. "Propositions (Stanford Encyclopedia of Philosophy)".*Plato.stanford.edu*. Retrieved . - ^
^{a}^{b}Groarke, Louis. "Aristotle: Logic -- From Words into Propositions".*Internet Encyclopedia of Philosophy*. Retrieved . **^**"Mathematics | Introduction to Propositional Logic | Set 1".*GeeksforGeeks*. 2015-06-19. Retrieved .**^**"The Definitive Glossary of Higher Mathematical Jargon -- Proposition".*Math Vault*. 2019-08-01. Retrieved .**^**Robinson, R. Clark (2008-2009). "Basic Ideas of Abstract Mathematics" (PDF).*math.northwestern.edu*. Retrieved .CS1 maint: date format (link)**^**Fitch, Greg; Nelson, Michael (2018), Zalta, Edward N. (ed.), "Singular Propositions",*The Stanford Encyclopedia of Philosophy*(Spring 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved**^**Structured Propositions by Jeffrey C. King**^**Quine, W. V. (1970).*Philosophy of Logic*. NJ USA: Prentice-Hall. pp. 1-14. ISBN 0-13-663625-X.

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