Proposition

A statement in the Aristotelian sense is a linguistic structure, from which it is reasonable to ask whether it is true or false (so-called Aristotelian principle of bivalence ). It is not necessary to be able to tell if the structure is true or false; it is sufficient that the question of truth ( " Applying " ) or falsity ("Not Applying " ) can be made useful, which is not the case, for example, in interrogative sentences, exclamations and wishes. Statements are thus sentences that describe situations and where you can assign a truth value.

• 2.1 assertion
• 2.2 value judgment
• 2.3 statement form
• 2.4 word
• 2.5 term
• 2.6 final

• 3.1 simple statements - Compound statements
• 3.2 analytic - synthetic statements ( sentences, phrases) 3.2.1 Criticism of the distinction

Ambiguity

The introduction lectured meaning of the expression statement is the dominant meaning.

The term phrase is used ambiguously.

This can be reduced to four basic meanings:

Statement within the meaning of declarative sentence ( sentence ) (or "a lot of successive sets " )

Statement within the meaning of the utterance ( Owner's setting) of a sentence;

Statement within the meaning of act of judgment (judgment )

Statement within the meaning of Proposition ( statement meaning, " the Fully Said the ( meant by the statement ) facts, the sense of judgment, the thought is thought, the proposition "), " objective rate ".

From the meaning of the expression statement depends on "what the subject matter of logic actually exactly " and what is actually a "carrier" of truth or falsity. For a technical use of logic there an answer to this question, however, is not required.

Declarative sentence and statement

According to widespread but controversial opinion statements are not exclusive but are declarative sentences (only ) the linguistic expression of statements. A declarative sentence is representative of a statement is merely a sign of a statement ( proposition) and only " the linguistic correlate of the statement".

For an equation of the statement and declarative sentence is argued that from the record type and its utterance the statement is to be distinguished, " which is made with this statement ."

Example 1: ( equally significant sentences ): " The house has three floors. " - " This residential building has three floors. " - " This house Has three floors. ": Three sets with a statement relating to facts.

Example 2: When Hans and Ina say "I 'm sick ", then both express the same set ( as defined record type ) and create different sentence-occurrences and make their comments is different statements.

According to Quine, it should not require the acceptance of propositions, so can that the term " statement" not to Fully Sagtes, but only declarative sentences relate to.

Set - Case - statement

Tugendhat speaks in a rough classification of a linguistic, psychological and ontological conception of logic: the verbal statement set corresponds to the judgment as a mental act and ontologically the statement that the thought ( Frege ); the facts ( Husserl, Wittgenstein I) or the proposition (English philosophy).

Between " set " - " judgment " - " statement" there is a proportionality and attributionsanaloges ratio. That the objective thought ( the statement Proposition ) is detected in thinking ( mental act of judgment ) and housed in a declarative sentence to the language. Statements about the declarative sentence therefore affect in an analogous sense, the objective statement content or the mental enunciation - and vice versa. Therefore, in most contexts, it does not depend on a more detailed distinction. Depending on the epistemological orientation, an appropriate terminology preferred. For the recipient, this means that will objectively talked about the same thing, what epistemological presuppositions whatsoever. Was formerly the term " judgment " naive ( Aristotle ) or psychologistically ( empiricism, Kant) dominant, dominated by the linguistic turn, the term " set " with which the term " statement" competes or is mixed. If you want to avoid the dazzling meaning of the term " statement", it is recommended that terminology to distinguish between declarative sentence and proposition. This is in the German language but not common.

Accruals

Assertion

With Frege 's statement of the contention that a statement is to be distinguished: " In a declarative sentence that is two things must be distinguished: the content that he has in common with the corresponding question of principle, and the claim That is the idea, or at least contains the thoughts it. . is therefore possible to express a thought without presenting it as true in an assertion sentence both are connected so that you can easily overlook the decomposability We therefore distinguish first grasping the idea -. . thinking, second the recognition of the truth of a thought - the judgments, the third manifestation of this judgment - say that. "

Value judgment

For propositional logic, it is irrelevant whether the property contains a score ie the statement a " value judgment " is.

Statement form

The statement ( the statement rate) must be distinguished from the statement form. A statement form is " an expression that contains one (or more ) free variable (vacancies ) and passes through the assignment of all free variables in a ( true or false ) statement. ". The statement form turns into a statement as soon as the variable is replaced.

In the mathematical logic of the syntactic structure of a statement is specified based on the formal character of a language L. Depending on the language are different atomic forms of expression allowed to be formed from those composed by connectives statement forms. In predicate logic, the possibility is added, the atomic statement forms contained variables by quantifiers ( " there is an x ​​for which true ", " for all x " ) to bind. A bound by any quantifier variable is called a free variable.

A logical statement is formally defined as a form of expression ( for definition, see there) over the language L in which no ( free ) variables occur.

Word

A single word that is not a statement, " shares nothing with ", " is not true or false." "Only if a word is an abbreviation for a set, we can speak of its truth or falsity, ...".

Term

The to distinguish from the word above applies accordingly ( actually ) for the term.

Behind each term are one or more statements that define its content and bring this term in a relation to others. "That's why leading the finding that the term in its content is a unit of features to the idea that each term is a conjunction of statements. " This was especially represented by Cohn and sounds also in Frege when it said that the word only in the set have a meaning.

Closing

" Any statement in an object something is awarded, can be seen as a kind of conclusion, define its premises the subject of the question and the statement it definition term deny a property on or off. "

Types of statements

Simple statements - Compound statements

Statements can be divided into simple statements and compound statements. Basic classification is whether statements made ​​distinguishable " separable " part statements are composed or not.

Example: " Berlin is a city " ( simple statement ); "Berlin is a city with more than 3 million inhabitants " (logically a composite statement with the partial statements " Berlin is a city " and "Berlin has more than 3 million people ").

The terminology varies: instead of "simple statement" is also called " unzusammengesetzter statement ", " atomic proposition ", " elementary proposition ", " elementary proposition " or " elementary proposition " ( Wittgenstein). Instead of compound statement is also talk of "statements linking " or "molecular evidence ". As atomic propositions statements referred to in the mathematical or formal logic that are not composed of other statements. Therefore exclude statements linking logical constants ( connectives ) as ∧ ( and ), ∨ (or) and ¬ (not ). The antonym is the composite statement or statements linking.

For example, if the statement The road is wet and it rains in two separate statements, which are linked by the and to make a statement, such a separation in the individual statements The road is wet and it rains no longer possible. Thus, it is because these statements are atomic propositions. In a propositional analysis of arguments, it is crucial to segment the formulations in atomic propositions, since only so important for the argument structure connectives can be formalized.

In a simple statement an object a single predicate is attributed to or.

If it is, a simple statement is not structured, so this is to be understood that the internal structure of a statement is not further specified.

The interpretation of atomic propositions is done by assigning truth values.

The symbols for simple statements are a matter of convention. Commonly, for example, the identification by capital letters A, B, C, possibly with indexed letters.

A compound statement is a statement that is created by combining several simple statements.

A link statements can be extensional ( extensional statements linking ) or intensional ( intensional statement linking ).

Extensional statements shortcuts are compound statements whose truth value is determined by the truth value of its sub- statements. The truth value of the overall presentation is therefore a function of the truth values ​​of the partial statements ( truth functionality).

Logical constants that cause a truth-functional compound statements, connectives are called.

The classical propositional logic is a Junktorenlogik ( Lorenzen ), a " logic of truth functions" ( Quine ) of statements. It is based on the extensionality.

A special case is the negation dar. This, however, from more terminological and practical reasons. In the negation no statements are linked and it is therefore no statements linking. It is nevertheless called terminological simplification reasons digit statement link. It provides the input value true the value false, or vice versa. Terminologically true appears herein, the term digit truth function.

For the combination of two statements, there are sixteen two-digit shortcuts ( connectives ). You specify a typical result for this link truth value for all possible combinations of truth values ​​. For example, linked with the conjunction statement a AND b is true only if both a and b is true; in any other case the conjunction is false.

Intensional statements shortcuts are non- truth-functional statements shortcuts. In these, the truth value of the whole statement does not depend on the truth value of the partial statements.

Example: " Anton reads a book on logic, because he finds logic incredibly exciting."

Analytic - synthetic statements ( sentences, phrases)

Statements are traditionally divided into analytic statements and synthetic statements. Instead of, statement ' is in the same direction also, set ' or ' judgment ' speech (see above Tugendhat'sche trichotomy ).

Analytical statements

• In the narrow sense are " statements that are necessarily ie in all possible worlds, are true simply because of their logical form, and whose truth can be determined without empirical verification. " They are therefore a logical tautology.

• In a broader sense " are those whose truth depends on their syntactic structure and the importance of their linguistic elements. They are based on semantic relations such as equality of meaning [ ... ] and importance of inclusion [ ... ] ". They are therefore a circular argument.

After Ernst Tugendhat found all analytic propositions on the law of contradiction. You do not have potential falsifiers.

Synthetic statements

• In the broader sense are, according to Aristotle, all statements ( judgments), ie a " synthesis of concepts ".

• In the narrow, ruling sense ( Kant) are "statements of fact conditions whose truth depends not only on their syntactic or semantic structure, but by extra-linguistic and thus empirically to be tested factors and experiences; [ ... ] ".

Criticism of the distinction

The authorization of the distinction between analytic and synthetic statements ( judgments ) was attacked by Quine. He represented a thesis of indeterminacy of meaning and put principle into question, that term meanings can be sharply distinguished from each other.

Statements in propositional logic

In SL, is only their formal and not their content, truth-value of significance for such statements. For example, one must be aware of the facts described to the truth value of the statement " Berlin is the capital city of Germany, and Rome the capital of Italy " to assess; This is not required in the statement " Madrid is the capital of Spain, and Madrid is not the capital of Spain," because after fixing (normalization ) of the use of logical or and not, these are a true statement regardless of whether Madrid really is or is not the capital of Spain. A in this sense formally true statement is generally valid or also called tautology.

Statements in predicate logic

A statement in the predicate logic is a form of expression without free variables. ( All variables contained in it are bound by quantifiers. )

In predicate logic, the truth value of a statement obtained due to the interpretation of the symbols contained in it. For example, the statement is determined as follows: For each x, the terms x and x x calculated. If there is an x such that both terms will have the same value (eg, for x = 0), then the statement is true, otherwise false. Thus, the truth value of a statement may come for the variables from the assignment depends on the basic amount (including the universe, domain, range, domain of individuals referred ) from.

Is a statement in any interpretation true, for example, it is called a universal or even tautology.

The model theory is the mathematical sub-discipline that deals with the question of which models are available for the quantities of statements.