Statement (logic)

A statement in the sense of Aristotelian logic is a linguistic structure, of which it makes sense to ask whether it is true or false (so-called Aristotelian two-valued principle ). It is not necessary to be able to say whether the structure is true or false. It is sufficient that the question of truth (“apply”) or falsehood (“not apply”) makes sense - which is not the case, for example, with questions, exclamations and wishes. Statements are therefore sentences that describe facts and to which one can assign a truth value .

ambiguity

The meaning of the expression statement given in the introduction is the dominant meaning.

However, the term statement is used ambiguously.

These can be reduced to four basic meanings:

1. Statement in the sense of a proposition ( sentence ) (or "a set of consecutive sentences")
2. Statement in the sense of the utterance (action) of a sentence;
3. Statement in the sense of an act of judgment ( judgment )
4. Statement in the sense of proposition ( meaning of statement, "what is said, the facts meant (by the statement), sense of judgment, thought, thought, proposition"), "objective sentence".

The meaning of the expression statement depends on "what exactly is the object of logic" and what actually is the "bearer" of truth or falsehood. However, this question does not need to be clarified for a technical use of logic.

Statements and statements

According to a widespread, but controversial view, statements are not sentences, but statements are (only) the linguistic expression of statements. A statement is representative of a statement, is only a sign for a statement (proposition) and only "the linguistic correlate of the statement".

The objection to equating a statement and a statement sentence is that the statement “which is made with this statement” must be distinguished from the sentence type and its utterance.

Example 1: (sentences with the same meaning): "The house is three-story." - "This residential building has three floors." - "This house has three floors.": Three sentences with one statement for a fact.

Example 2: When Hans and Ina say “I'm sick”, then both utter the same sentence (in terms of sentence type) and generate different sentence occurrences and make different statements with their utterances.

According to Quine , the assumption of propositions should not be necessary, so that the term "statement" should not refer to what has been said, but only to statements.

Sentence - judgment - statement

Tugendhat speaks roughly of a linguistic, psychological and ontological basic conception of logic: the linguistic statement corresponds to the judgment as a psychological act and ontologically to the statement, the thought (Frege); the facts (Husserl: Wittgenstein I ) or the proposition (English philosophy).

Between “sentence” - “judgment” - “statement” there is a relationship analogous to proportionality and attribution . I.e. the objective thought (the statement, proposition) is captured in the thinking (psychic act of judgment) and brought up in a statement. Statements about the statement therefore also relate in an analogous sense to the objective statement content or the psychological statement act - and vice versa. In most contexts, a closer distinction is therefore not important. Depending on the epistemological orientation, a corresponding terminology may be preferred. For the recipient, this means that the same thing is factually talked about, regardless of the epistemological prerequisites. While the term “judgment” used to be naive ( Aristotle ) or psychologistic (empiricism, Kant), the term “sentence” dominates after the linguistic turn , with which the term “statement” competes or is mixed up. If you want to avoid the ambiguous meaning of the expression "statement", it is advisable to distinguish terminologically between statements and proposition . However, this is not common in German-speaking countries.

Delimitations

claim

With Frege the statement is to be distinguished from the assertion of a statement: “In an assertion sentence two things have to be distinguished: the content, which it has in common with the corresponding sentence question, and the assertion. That is the thought, or at least contains the thought. So it is possible to express a thought without making it true. In a statement of assertion both are connected in such a way that one easily overlooks the decomposability. We differentiate accordingly

1. grasping the thought - thinking,
2. the recognition of the truth of a thought - the judgment,
3. the manifestation of this judgment - claim that. "

Value judgment

For propositional logic, it does not matter whether the property contains a score; H. the statement is a value judgment .

Form of statement

The statement (the statement) is to be distinguished from the statement form . A statement form is "an expression that contains one (or more) free variables (spaces) and which changes into a (true or false) statement by occupying all free variables." The statement form changes into a statement as soon as the variable is replaced .

In mathematical logic, the syntactic structure of a statement is formally specified based on the characters of a language L. Depending on the language, different atomic forms of expression are allowed, from which combined forms of expression are formed by means of joiners. In the case of predicate logic, there is also the option of binding variables contained in the atomic statement forms using quantifiers (“there is an x ​​for which applies”, “for all x applies”). A variable that is not bound by a quantifier is called a free variable .

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

word

A single word that does not stand for a statement “does not convey anything”, “is not true or false”. "Only when a word is an abbreviation for a sentence can we speak of its truth or falsehood ...".

term

What was said about the delimitation of the word applies accordingly (actually) to the term .

Behind each term are one or more statements that define its content and relate this term to others. "Therefore, the statement that the concept is a unit of characteristics in its content leads to the idea that every concept represents a connection of statements." This was particularly advocated by Cohn and is also echoed by Frege when he said that the word only has a meaning in a sentence.

Enough

"Every statement in which something is ascribed to an object can be viewed as a kind of inference, the premises of which define the subject of the statement in question and attribute or deny a property to the term that defines it."

Types of statements

Simple - put together

Statements can be divided into simple statements and compound statements . The reason for classification is whether or not statements are made up of distinguishable “separable” partial statements.

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

The terminology varies: instead of “simple statement” one also speaks of “unassembled statement”, “atomic statement”, “elementary statement”, “elementary statement” or “elementary proposition” (Wittgenstein). Instead of a compound statement, there is also talk of “statement combination” or “molecular statement”. In mathematical or formal logic, statements that are not composed of other statements are called atomic statements . Therefore they do not contain statements linking logical operators ( connectives ) as ∧ ( and ), ∨ ( or ) and ¬ ( not ). The opposite term is the compound statement or combination of statements .

Is z. For example, separating the statement “The road is wet and it's raining” into two statements that are linked by and to form one statement is no longer such a separation for the individual statements “The road is wet” and “It's raining” possible. Thus, these statements are atomic statements. In a propositional analysis of arguments, it is essential to subdivide the formulations into atomic statements, since this is the only way to formalize the junctions that are important for the argument structure.

Examples of atomic statements (predicate logic)
${\ displaystyle t = s}$for any terms ,${\ displaystyle t}$${\ displaystyle s}$
${\ displaystyle R (t_ {1}, \ dots, t_ {n})}$for every n-place relation and any terms up to${\ displaystyle R}$${\ displaystyle t_ {1}}$${\ displaystyle t_ {n}}$

In a simple statement, an object is given or denied a single predicate.

If it is said that a simple statement is not further structured, then this is to be understood to mean that the inner structure of a statement is not further specified.

The interpretation of the atomic statements is done by assigning truth values.

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

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

A statement link can be made extensionally (extensional linkage of statements) or intensional (intensive statement linkage).

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

Logical constants that create a truth-functional statement connection are called joiners .

Classical propositional logic is a join logic (Lorenzen), a "logic of truth functions" (Quine) of statements. It is based on the extensionality principle .

Negation is a special case . However, this is for more terminological and practical reasons. In the case of negation, no statements are linked and therefore it is not a statement combination. Nevertheless, for reasons of terminological simplification, it is called a single-digit combination of statements. With the input value true it delivers the value false and vice versa. The term one-digit truth function appears to be more appropriate in terms of terminology.

There are sixteen two-digit links (junctions) for combining two statements. For all possible combinations of truth values, you specify a typical result truth value for this link. For example, the is conjunct associated statement a AND b only true if both a and b is true; in any other case the conjunction is wrong.

Intensional statement links are non-truth-functional statement links. With these, the truth value of the overall statement does not depend on the truth value of the partial statements.

Example: "Anton is reading a book about logic because he finds logic incredibly exciting".

analytical - synthetic

Statements are traditionally divided into analytical statements and synthetic statements. Instead of “statement”, “sentence” or “judgment” is also used in the same sense (see Tugendhat's three-way division above).

Analytical statements

• in the narrower sense are “statements that necessarily, i. H. in all possible worlds, are true solely because of their logical form, and whose truth can be determined without empirical verification ”. They thus correspond to a logical tautology .
Example: The sun is shining or the sun is not shining.
• in a broader sense, “are those whose truth depends on their syntactic structure and the meaning of their linguistic elements. They are based on semantic relations such as equality of meaning [...] and inclusion of meaning [...] ”. They thus correspond to a circular argument .
Example: "Siblings are related".

According to Ernst Tugendhat , all analytical propositions are based on the principle of excluded contradiction . You don't have any potential falsifiers .

Synthetic statements

• In the broader sense , according to Aristotle, all statements (judgments), i.e. H. a “synthesis of concepts”.
• in the narrower, dominant sense ( Kant ) are “statements about factual relationships, the truth of which depends not only on their syntactic or semantic structure, but on extra-linguistic and thus empirically verifiable factors and experiences; [...] ".
Criticism of distinction

The justification of the distinction between analytical and synthetic statements (judgments) was attacked by Quine . He advocated a thesis of the indeterminacy of meaning and fundamentally questioned whether conceptual meanings can be sharply delimited from one another.

For example: “All black horses are black.” Versus “All crows are black.” Since blackness is a key feature to distinguish black horses from other horses, it is an analytical statement. According to Quine, it is not possible to determine whether the concept of 'crows' can be used without blackness - that is, white (e.g. albino ) birds with otherwise the same characteristics are also included in the term - or whether crows are also (also) defined by blackness clearly determine. In fact, crow is used as a term for a natural species, so whether all specimens of this species will appear black is an empirical question.

Statements in propositional logic

In propositional logic, only their formal truth value and not their content-related truth value is important for such statements. For example, one must have knowledge of the facts described in order to be able to judge the truth value of the statement "Berlin is the capital of Germany and Rome the capital of Italy"; this is not necessary for the statement "Madrid is the capital of Spain, or Madrid is not the capital of Spain", because according to the definition (standardization) of the use of the logical or and not , this is a true statement regardless of whether Madrid is really the capital of Spain or not. A statement that is formally true in this sense is called universally valid or also called a tautology .

Statements in predicate logic

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

In predicate logic, the truth value of a statement results from the interpretation of the symbols it contains. For example, the statement can be determined as follows: the terms x and x + x are calculated for each x . If there is an x ​​such that both terms have the same value (e.g. for x = 0), then the statement is true, otherwise it is false. Thus, the truth value of the statement depends on the basic set (also called universe, domain, range of values, range of individuals) from which assignments for the variables may come. ${\ displaystyle \ exists x: x = x + x}$

Is a statement true for every interpretation, e.g. B. , that's what they call universal or tautology . ${\ displaystyle \ forall x: f (x) = f (x)}$

The model theory is the mathematical part of discipline that deals with the question of which models it for what amounts are statements.

Individual evidence

1. Hoyningen-Huene: Logic . 1998, p. 32 f. gives five meanings: [1] the utterance of the statement; [1.1] the scheme of the utterance; [2] an act of judgment; [2.1] the scheme of the act of judgment; [3] the sense of meaning.
2. Joseph Verguin: Statement . In: André Martinet (Ed.): Linguistics . 1973, p. 60.
3. Judgment is an older expression for statement, cf. Strobach: Introduction to Logic . 2005, p. 49.
4. ^ A b Hoyningen-Huene: Logic . 1998, p. 33.
5. Bochenski: The contemporary methods of thinking . 10th edition. 1993, p. 13.
6. Hoyningen-Huene: Logic . 1998, p. 34.
7. ^ Judgment . In: Regenbogen, Meyer: Dictionary of Philosophical Terms . 2005.
8. a b Tatievskaya: propositional logic . 2003, p. 65.
9. a b Beckermann: Introduction to Logic , 2nd edition. 2003, p. 17.
10. statement, sentence . In: Seiffert: Theory of Science IV . 1997.
11. Strobach: Introduction to Logic . 2005, p. 49 f.
12. Tugendhat, Wolf: Logical-semantic propaedeutics . 1983, p. 17.
13. ^ Weingartner: Theory of Science I: Introduction to the main problems . 2nd Edition. 1978, p. 28 footnote 1: "One can show that with suitable definitions for the expressions 'statement', 'judgment' (where 'judgment' denotes a psychological act in which something is recognized or rejected) and 'proposition' the expressions 'true' in the meta-statements 'statements are true', 'judgments are true' and 'propositions are true' are related to one another in a proportionality-analogous and attribution-analogous relationship. "
14. ^ Brandt, Dietrich, Schön: Linguistics. 2nd Edition. 2006, p. 292 fn. 16.
15. Gottlob Frege : The thought. P. 34 f., Quoted from Patzig, this from Tugendhat / Wolf: Logisch-semantische Propädeutik . 1983, p. 27.
16. ^ Herberger, Simon: Theory of Science for Jurists. 1980, p. 34.
17. Thomas Zoglauer: Introduction to formal logic for philosophers. 1999, p. 24 (Example: "All S are P".)
18. Bucher: Logic. 1987, p. 43. Menne: Logic. 6th edition. 2001, p. 59 (or is quantified).
19. a b Reichenbach: Fundamentals of symbolic logic . 1999, p. 5.
20. Tatievskaya: propositional logic . 2003, p. 66.
21. Hoyningen-Huene: Logic . 1998, p. 35.
22. Hoyningen-Huene: Logic . 1998, p. 37 f.
23. Hoyningen-Huene: Logic . 1998, p. 45.
24. Hoyningen-Huene: Logic . 1998, p. 38.
25. a b c Bußmann: Lexicon of Linguistics . 3. Edition. 2002. Analytical vs. Synthetic sentences.
26. Tugendhat, Wolf: Logical-semantic propaedeutics. 1983, p. 65
27. ^ Jürgen Bortz : Statistics for human and social scientists . 6th edition. Springer Medizin Verlag, Heidelberg 2005, pp. 4–5.
28. de Vries: Synthesis . In: Brugger: Philosophielexikon . 1976.
29. ^ Jacob Rosenthal: Induction and confirmation . In: Andreas Bartels, Manfred Stöckler (Ed.): Theory of Science . mentis Verlag, Paderborn 2009, p. 111.