Term logic

Term logic or terminological logic (English Terminological logic or term logic), also term logic or traditional logic, sometimes classical logic called ( " classic " as a historical concept in the sense of: logic of the ancient world, not to be confused with classical logic in modern parlance ) is one kind or view of logic, in which the terms of their content and sizes and their relationships to each other at the center or at the beginning of the viewing stand.

Description

Formally, a logical system is precisely then a term logic, if the atomic characters, be they are constant or variable, for concepts. In the philosophical and conceptual logical tradition only such systems are called conceptual logic generally, in which the atomic characters are available only for items, that is, where there is no other category of basic characters.

The question of what exactly a concept is being discussed intensively in the tradition of the term logic, but turns out to be philosophically relatively elusive and is therefore interpreted quite differently ( see Terminology (philosophy) ). For the conceptual logical reasoning itself the particular interpretation of the term 'concept' is, however, in practice, is of secondary importance. Universally accepted examples of terms " person" or " mammal". Whether proper names, such as " Socrates " or " Aristotle ", and relations (relations) between things, such as existing between Berlin and Paderborn relationship, the greater than one ( Berlin is greater than Paderborn ) or between the three figures 10, 4 and 6 -existing relationship that the first is the sum of the two is the latter, can also be understood as concepts, has been answered in different ways in the tradition.

Of greater importance for the conceptual logical practice, the distinction between the scope and content ( "Extension and Intension " ) of a term. The scope of a term, its extension, is generally considered as the totality of things that fall under the term - as is the scope of the term " person " means all people. The content of a concept, its intension is understood differently in the tradition. Can be roughly located under the content of a concept to present the totality of features or characteristics that make up this term - in the case of the term "man" among many other properties, the properties of being a mammal to be able to think rationally and to be a linguist. Depending on begriffslogischem system, the variables for either extents or connotations - or they can be interpreted in either of two ways.

In a term logic declarative sentences are from the terms ( veraltend also called judgments) formed, which meet each other a statement about the relationship between two or more concepts. The most frequently mentioned ratio of two terms is the type - genus ratio, that is the finding that a term is kind of by another term expressed genus. An example of a statement ( an opinion ) that ( it) expresses a type - genus ratio, " ( all) humans are mammals ": This statement is expressed that " man " is a species of the genus " mammal" is.

The judgments formed from the words are put together in the term logic circuits ( arguments). For example, can be calculated from the two judgments " ( all) humans are mammals " and " (All) logicians are people " on the judgment " (All) logicians are mammals " close and form the following argument:

The commonly used and chosen here formulations " ( all) humans are mammals, " "Some people are not Logikerinnen " etc. are so far somewhat unfortunate, as they can easily be understood as statements about individuals, for example in the sense of " Every individual, this is a man who is also a mammal. " As a conceptual logical statements but they are just not that, but they express the ratio of two terms. Unmistakable it would be to select a clearer formulation, for example, " Man is kind of the kind mammal" or "mammal comes to every person ," as that was also handled in the tradition often. If, nevertheless, the ambiguous wording is chosen in the following, so that happens in terms of their ordinariness and linguistically simpler readability and trusting that the reader interpret them in the context of this article in the conceptual logical sense.

In contrast, the concept of logic not in terms of modern logic considered as basic elements, but - depending on the system - statements ( in propositional logic), predicates ( in predicate logic) or functions ( in the lambda calculus ). In conceptual logical tradition sometimes did not understand all logical systems are called logic judgment; content of this generalization is wrong from a modern perspective.

Syllogistic

Historical starting point of the concept of logic are the works of Aristotle, who presented a formal logical system in the modern sense in the form of his syllogistic. In the syllogistic arguments are considered in a rigid form, which consist of exactly three sentences, two premises and a conclusion. Premises and conclusion press in each case is the relationship between exactly two terms. Aristotle distinguishes four kinds of judgments:

Proper names (such as " Socrates " ) Aristotle considered not as terms in this sense.

Leibniz's concept of logic

Leibniz developed in the 17th century a logical system, which in its formal features already with the later system of Boole (see next section ) has similarities. In this sense, Leibniz's work be considered as an anticipation of the algebraization of logic, although his work historically probably remained without much influence, and only in the 20th century - after the completion of development of the formal algebra - greater attention has been given and appreciated to its full extent were.

Leibniz developed in the course of his work several formal systems and thereby uses different characters to which will not be discussed further here. Common to all stages of Leibniz's development, that in terms of their intension, that is the connotation in the center of the viewing stands. Conceptual content is defined as the totality of characteristics that make up the term. The content of the concept of man has in this sense, for example, features such as " rational ", " linguist " or " bipedal " ( but of course by the three characteristics are not fully determined ).

Leibniz already sees the relationship between intensional and extensional interpretation of formal concept logic and is aware that the valid values ​​that makes his systems via extensions of concepts and their relationships, are given a suitable interpretation of the characters used to valid statements about the concept of content and their relationships.

In an early system Leibniz assigns to each word or each word atomic variable a prime, for example, the term A is the number 3, the term B is the number 5, and the term the variable C 7 Combining terms in this system corresponds to the numerical formally multiplication. The term AB would increase the number assigned to 3 x 5 = 15 in this example, the term ABC is the number 3 × 5 × 7 = 105 According to this method, it is possible mathematically to determine whether a term falls under a different term: General falls a term S if and only under a concept P when the numerical value of S integer ( that is, with remainder 0) by the numerical value of P is divisible. Here are two examples:

The analogy to calculations with primes becomes more difficult when it comes to negative ( negative ) and to particulate statements. In order to deal with negative statements adequately must assign a second, negative prime Leibniz each atomic concept. Due to the associated complications are Leibniz this first system early on.

Algebraization of logic: Boole's logic term

Your technical highlight is experiencing the traditional logic in the sense the term logic with their algebraization by George Boole and Augustus De Morgan in the 19th century.

In Boole's system, the variables for terms, however, expressly for its scope (extension), not for its content (even if by a suitable reinterpretation of the concatenation character a conceptual substantive interpretation is possible). Boole's system uses capital letters for terms, the character 0 ( zero) falls for the empty concept under which nothing, and the number 1 (one) for the universal term which falls under the everything. Be linked term character by mere juxtaposition writing or by one of the characters " " (plus ) and "- " (minus ):

• The juxtaposition of writing, such as " AB " is interpreted as forming intersections or (more the conceptual logical mindset accordingly) as formation of a term that fall under the only things that fall under both " A" and under "B". If, for example, A for the term " philosopher " and B for the term " logician, " then AB represents the term " logician and philosopher ," meaning for the term, fall below all persons who are also Logikerinnen and philosophers.
• The "addition, " " A B" is interpreted as the term that includes everything that falls under either A or B below. Are there things that fall under both A and B below, the expression " A B" is undefined - that's the big difference between Boole's system and later realized logical systems. If, for example, A for the term " man " and B for the term " book", then " A B", the term covered by the people as well as books. Where it is A for the term " logician " and B for the term " philosopher, " then the expression " A B" is undefined because it is very well Logikerinnen that philosophers are also (and vice versa).
• The " subtraction, " "AB", is interpreted as the formation of a concept covered by whom are all things that fall under A but not B. If, for example, A for the term " man " and B for the term " logician " then is "AB" for the concept of people who are not logicians.

To express the relationship between two terms, Boole used different equivalent notations. The statement (the " Judgment " ) " All A are B" for example, can be expressed in his system, among others, as AB = A and as A ( 1-B) = 0.

Boole's logical conceptual system is the first, which is formally drawn up so far that it also allows a propositional interpretation. If one interprets the variables are not as concepts but as statements preceded by " multiplication " as the set link ( the conjunction ) "and" (conjunction ) and the addition as the exclusive or ( " either ... or ... " in modern speech: XOR operation ), then all valid conceptual logical statements of Boole's system to valid propositional statements. This observation of the structural equivalence of content completely different logical systems ( term logic and propositional logic ) founded the discipline of formal algebra, also called abstract algebra.

Relations in the logic: Augustus De Morgan and Charles Sanders Peirce

The deficiency in Boole begriffslogischem system as well as in the traditional term logic in the sense of syllogism is the lack of options for the treatment and representation of relations. Relations are relations between individuals (or relationships between concepts ), for example the relationship of greater than one, such as those between the two figures 5 and 2, (5 is greater than 2). They are not only in mathematics is of great importance, but almost everywhere in everyday and scientific closing, so that it is almost surprising from today's perspective, that they were not considered in detail in the long tradition of Aristotelian logic justified.

De Morgan has the merit to have pointed to the importance of the relations for closing in general ( and for the mathematical Close in particular). He is often - may not be right - the now classic objection to the traditional term logic attributed to the is to formulate the following argument:

This argument, although intuitively clear valid, can be the means of the traditional term logic does not adequately formulate or even inferred.

It was Charles Sanders Peirce, which there Resulting from succeeded in his 1870 published article Description of a Notation for the Logic of Relatives of Amplification of the Conceptions of Boole 's Calculus of Logic, the ideas of Boolean algebra to relations (not only relative of him but also relative terms - " relational concepts " - called ) to use and extend.

The transition to the quantifiers: Peirce, Schröder, Tarski

Peirce already in use quantifiers, as they also occur in the logic of Ernst Schröder. But with two authors, it is uncertain whether they considered the quantifiers as a mere tool, can be expressed with the specific complex issues simple, or whether they considered the quantifiers to be necessary for full expression of strength; that is, whether they assumed that there are matters that without the use of quantifiers - in a purely algebraic system - can not be expressed.

Content answered this question Alfred Tarski: He manages to show that the full expressiveness quantifiers are essential.

Term logic from a modern perspective

From a modern perspective is traditional term logic is equivalent to a special case of predicate logic, namely the digit ( " monadic " ) predicate logic. Place predicate logic is limited to the use of single-digit predicates, such as " _ is a man" or "_ is a logician ." Translate can be made between traditional term logic and predicate logic -digit by each term by the X -place predicate "_ is X" is replaced and vice versa ( for example, the term "man" by the predicate " _ is a man" ). It is therefore no matter for abstract formal point of view, whether one operates term logic or place predicate logic. The place predicate logic - and thus the traditional term logic - is decidable.

Term logic with relational extensions, as proposed by De Morgan and implemented by Peirce, need for general predicate logic representation, ie multi-digit predicate logic. Relations ( in conceptual logical terminology: Relation terms ) are expressed by two -place predicates. For example, the mathematical relation is greater in the predicate logic expressed by the binary predicate "_1 _2 is greater than ". An additional advantage of predicate logic is that beliebigstellige relations can be expressed naturally, as the three -place relation "_1 _2 and _3 is between ".

With relations alone, for example with the relational extended conceptual logic system of Peirce, You still can not cover the full scope of predicate logic - this is needed as the term logic using quantifiers that there - see the above comments about Peirce - actually were introduced early.

Decline of the concept of logic

Until well into the 20th century conceptual logical systems and modern logical systems such as propositional logic or predicate logic were used in parallel, the frequency and intensity of use of term logic went back to the same degree as the frequency and intensity of use of modern logical systems increased. This change is mainly explained by the fact that the modern logical systems address the needs of the predominantly mathematical users better satisfied than the classical, understood logical systems, while at the same time became more and more the influence of the philosophy with its strong tradition oriented arrest in the Aristotelian term logic in the background. In blunt formulation: The modern logical formalisms for example, the propositional and predicate logic were by the users mostly as a simple and problem- adequate perceived as the - even extended to relations and quantifiers, and thus abstract- formal equally powerful - term logic in the formation of Peirce.

Modern recourse to term logic

Despite the actual ( practical application ) complete detachment concept of logical systems by modern logical systems exist - even away from purely historically motivated investigations - occasional recourse to conceptual considerations and logical systems. Such recourse done rarely or never in formal logic or mathematics, but mainly in the philosophical field. In fact, the individual recourse to the concept of logic are usually motivated on one of the following ways:

• Occasionally, a primacy of the concept is claimed or demanded by other logical categories such as functions, predicates or statements; in the English language is for that ( philosophischen! ) Position of the label terminist philosophy common. Under such a setting is to work with a system whose basic concepts are precisely any terms, at least satisfactory, and is an effort to be able to bring the primacy of the notion within a logical system for expression. From a purely formal point of view and from a logical side, this objection does not have much weight to, because the relevant systems are equivalent in form and the choice of one or the other system is thus reduced to a mere matter of taste.
• Many a time it is pointed to the structural discrepancy between the formulas of modern logical systems, most of predicate logic, and their natural language equivalents. It is argued that in some or in many important cases a conceptual logical formula of a natural language statement is structurally similar as, for example, a statement of predicate logic. In the present state of modern logic, and modern linguistic theory of grammar and semantics of natural languages ​​that position is not a majority, because the grammatical structure, at least as far as it is visible, it is generally not at all considered to be adequate representation of the underlying logical form.
• It is often purely practical argues that modern formal logic is hard to learn and that for expressing simple relationships as a daily life - possibly in daily scientific life - encounter, a simple conceptual logical system - such as in the sense of syllogistic - was sufficient, it was also easier to learn. Course can be on both assumptions - both the greater complexity digit predicate logic over the syllogistic and their sufficiency for everyday (scientific ) work - discuss; but this access is the most comprehensible from a formal point of view.

In the first category fall logical systems as they were propagated, for example, by Bruno Freytag- Löring Hoff in the 1960s. Rather, in one of the latter two categories fall systems such as the TFL (term - functor logic) by Fred Sommers, also formed in the 1960s. From a formal point of view, both systems are such in their full expression to the modern predicate logic equivalent, that any statement of these systems can be unambiguously translated into a predicate logic statement, and vice versa.

The most important application of the term logic in modern times is John Corcoran's formalization of Aristotelian logic by Natural Deduction in 1973. Precursor, Jan Łukasiewicz, who claimed the first term logical formalization of Aristotelian logic in his book. Both systems have the advantage that the entire Aristotelian syllogistic without additional assumptions, which are not present in Aristotle leaves ( existence assumptions) derive. In contrast to Corcoran Łukasiewicz used in its formalization of Aristotelian logic propositional logic, which has since been frequently criticized and can be avoided by Corcoran's work. Corcoran's theory is estimated at philosophers and historians of logic, because the evidence reproduce almost verbatim by Natural Close the reasoning of Aristotle in his Analytica Priora.

Hans Hermes in 1965 established a Term logic with choice operator.

To date, none of the modern conceptual logical systems has had a significant influence on the practice in logic, mathematics, philosophy or scientific theory, should read: Without prejudice to their performance as logical systems and their declaration character in particular with regard to the syllogistic of Aristotle is where not logic itself the object of research is, but where logic is used for example to describe or to solve problems, only rarely resorted to understand logical systems.

Swell