Conceptual logic
Conceptual logic or terminological logic ( English terminological logic or term logic ), also term logic or traditional logic , sometimes called classical logic ("classical" as a historical term in the sense of: logic of antiquity, not to be confused with classical logic in modern parlance) is a type or view of logic in which the concepts , their content and scope and their relationships to one another, expressed in particular through a subject-predicate relationship, are the focus or at the beginning of consideration. The term conceptual logic particularly characterizes the tradition of Aristotelian logic, which dominated until the 19th century, including its further developments, for example, by Avicenna and the algebraization by George Boole . It marks both a delimitation of logics such as stoic logic or modern propositional logic , which are based on the relationship of statements to one another, and, on another level, a delimitation of a predicate logic established by God, Frege , which determines the relationship of concepts to objects subordinate to them ( or of predicates / relations to objects connected by them) as fundamental and, with the help of quantification via objects, traces conceptual relationships back to the relationship between concepts and objects. Extensions of the classic conceptual logic, which are structurally similar to this, but in the succession of Augustus De Morgan also allow multi-digit relations in place of single-digit terms, are sometimes also called concept logics (as logic of relative terms instead of logic of absolute terms ), and sometimes by these but delimited as relational logics.
description
Formally, a logical system is a conceptual logic if and only if the atomic signs, be they constants or variables, stand for concepts. In the philosophical and conceptual-logical tradition, as a rule, only those systems are designated as conceptual-logical in which the atomic signs only stand for concepts, that is, in which there is no other category of basic signs.
The question of what exactly a concept is is discussed intensively in the tradition of conceptual logic, but proves to be relatively difficult to grasp philosophically and is therefore interpreted quite differently (see concept (philosophy) ). For the conceptual inference itself, however, the respective interpretation of the term “term” is of subordinate importance in practice. Commonly accepted examples of terms are “human” or “mammal”. Whether proper names, for example "Socrates" or "Aristotle", and relationships (relations) between things, e.g. B. the relationship between the cities of Berlin and Paderborn of being larger (Berlin is larger than Paderborn) or the relationship between the three numbers 10, 4 and 6 that the first is the sum of the latter two can also be understood as terms, was answered differently in the tradition.
The distinction between scope and content (" extension and intension ") of a term is of greater importance for conceptual practice . The scope of a term, its extension, is generally considered to be the totality of the things that fall under the term - so the scope of the term "man" is the totality of all people. The content of a term, its intention, is understood differently in the tradition. Roughly one can imagine the totality of all features or properties that make up this term under the content of a term - in the case of the term “human”, among many other properties, the properties of being a mammal, being able to think sensibly and being able to speak. Depending on the conceptual logic system, the variables either stand for the scope of the concept or for the content of the concept - or they can be interpreted in either of the two ways.
In a conceptual logic, propositional sentences (also called judgments) are formed from the concepts, which make a statement about the relationship between two or more concepts. The most frequently mentioned relationship between two terms is the species-genus relationship, that is, the statement that a term is a species of the genus expressed by another term. An example of a statement (a judgment) that expresses a species-genus relationship is “(All) humans are mammals”: This statement expresses that “human” is a species of the genus “mammal”.
The judgments formed from the concepts are also put together in the concept logic to form conclusions (arguments). For example, from the two judgments “(All) humans are mammals” and “(All) logicians are humans”, the judgment “(All) logicians are mammals” can be inferred and the following argument can be made:
(All) humans are mammals | |
(All) logicians are human | |
So | (All) logicians are mammals |
The formulations commonly used and chosen here, "(All) people are mammals," "Some people are not logicians", etc. are somewhat unfortunate in that they can easily be understood as statements about individuals, for example in the sense of "Every individual, that is a human is also a mammal ”. As conceptual statements, however, they are not exactly that, but rather they express the relationship between two concepts. It would be more unambiguous to choose a more unambiguous formulation, for example “ Man is a species of the genus mammal ” or “ Mammal belongs to every human being”, as was often done in the tradition. If the ambiguous formulation is chosen in the following, this is done with a view to its usability and linguistically easier readability and in the trust that the reader will interpret it in the context of this article in the logical sense.
In contrast to conceptual logic, modern logic does not consider concepts as basic elements, but - depending on the system - statements (in propositional logic ), predicates (in predicate logic ) or functions (in lambda calculus ). In the tradition of conceptual logic, all non-conceptual systems are sometimes called judgment logic; In terms of content, this generalization is wrong from a modern point of view.
Syllogistics
The historical starting point of the conceptual logic are the work of Aristotle , who presented a formal logical system in the form of his syllogistics in the modern sense. In syllogistics, arguments are viewed in a rigid form, which consists of exactly three judgments, two premises and one conclusion. Premises and conclusions express the relationship between exactly two terms. Aristotle distinguishes four types of judgments:
- Universal affirmative: "All A are B" (A is a species of genus B, e.g. "All humans are mammals.")
- Universal negative: "No A is B" or "All B are not-A" (where "Not-A" is the conceptual negation of A, i.e. the term that includes everything that does not fall under A)
- Particularly affirmative: "Some A are B" (e.g. "Some people are logicians")
- Particularly negative: "Some A are not B" (e.g. "Some people are not logicians")
Aristotle does not regard proper names (e.g. "Socrates") as concepts in this sense.
Leibniz's conceptual logic
Leibniz developed a logical system as early as the 17th century whose formal features were already similar to the later Boolean system (see next chapter). In this sense, Leibniz's work can be viewed as a foretaste of the algebraization of logic, even if his work probably remained without much influence historically and only in the 20th century - after the development of formal algebra - received greater attention and was fully appreciated were.
Leibniz developed several formal systems in the course of his work and used different symbols, which will not be discussed in more detail here. What all stages of Leibniz's development have in common is that in the case of concepts, their intent , i.e. the conceptual content, is the focus of consideration. The term content is defined as the totality of all features that make up the term. In this sense, the content of the term human includes, for example, characteristics such as “reasonably gifted”, “linguistically gifted” or “two-legged” (but is of course not fully determined by these three characteristics).
Leibniz already sees the connection between intensional and extensional interpretation of formal conceptual logic and is aware that the valid statements that his systems make about the scope of concepts and their relationships, with a suitable interpretation of the signs used, become valid statements about the conceptual contents and their relationships .
In an early system, Leibniz assigned a prime number to each atomic concept or concept variable, for example the number 3 for the concept A, the number 5 for the concept B and the variable 7 for the concept C. The combination of concepts in this system corresponds formally to the numerical one Multiplication. In this example, the term AB would be assigned the number 3 × 5 = 15, the term ABC the number 3 × 5 × 7 = 105. With this method it is possible to decide purely arithmetically whether a term falls under another term: In general, a term S falls under a term P if and only if the numerical value of S is an integer (i.e. with remainder 0) through the numerical value of P is divisible. Here are two examples:
- “All AB are B” (“All pink pigs are pigs” if A stands for the term “pink” and B for the term “pig”): To check the validity of this statement, divide the numerical value of AB the above assignment 15, by the numerical value of B, i.e. 3. The result of this division is 5, the remainder is 0. Since the remainder is 0, the statement “All AB are B” is valid.
- “All AB are C” (“All pink pigs are motor vehicles”): If you divide the numerical value of AB, 15, by the numerical value of C, 7, the result is the number 2 and remainder 1. Since this remainder of 0 is different, the statement “All AB are C” is not valid.
The analogy to calculating with prime numbers becomes more difficult as soon as it comes to negative (negative) and particular statements. In order to be able to handle negative statements adequately, Leibniz has to assign a second, negative prime number to every atomic term. Due to the associated complications, Leibniz abandoned this first system early on.
Algebraization of Logic: Boolean Conceptual Logic
Traditional logic in the sense of conceptual logic experienced its technical climax with its algebraization by George Boole and Augustus De Morgan in the 19th century.
In the Boolean system, the variables stand for terms, but expressly for their scope (extension), not for their content (even if the content of the term can be interpreted by appropriately reinterpreting the linking symbols). Boolean system uses uppercase letters for terms, the character 0 (zero) for the empty term that does not include anything, and the character 1 (one) for the universal term that includes everything. Conceptual signs are linked by simply writing them next to each other or by using one of the signs "+" (plus) and "-" (minus):
- Writing side by side, e.g. B. “AB” is interpreted as an intersection or (more in line with the conceptual logic) as the formation of a term that only includes things that fall under both “A” and “B”. For example, if A stands for the term “philosopher” and B for the term “logician,” then AB stands for the term “logician and philosopher,” that is, the term that includes all persons who are logicians and philosophers at the same time.
- The “addition”, “A + B”, is interpreted as the term that encompasses everything that falls under either A or B. If there are things that fall under both A and B, then the expression “A + B” is undefined - that is the big difference between the Boolean system and later conceptual systems. For example, if A stands for the term “man” and B for the term “book”, then “A + B” is the term that includes both people and books. If, on the other hand, A stands for the term “logician” and B for the term “philosopher”, then the term “A + B” is undefined because there are logicians who are also philosophers (and vice versa).
- The “subtraction”, “A − B”, is interpreted as the formation of a term that includes all things that come under A but not under B. For example, if A stands for the term “man” and B for the term “logician”, then “A − B” stands for the concept of people who are not logicians.
To express the relationship between two terms, Boole uses different, equivalent spellings. The statement (the “judgment”) “All A are B”, for example, can be expressed in his system as AB = A and A (1-B) = 0, among other things.
Boole's conceptual logic system is the first that is formally so elaborated that it also allows a propositional interpretation. If one interprets the variables not as concepts, but as statements, the "multiplication" as the sentence link (the connecting word) "and" ( conjunction ) and the addition as the exclusive or ("either ... or ..."), then all valid conceptual logic Statements from the Boolean system on valid propositional statements. This observation of the structural equivalence of completely different logical systems (conceptual logic and propositional logic) established the discipline of formal algebra , also known as abstract algebra.
Relations in logic: Augustus De Morgan and Charles Sanders Peirce
The deficiency in Boole's conceptual logic system as well as in traditional conceptual logic in the sense of the syllogistics is the lack of possibilities for the treatment and representation of relations . Relations are relationships between individuals (or relationships between concepts), for example the relationship of being larger, as it exists between the two numbers 5 and 2 (5 is greater than 2). They are not only of great importance in mathematics, but almost everywhere in daily and scientific reasoning, so that from today's perspective it is almost surprising that they were not considered in the long tradition of Aristotelian-based logic.
De Morgan deserves the credit for pointing out the importance of relations for reasoning in general (and for mathematical reasoning in particular). He is often - possibly not rightly - ascribed the objection to the traditional conceptual logic that has become classic, which consists in the formulation of the following argument:
- All horses are animals.
- So all horse heads are animal heads.
This argument, although clearly intuitively valid, cannot be adequately formulated with the means of traditional conceptual logic.
It was Charles Sanders Peirce who, in his article Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic , published in 1870 , succeeded in applying the ideas of Boolean algebra to relations (not just relatives , but also relative terms - called "relation terms " - to apply and expand.
The transition to the quantifiers: Peirce, Schröder, Tarski
Peirce already uses quantifiers as they appear in Ernst Schröder's logic . With both authors, however, it is uncertain whether they viewed the quantifiers as a mere aid with which certain complex facts can be expressed more easily, or whether they regarded the quantifiers as necessary for full expressiveness; that is, whether they assumed that there are facts that cannot be expressed without the use of quantifiers - in a purely algebraic system.
Alfred Tarski answers this question in terms of content: He succeeds in showing that quantifiers are essential for full expressiveness.
Conceptual logic from a modern point of view
From a modern point of view, traditional conceptual logic is equivalent to a special case of predicate logic , namely to single-digit (“monadic”) predicate logic. One-digit predicate logic is limited to the use of single-digit predicates, e.g. B. "_ is a person" or "_ is a logician". It is possible to translate between traditional conceptual logic and single-digit predicate logic by replacing each concept X with the single-digit predicate “_ is X” and vice versa (e.g. the term “human” with the predicate “_ is a human”). Therefore, from an abstract formal point of view, it is irrelevant whether one uses conceptual logic or single-digit predicate logic. The single-digit predicate logic - and thus the traditional conceptual logic - can be decided .
Conceptual logic with the relational extensions, as proposed by De Morgan and implemented by Peirce, requires general, that is, multi-digit predicate logic, for the predicate logic representation. Relations (in conceptual terminology: relational concepts) are expressed by two-place predicates. For example, the mathematical magnitude relation in the predicate logic is expressed by the two-digit predicate "_ _{1} is greater than _ _{2} ". An additional advantage of the predicate logic is that arbitrary relations can of course be expressed, for example the three-digit relation "_ _{1} is between _ _{2} and _ _{3} ".
With relations alone, for example with the relationally extended conceptual logic system of Peirce, the full scope of predicate logic can still not be covered - this also requires the use of quantifiers in the conceptual logic, which there - see the above remarks about Peirce - actually were introduced early.
Decline of conceptual logic
Conceptual logic systems and modern logic systems such as propositional logic or predicate logic were used in parallel well into the 20th century , with the frequency and intensity of the use of conceptual logic falling to the same extent as the frequency and intensity of the use of modern logical systems increasing. This change is mainly explained by the fact that the modern logical systems satisfied the needs of the predominantly mathematical users better than the classical, conceptual-logical systems, while at the same time the influence of philosophy with its strongly tradition-oriented attachment to the Aristotelian conceptual logic faded more and more into the background. To put it bluntly: The majority of users found the modern logical formalisms, for example propositional and predicate logic, to be simpler and more appropriate to the problem than the conceptual logic in the form of Peirce, which was itself expanded to include relations and quantifiers and thus abstract and formally equally powerful.
Modern recourse to conceptual logic
Despite the factual (practical application) complete replacement of conceptual-logical systems by modern logical systems, there are isolated recourse to conceptual-logical considerations and systems - even apart from purely historically motivated investigations. Such recourse rarely or never occurs within formal logic or mathematics, but predominantly in the field of philosophy. In fact, the individual recourse to conceptual logic is mostly motivated in one of the following ways:
- Occasionally a primacy of the term over other logical categories such as functions, predicates or statements is asserted or demanded; In the English-speaking world, the term terminist philosophy is used for this (philosophical!) point of view . With such an attitude, working with a system whose basic concepts are not concepts is at least unsatisfactory, and there is an effort to be able to express the primacy of the concept within a logical system.
- The structural discrepancy between the formulas of modern logical systems, mostly predicate logic, and their natural language equivalents is often pointed out. It is argued that in some or in many important cases a conceptual logic formula is structurally more similar to a natural language statement than, for example, a statement of predicate logic.
- Often it is also argued in a purely practical manner that modern formal logic is difficult to learn and that a simple conceptual logic system - for example in the sense of the - is used to express simple connections as they come across in everyday life - possibly also in everyday scientific life Syllogistics - enough that is also easier to learn. Of course, both assumptions can be discussed - both the greater complexity of single-digit predicate logic compared to syllogistics and their sufficiency for everyday (scientific) work.
Logical systems, such as those propagated by Bruno Freytag-Löringhoff in the 1960s , fall into the first category . Systems like the TFL ( term-functor logic ) by Fred Sommers , also developed in the 1960s , fall into one of the latter two categories . From a formal point of view, both systems in their full expression are equivalent to modern predicate logic in such a way that every statement from one of these systems can be clearly translated into a predicate logic statement and vice versa.
The most important application of term logic in recent times is John Corcoran's formalization of Aristotelian logic through natural deduction in 1973. The forerunner is Jan Łukasiewicz , who gave the first terminological formalization of Aristotelian logic in his book. Both systems have the advantage that the entire Aristotelian syllogistics can be derived without additional assumptions that Aristotle does not have (existence assumptions). In contrast to Corcoran, Łukasiewicz used propositional logic in his formalization of Aristotelian logic , which has been criticized frequently since then and can be avoided through Corcoran's work. Corcoran's theory is valued by philosophers and historians of logic because the evidence through Natural Inference reproduces almost verbatim Aristotle's reasoning in his Analytica priora .
In 1965, Hans Hermes set up a term logic with a selection operator.
See also
literature
Secondary literature
- William Kneale , Martha Kneale : The Development of Logic . Clarendon Press, 1962, ISBN 0-19-824773-7
- Jan Łukasiewicz : Aristotle's Syllogistic from the Standpoint of Modern Formal Logic . 2nd Edition. Clarendon Press, Oxford 1957
- Otto Bird: Syllogistic and Its Extensions . Prentice-Hall, Englewood Cliffs 1964
- Logic . In: Encyclopaedia Britannica , 15th edition. Britannica, Chicago 1974 (2003), ISBN 0-85229-961-3 , Volume 23, pp. 225-282.
- Albrecht Heinekamp, Franz Schupp (Ed.): Leibniz Logic and Metaphysics . Scientific Book Society, Darmstadt 1988
Primary sources
- Charles Sanders Peirce : Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic . In: Memoirs of the American Academy of Sciences , 9, 1870, pp. 317-78
- Charles Sanders Peirce: On the algebra of logic. A contribution to the philosophy of notation . In: The American Journal of Mathematics , 7, 1885, pp. 180-202
- Translated from English, reprinted in Karel Berka , Lothar Kreiser: Logic texts. Annotated selection on the history of modern logic . 4th edition. Akademie, Berlin 1986, pp. 29-51
- Alfred Tarski : On the Calculus of Relations . In: Journal of Symbolic Logic , 6, pp. 73-89
- George Boole : Investigation of The Laws of Thought On Which Are Founded the Mathematical Theories of Logic and Probabilities . Dover, New York 1958, ISBN 0-486-60028-9
- George Boole: The mathematical analysis of logic: being an essay towards a calculus of deductive reasoning . 1847, ISBN 1-85506-583-5
- Translated from English, annotated and with an afterword by Tilman Bergt: The mathematical analysis of logic . Hallescher Verlag, 2001, ISBN 3-929887-29-0
- Abridged and translated from English, reprinted in Karel Berka , Lothar Kreiser: Logic texts. Annotated selection on the history of modern logic . 4th edition. Akademie, Berlin 1986, pp. 25-28
- Bruno Freytag called Löringhoff: Logic I. The system of pure logic and its relationship to logistics . 5th edition. Kohlhammer, Stuttgart 1972, ISBN 3-17-232221-1 (= Urban books 16)
- Bruno Freytag called Löringhoff: New system of logic. Symbolic-symmetrical reconstruction and operative application of the Aristotelian approach . Meiner, Hamburg 1985, ISBN 3-7873-0636-6 (= Paradeigmata 5)
- Gottfried Wilhelm Leibniz : Generales Inquisitiones de Analysi Notionum et Veritatum , 1686: lt.-dt. = General research on the analysis of concepts and truths , edited, translated and commented on by Franz Schupp. Meiner, Hamburg 1982
- Fred Sommers: The Calculus of Terms . In: Mind , vol. 79, 1970, pp. 1-39; Reprint: George Englebretsen (Ed.): The new syllogistic . Peter Lang, New York 1987, ISBN 0-8204-0448-9
- Fred Sommers: Predication in the Logic of Terms . In: Notre Dame Journal of Formal Logic , Volume 31, Number 1, Winter 1990, pp. 106-126, projecteuclid.org
- Fred Sommers, George Englebretsen: An invitation to formal reasoning. The logic of terms . Ashgate, Aldershot / Burlington / Singapore / Sydney 2000, ISBN 0-7546-1366-6
Web links
- A good text about Leibniz's logical system (PDF, 223 kB)
- Web presence of a representative of Freytag's point of view
- George Boole: The Calculus of Logic in the English original
Individual evidence
- ↑ For more information on Leibniz's logical systems, see z. B. Glashoff (PDF)
- ^ Logic . In: Encyclopaedia Britannica , 15th edition. Britannica, Chicago 1974 (2003), ISBN 0-85229-961-3 , Volume 23, p. 270
- ^ William Kneale, Martha Kneale: The Development of Logic . Clarendon Press, 1962, ISBN 0-19-824773-7 , p. 330
- ^ William Kneale, Martha Kneale: The Development of Logic , Clarendon Press 1962, ISBN 0-19-824773-7 , p. 338
- ^ Logic . In: Encyclopaedia Britannica , 15th edition. Britannica, Chicago 1974 (2003), ISBN 0-85229-961-3 , Volume 23, p. 272
- ↑ For its logical ideas s. Entry in Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
- ^ Logic . In: Encyclopaedia Britannica , 15th edition. Britannica, Chicago 1974 (2003), ISBN 0-85229-961-3 , Volume 23, p. 273
- ^ Logic . In: Encyclopaedia Britannica , 15th edition. Britannica, Chicago 1974 (2003), ISBN 0-85229-961-3 , Volume 23, p. 273
- ^ J. Corcoran: Completeness of an Ancient Logic . In: The Journal of Symbolic Logic , Vol. 37, Number 4, December 1973
- ↑ Jan Łukasiewicz: Aristotle's syllogistic. From the standpoint of modern formal logic . Clarendon Press, Oxford 1951.
- ↑ George Boger: Completion, Reduction and Analysis: Three Proof-theoretic Processes in Aristotle's Prior Analysis . In: History and Philosophy of Logic , 19, 1998, pp. 187-226