Syllogism
The syllogisms ( AltGr. Συλ - λογισμός syllogismos " ( the ) aggregation, logical conclusion " ) are a catalog of types of logical arguments. They form the core of the fourth century before our era incurred ancient logic of Aristotle and the traditional logic until the 19th century. As the syllogistic theory is described by the syllogisms in general. It examines in particular the conditions under which syllogisms are valid.
Syllogisms are always built on the same pattern. Two premises ( preconditions ), major premise and minor premise called, lead to a conclusion (Conclusion ). The premises and the conclusion are statements of a certain type, in each of which a term, the syllogistic subject, another term, the syllogistic predicate (not synonymous with subject and predicate in the grammar) agreed, in a certain way on or off. Depending on the point at which they occur in the syllogism, the terms occurring generic term, medium term and narrower term are called.
An example of a valid syllogism is the following:
- 3.1 modes ( combinations ) and their wishlist words
- 3.2 Reduction to the first figure 3.2.1 Simple conversion
- 3.2.2 Conversion by limitation
- 3.2.3 interchange of the premises
- 3.2.4 Indirect evidence
- 3.2.5 Different representations
- 3.3.1 The first figure of the categorical syllogism 3.3.1.1 AAA - mode Barbara
- 3.3.1.2 EAE - mode Celarent
- 3.3.1.3 AII - mode Darius
- 3.3.1.4 EIO - mode Ferio
- 3.3.1.5 AAI - mode Barbari
- 3.3.1.6 EAO - mode Celaront
- 3.3.2.1 - AOO mode Baroco
- 3.3.2.2 EAE - mode Cesare
- 3.3.2.3 AEE - mode Camestres
- 3.3.2.4 EIO - mode Festino
- 3.3.3.1 OAO - mode Bocardo
- 3.3.3.2 AII - mode Datisi
- 3.3.3.3 IAI - mode Disamis
- 3.3.3.4 EIO - mode Ferison
- 3.3.3.5 AAI - mode Darapti
- 3.3.3.6 EAO - mode Felapton
- 3.3.4.1 AAI - mode Bamalip
- 3.3.4.2 AEE - mode Calemes
- 3.3.4.3 IAI - mode Dimatis
- 3.3.4.4 EAO - mode Fesapo
- 3.3.4.5 EIO - mode Fresison
History
The Latin term syllogism goes back to the Greek syllogismos ( συλλογισμός ). With syllogismos Aristotle called a deductive argument, which he defined as the first as follows:
" A deduction ( syllogismos ) is therefore an argument in which, if something has been set, something other than the laws of necessity by the laws results. "
In this broader sense, ie as a synonym for the word "argument", the word " syllogism " in everyday language was used until the 20th century. In modern parlance, these wide use is no longer common and can be found ( a collective term for certain consideration in the tradition of propositional inferential ) only in compound words such as hypothetical syllogism.
Syllogism called confusingly traditionally now exclusively a special form of deductive argument ( syllogismos ), namely, in Aristotle's First Analytics deduction treated, consisting of exactly two premises and a conclusion three terms. Since the definition of deduction does not have this limitation, although every syllogism is a syllogismos, but not everyone syllogismos a syllogism.
After the position of the middle term - that is, of that concept, which occurs only in the premises - Aristotle distinguishes three types of circuits, called figures (see figures). The introduction of a fourth figure whose conclusions Aristotle already recognizes as valid, is ascribed by Avicenna and Galen other, although there is no direct evidence in the traditional works of Galen of this write-up and this they even expressly denies, in fact. Until the introduction of the fourth figure syllogisms her in the tradition of Theophrastus are often attributed to the first figure.
In the Latin Middle Ages, which first took up the logical works of Aristotle from translations and commentaries of Boethius, the traditional Latin names for the quantity and quality of the judgments were (see section types of statements ) by Peter Hispanus common. In the scholastic syllogistic received the form, which was then handed down for centuries in the textbooks, the authentic content of the Aristotelian syllogistic had been lost since antiquity and it has undergone since the Renaissance increasingly sharp criticism ( is famous about the criticism of René Descartes). It was not until January Łukasiewicz discovered Aristotle's logic in a pioneering work new and reconstructed from the standpoint of modern logic of axiomatic; among other things because of the high number of this recognized axioms, however, doubts that this reconstruction has failed adequately to state adequately. On Łukasiewicz connects to recent research that has found her German -language standard work in Günther Patzig representation ( 1959).
Since then, a distinction between the Aristotelian and the traditional syllogistic. The most noticeable external difference is that Aristotle syllogisms not writing down as a series of three sets, but as a sentence of the form "If ( premise 1 ) and ( premise 2), as necessary ( conclusion )"; there is disagreement as to whether this formulation could be explained as a metalinguistic statement about a syllogism in the traditional sense, or whether the view Łukasiewicz was to follow that Aristotle consider a syllogism as a composite statement. The two readings can be easily transformed into one another; the present article gives concrete syllogisms in the sense of the former reading consistently as a result of three sets again. Even apart from this there is disagreement between the Aristotelian and the traditional syllogistic numerous differences in the logical- semantic conception, so that today many argue Aristotle stand of modern logic basically a lot closer than the traditional syllogistic. Already on Augustus De Morgan goes elaborated among others by Dittes view of Aristotle's syllogistic theory of certain double-digit returns as relations between concepts and the relative product of such relations. A syllogism is then a relation product, which again is itself a relation in that particular form, which is expressed in the four record types A, E, I or O ( for A, E, I, O see types of statements ).
The indiscriminate equation of Aristotelian and traditional syllogistic in the earlier history of logic ( Carl Prantl, Henry Maier ), however, has numerous errors - such as the alleged metaphysical presuppositions of Aristotle's logic - produced, of which Aristotle interpretation could free only with difficulty.
General presentation
Syllogistic arguments are always built on the same pattern. Two premises ( preconditions ), called major premise (Latin propositio major) and subset (Latin propositio minor), lead to a conclusion (Conclusion, latin conclusio ). In the categorical syllogism shown here (also called assertoric syllogism ) are premises and conclusion categorical judgments, that is, statements in which a term (Greek ὅρος - horos, latin terminus), the subject, another term, the predicate, in certain manner is attributed to or. For example, in the categorical judgment " All men are mortal " the subject "man" the predicate awarded " mortal". Please note - and be seen in this example - is that the words "subject" and " predicate " in the context of syllogistic be used differently than in traditional grammar, where the grammatical subject of the phrase "all men " and the grammatical predicate - the on the perspective - the word "are" or the phrase " are mortal " would be.
Within a syllogism total of three different terms are used:
Following in the footsteps of John Philoponus the names " preamble " and " concept " since the 17th century is largely attributed to any substantive meaning and they are explained solely from their appearance in the major premise or in the base and as a predicate or subject of the conclusion. Occasionally, the lower and upper term are also referred to as the subject or predicate of the syllogism.
An example of a valid syllogism is the following:
The middle term in a syllogism, the term " rectangle "; in the major premise of this syllogism the middle term occurs as a subject in its pedestal as the predicate. The narrower term of this syllogism, the term " square, " he steps in the pedestal on as a subject. The preamble of this syllogism is, after all, the term "circle; " it occurs in the upper set on a predicate.
As an alternative to phrases such as " No S is P" or " All S are P" are also synonymous terms such as " P is not S to" and "P comes all S to" used. In this expression, the above syllogism is as follows:
The two notations are equivalent and equal. While Aristotle himself in his Posterior Analytics mostly variants of the second formulation, " is P all S to" selected (usually " τὁ P κατηγορεῖται τοῦ S " - " P is said about the S"), is since of scholasticism variants of the first spelling, " All S is P, " given preference. More than in the traditional occurs in Aristotle's formulation revealed the difference between grammatical and syllogistischem subject and predicate; so has the phrase " P comes mainly south to" the syllogistic predicate "P", the function of the grammatical subject and the subject syllogistic, "S", the function of the grammatical predicate.
There are, however, in the wake of Jan Łukasiewicz the opinion that the Aristotelian syllogisms in contrast to those of no arguments from two premises and a conclusion be upon him professional tradition going, but composite individual records. From this perspective, the Aristotelian variant of the above example would read as follows:
The correct classification of the Aristotelian syllogisms is controversial to this day. Since the conversion between the two readings is simple and since Aristotle " if-then " used his syllogisms despite their formulation in form as final rules, the present Article concrete syllogisms consistently in its traditional formulation as a composite of three statements arguments dar.
As a further development of the categorical or assertoric syllogistic Aristotle, there are already signs of a modal syllogistic, when in - apart from this difference, the same construction - modal syllogisms statements such as " All men are mortal may " are allowed.
Logical systems that work like the syllogistic with statements in which concepts are related to each other, are commonly called term logics.
Types of statements
A statement in a syllogism, a categorical judgment, always requires two terms in a relationship. Only four types of judgments regarding the relationship between a subject ( S) and a predicate (P) are considered:
The vowels are drawn from the Latin words " affirmo " ( I affirm ) and " nego " ( I deny ), each of the first vowel represents a general, the second for a particulate judgment.
Quantity and quality
The property of a statement about how many items she speaks, is traditionally called the quantity of this statement. In this sense there is in the syllogism two quantities, namely ( a) particulate and ( b ) universal or general. The property of a statement, a subject, a predicate to deny or, is traditionally called the quality of this statement. Speaks a statement to a subject a predicate, they are called affirmative statement, she speaks it from him negative decision. The types of statements are broken down in the following table according to their quality and quantity:
Logical square
Under the assumption that their subjects are not empty concepts exist between the different types of statements different relationships:
- Two statements form a contradictory opposite if and only if both not simultaneously be true nor false at the same time, in other words, if both have different truth values must have. Which in turn is exactly the case when a statement is the negation of the other (and vice versa). For the syllogistic statements types meets the adversarial relationship with the pairs A -O and I-E.
- Two statements form a contrary contrast iff they not both at the same time true, but probably both be wrong. In the syllogistic is only the pair of statements A-E in contrary juxtaposition.
- Two statements form a contrast subkonträren if and only if both ( probably true but both at the same time ) may be wrong at the same time. In the syllogistic is only the pair of statements I- O in subkonträrem contrast.
- Between the statement types A and I on the one hand and E and O on the other hand an inference context ( traditionally, this inference relation mentioned in the logical square subalternation ): A implies I, that is, if all SP are, then there is actually S, P are; and from E follows O, that is, if no SP, then there are actually S that are not P.
These relationships are often in a scheme under the name of " logical square " was announced summarized ( see figure). The oldest known writing of the logical square dates from the second century AD and Apuleius attributed by Madauros.
Existential conditions
As in the logical square seen many of the traditional laws of syllogistic valid only under the condition that at least the subject concerned statements is not empty. In general, it is therefore assumed that syllogistic statements actually make existential statements about the subject, that is, assume that the subject is not an empty phrase:
- The statement " All S is P " means this: " There are S, and all of them are P".
- The statement " No S is P " means this: " There are S, and none of them are P".
- The statement " Some S are P " means this: " There are S, and some of them are P. "
- The statement " Some S are not P " means this: " There are S, and some of them are not P. "
The existence statement " There are S" is not usually understood as part of the respective syllogistic judgment, but as its presupposition, ie, as a prerequisite for the respective ruling on syllogistic reasoning can be used at all. While making The existence theorem for part of the syllogistic judgment is possible, but formally quite complicated, and is judged differently in terms of its adequacy.
Depending on the interpretation of syllogistic statements and laws, the view is possible that syllogistic Close only with non-empty terms was at all possible, that is, that the predicates can not be empty. The question of which authors have represented the tradition which view is perceived differently and is still a matter of philosophical and philological studies.
Although existential conditions similar to the natural language (usually one feels only generalizations about actually existing things as meaningful ), it is important to be aware of, because there is just logical systems that do not make these requirements.
Distribution
In the syllogism is of the distribution ( from the Latin distributio, distribution) spoke of a term within a statement. A term is within a statement if and only distributed if this statement follows from any other statement that comes from the original statement by the original term is replaced by a real sub- term. An often used and equivalent with proper understanding formulation is: A term is within a syllogistic statement if and only distributed when he refers in the statement to all items on which the concept.
For example, in the syllogistic A- statement " All philosophers (subject) are people ( predicate ) " the term " philosopher " distributed: From the fact that all philosophers of people, it follows that all philosophers of language ( a narrower term of " philosopher " ) People are that all existentialists (another sub- concept of " philosopher " ) people, etc. not distributed in this statement, however, the term " person": from the fact that all philosophers people are not followed, for example, for a long time, that all philosophers Europeans are ( a sub- concept of man ).
For an overview of to what type of statement which term is distributed, gives the following table.
Syllogisms from a modern perspective
The classical syllogisms can be represented both as a modern application of a subsystem of predicate logic, namely the monadic predicate logic, as well as a lot of relationships.
In the representation of the amount of each term relationships is interpreted as its scope ( technical language extension), that is, as the set of objects that fall under this term. The term " person," for example, is interpreted set-theoretically as the set of all people.
In predicate logic interpretation of each term is represented as a predicate in the sense of predicate logic, that is, as a unary function in the mathematical sense, which can be applied to concrete individuals and the information for each individual provides, whether it under this term falls or not. For example, would the term "man" as the predicate "_ is a human being " interpreted. If we apply this predicate to a man to, for example, Socrates, then it returns the truth value "true"; turns you put it on an item that that is not a human - for example, to an animal, to a planet or to a number - then it returns the truth value "false".
Rules for the validity of syllogisms
Valid syllogisms have certain characteristics in terms of quality, quantity and distribution of the terms occurring in them; for example, may be a valid syllogism never when its premises are particulate statements, but his conclusion is a general statement.
As a function of the particular interpretation of many syllogistic modes are different valid, there is the tradition, different sets of rules. The following are the most common today rules are displayed. You go into this simple form back to the late Middle Ages and are not part of the ancient, Aristotelian syllogistic. The said control system is the simplicity redundant, that is, some of the rules can be expressed by others.
Rules of quality
Rules of quantity
Rules of Distribution
Figures
Which of the three terms S, P and M must occur in which statement of the syllogism is set: The upper set consists of P and M, the subset of S and M, the conclusion of S and P. The conclusion always has the form S - P, the arrangement of the terms in the premises can be chosen freely. The order in which the premises are written is indeed irrelevant to the validity of a syllogism, but since Aristotle the major premise and following the subset is called first.
Depending on the arrangement of the terms in the premises, we distinguish between four possible figures ( σχἠματα, schemes ):
Example:
Modes ( combinations ) and their watch words
Since each of the three statements can be in a syllogism of one of the four types A, E, O, I, there are ways to combine statements to a syllogism of the respective figure per character. Each of these options is a mode (plural: modes) called or a combination of the respective figure. A total of four different figures there are so total possible combinations, ie 256 types of syllogisms. Among these 256 modes are valid 24 and 232 is not valid syllogisms.
One mode is described by three letters. Here are the first two letters for the types of the premises, the third letter for the type of conclusion.
Example:
The 24 valid modes are traditionally referred to with the following features of the words:
In these words the vowels Wish describe the types of statements in the order upper set - subset -conclusion; for example, called mode Darius a syllogism of the first figure and the type A -I -I. The consonants indicate on which the first figure syllogism (first consonant ) of the respective syllogism can be recycled and what change (in each case to the following vowel consonant ) this reduction is possible (see section reduction to the first figure ).
It should be noted that in the tradition of circulating different versions of the watch words. The earliest known versions of these mnemonic syllogistic come from the scholastic logicians William of Sherwood and Peter Hispanus to 1240/1250, where the priority is uncertain.
The five non- bold modes are subject to a "weak " conclusions of a bold "strong" mode of each figure. "Strong " means that the conclusion is a general statement (A or E); "Weak" means that the conclusion is a particulate statement (I or O), which is a direct consequence of the respective strong statement. It is assumed that weak modes were first 50 BC addressed by Ariston of Alexandria.
Examples:
- Mode Barbara (strong): All Munich is Bavaria, all Munich Schwabing are, it follows: All are Schwabinger Bavaria.
- Mode Barbari (weak): All Munich is Bavaria, all Munich Schwabing are, it follows: Some Schwabinger are Bavaria.
- Mode Celarent (strong): No Munich is Passau, all Schwabing Munich are, it follows: No Schwabing is Passau.
- Mode Celaront (weak): No Munich is Passau, all Schwabing Munich are, it follows: Some Schwabinger are no Passau.
The weak conclusions are logically valid, provided certain Zusatzbedinungen are met: Each specific terms ( subject, predicate or middle term ) must not be empty ( see also section existential conditions ).
Reduction to the first figure
With some simple transformations that are encoded in the consonants of the traditional reminder words, the modes of all the figures suggest a mode of the first figure traced ( " reduce "). This fact was already known to Aristotle, who also formulated corresponding transformation rules and the first figure as the perfect syllogisms of the first figure as a perfect syllogism - designated ( τέλειος συλλογισμός teleios syllogismós ).
The first letter of each word indicates traditional shopping, may be on the mode of the first figure, the selected mode is returned: modes whose name starts with "B" can be traced back to the mode Barbara; Modes whose name starts with "C", can be traced back to the mode Celarent; and also can be modes whose name starts with " D" or "F", attributed to the mode of Darius and Ferio.
The transformations of the syllogism are rules of inference in a formal sense, that is, the result of each syllogistic transformation of a statement or a syllogism follows from the re-formed statement or from the formed syllogism.
The transformations required for the reduction are described in more detail below; In addition, an example is mentioned and shown its reduction to the first figure in the Examples section and reduction to the first figure for each syllogistic mode.
Easy conversion
In the simple conversion (Latin conversio simplex) subject and predicate of each statement are interchanged; then from the statement " Some philosophers are Greeks," according to the simple conversion, the statement " Some Greeks are philosophers '. Add to wish words to easily convert to a statement by the letter " s" appears after the statement of the relevant associated vowel; For example, the first premise, an e- statement, a simple conversion must be subjected in reducing the Cesare mode.
Easy conversion is only possible with statements of types E and I: If no pigs are sheep, then no sheep pigs ( e- statement ); and if some Greeks philosophers, then so are some philosophers are Greeks (I statement ). For the A and O statement not a simple conversion is possible: if all men are philosophers, that is namely not long, that all men are philosophers (A- statement ); and if some people are not politicians, that does not mean that some politicians are not human (O- statement ). In fact, among the traditional watch words are only those in which the following "s" to an "e" or "i".
Normally, the simple conversion is applied to each of the premise to be reduced syllogism. If, however, the " s" at the end of the watch word, then do not the conclusion of the syllogism to be reduced to the simple conversion is subjected, but the conclusion of that syllogism of the first figure to be reduced to the. An example of this special case is the Dimatis mode: it is attributed to a mode Datisi, in its conclusion subject and predicate are reversed, ie a syllogism of the form " All P are M. Some M are S. So are some P p "
Conversion by limitation
In the conversion by limitation (Lat. conversio per accidens ) in addition to the interchange of subject and predicate of each statement changed their type from A to I or from E to O. Thus, for example, is from the A- statement, " All pigs are pink " after conversion by limiting the I- statement " Some pink ( things ) are pigs " and is from the e- statement " No Pigs are sheep," the O- statement " Some sheep are no pigs ". Add to wish words is the conversion by restricting by the letter " p" is displayed after the statement of the relevant associated vowel.
Also in this transformation is a special case before, when the "p" in the mark-word after the third vowel - ie at the end of the word - is: In this case it refers not, as in the simple conversion to the conclusion of the reducible syllogism, but on the conclusion of the resulting syllogism of the first figure.
Permutation of the premises
Permutation of the premises (Latin mutatio praemissarum ) is required for the reduction of all those modes in which watch words of the consonant "m " occurs at any point. Regardless of the position of the consonant "m " in the respective mark-word interchanging the premises after each as may be required simple conversion and after each conversion that might be necessary should be carried out by restriction.
Indirect evidence
Modes in which watch words of the consonant "c" exists, but is not in word-initial, - therefore, only the modes Baroco and Bocardo - can be traced back to the first figure only by indirect evidence (Latin reductio ad absurdum ). At this Behuf the truth of A is the premise to reducing syllogism (in the case of Baroco ie the first, the second premise in the case of Bocardo ), that is, the negation of the conclusion adopted as well as the contradictory opposite. In this way, a mode Barbara, contradicts the conclusion of the O- premise of reducing to syllogism formed. Since the assumption that the conclusion is not correct to say of such has led to a contradiction is shown that the conclusion must be true.
Is carried out in detail the indirect evidence in sections - AOO mode Baroco and OAO - mode Bocardo.
Divergent views
With regard to the exact formulation of the transformation rules can be found at the individual authors differences; in particular, it is customary to omit the results presented here a special case in the simple conversion and the conversion by limitation and the consonant "s" and "p" also refer to the end of a word to be converted syllogism and not - as shown here - on the target syllogism. However, this formulation would be the reduction of the two modes ' Bamalip "and" Camestrop " in the illustrated form, impossible, because neither a I message nor an O- statement conversion, by limiting possible.
Examples and reduction to the first figure
The first figure of the categorical syllogism
The first figure has the following form:
Your valid modes are Barbara, Celarent, Darius, Ferio, Barbari and Celaront.
AAA - mode Barbara
EAE - mode Celarent
AII - mode Darius
EIO - mode Ferio
AAI - mode Barbari
EAO - mode Celaront
The second figure of the categorical syllogism and its reduction to the first figure
The second figure has the following form:
The valid modes of the second figure are Baroco, Cesare, Camestres, Festino, Camestrop and Cesaro.
AOO - mode Baroco
EAE - mode Cesare
AEE - mode Camestres
EIO - mode Festino
The third figure of the categorical syllogism and its reduction to the first figure
The third figure has the following form:
The valid modes of the third figure are Bocardo, Datisi, Disamis, Ferison, Darapti and Felapton.
OAO - mode Bocardo
AII - mode Datisi
IAI - mode Disamis
EIO - mode Ferison
AAI - mode Darapti
EAO - mode Felapton
The fourth figure of the categorical syllogism and its reduction to the first figure
The fourth figure has the following form:
The valid modes of the fourth figure Calemes, Dimatis, Fresison, Bamalip, Calemop and Fesapo.