"Socrates" at the Louvre

A syllogism (Greek: συλλογισμός, syllogismos, 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.

In its earliest form (defined by Aristotle in his 350 BC book Prior Analytics), a deductive syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across.[1] For example, knowing that all men are mortal (major premise) and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.[2]

In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.[3] From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably. This article is concerned only with this historical use. The syllogism was at the core of historical deductive reasoning, whereby facts are determined by combining existing statements, in contrast to inductive reasoning in which facts are determined by repeated observations.

Within some academic contexts, syllogism has been superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift (Concept Script; 1879). Syllogism, being a method of valid logical reasoning, will always be useful in most circumstances and for general-audience introductions to logic and clear-thinking.[4][5]

Early history

In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.[3]

Aristotle

Aristotle defines the syllogism as "a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so."[6] Despite this very general definition, in Prior Analytics Aristotle limits himself to categorical syllogisms that consist of three categorical propositions, including categorical modal syllogisms.[7]

The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle. Before the mid-12th century, medieval logicians were only familiar with a portion of Aristotle's works, including such titles as Categories and On Interpretation, works that contributed heavily to the prevailing Old Logic, or logica vetus. The onset of a New Logic, or logica nova, arose alongside the reappearance of Prior Analytics, the work in which Aristotle developed his theory of the syllogism.

Prior Analytics, upon rediscovery, was instantly regarded by logicians as "a closed and complete body of doctrine", leaving very little for thinkers of the day to debate and reorganize. Aristotle's theory on the syllogism for assertoric sentences was considered especially remarkable, with only small systematic changes occurring to the concept over time. This theory of the syllogism would not enter the context of the more comprehensive logic of consequence until logic began to be reworked in general in the mid-14th century by the likes of John Buridan.

Aristotle's Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise, that is, a premise containing the modal words necessarily, possibly, or contingently. Aristotle's terminology in this aspect of his theory was deemed vague and in many cases unclear, even contradicting some of his statements from On Interpretation. His original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use and would be replaced by new distinctions and new theories altogether.

Medieval syllogism

Boethius

Boethius (c. 475–526) contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the 12th century, his textbooks on the categorical syllogism were central to expanding the syllogistic discussion. Rather than in any additions that he personally made to the field, Boethius' logical legacy lies in his effective transmission of prior theories to later logicians, as well as his clear and primarily accurate presentations of Aristotle's contributions.

Peter Abelard

Another of medieval logic's first contributors from the Latin West, Peter Abelard (1079–1142), gave his own thorough evaluation of the syllogism concept and accompanying theory in the Dialectica—a discussion of logic based on Boethius' commentaries and monographs. His perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelard's distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a more coherent concept of Aristotle's modal syllogism model.

Jean Buridan

The French philosopher Jean Buridan (c. 1300 – 1361), whom some consider the foremost logician of the later Middle Ages, contributed two significant works: Treatise on Consequence and Summulae de Dialectica, in which he discussed the concept of the syllogism, its components and distinctions, and ways to use the tool to expand its logical capability. For 200 years after Buridan's discussions, little was said about syllogistic logic. Historians of logic have assessed that the primary changes in the post-Middle Age era were changes in respect to the public's awareness of original sources, a lessening of appreciation for the logic's sophistication and complexity, and an increase in logical ignorance—so that logicians of the early 20th century came to view the whole system as ridiculous.[8]

Modern history

The Aristotelian syllogism dominated Western philosophical thought for many centuries. Syllogism itself is about drawing valid conclusions from assumptions (axioms), rather than about verifying the assumptions. However, people over time focused on the logic aspect, forgetting the importance of verifying the assumptions.

In the 17th century, Francis Bacon emphasized that experimental verification of axioms must be carried out rigorously, and cannot take syllogism itself as the best way to draw conclusions in nature.[9] Bacon proposed a more inductive approach to the observation of nature, which involves experimentation and leads to discovering and building on axioms to create a more general conclusion.[9] Yet, a full method of drawing conclusions in nature is not the scope of logic or syllogism, and the inductive method was covered in Aristotle's subsequent treatise, the Posterior Analytics.

In the 19th century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Immanuel Kant famously claimed, in Logic (1800), that logic was the one completed science, and that Aristotelian logic more or less included everything about logic that there was to know. (This work is not necessarily representative of Kant's mature philosophy, which is often regarded as an innovation to logic itself.) Although there were alternative systems of logic elsewhere, such as Avicennian logic or Indian logic, Kant's opinion stood unchallenged in the West until 1879, when Gottlob Frege published his Begriffsschrift (Concept Script). This introduced a calculus, a method of representing categorical statements (and statements that are not provided for in syllogism as well) by the use of quantifiers and variables.

A noteworthy exception is the logic developed in Bernard Bolzano's work Wissenschaftslehre (Theory of Science, 1837), the principles of which were applied as a direct critique of Kant, in the posthumously published work New Anti-Kant (1850). The work of Bolzano had been largely overlooked until the late 20th century, among other reasons, because of the intellectual environment at the time in Bohemia, which was then part of the Austrian Empire. In the last 20 years, Bolzano's work has resurfaced and become subject of both translation and contemporary study.

This led to the rapid development of sentential logic and first-order predicate logic, subsuming syllogistic reasoning, which was, therefore, after 2000 years, suddenly considered obsolete by many. The Aristotelian system is explicated in modern fora of academia primarily in introductory material and historical study.

One notable exception to this modern relegation is the continued application of Aristotelian logic by officials of the Congregation for the Doctrine of the Faith, and the Apostolic Tribunal of the Roman Rota, which still requires that any arguments crafted by Advocates be presented in syllogistic format.

Boole's acceptance of Aristotle

George Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought.[10][11] Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought.[12] According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by:[12]

  1. providing it with mathematical foundations involving equations;
  2. extending the class of problems it could treat, as solving equations was added to assessing validity; and
  3. expanding the range of applications it could handle, such as expanding propositions of only two terms to those having arbitrarily many.

More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced Aristotle's four propositional forms to one form, the form of equations, which by itself was a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments, whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce: "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle."

Pure Strict syllogistic

There exists a couple of modern theories concerning syllogistic. One of them is able to explain the whole theory and makes it possible to deduce all syllogisms: Strict logic is a logic by Walther Brüning, stated in his book Grundlagen der Strengen Logik ("Foundations of Strict Logic") published in 1996. The part of "Strict general logic", which he refers to circumstances (or cases, German: Sachverhalte) and connected circumstances, presumes only the principle of limitation and the principle of identity, as well as affirmation and negation (not to be confused with the negation operation "" or the complementary-term sign "~"). This text deals with pure strict syllogistic logic in particular, which relies also on findings of Albert Menne. Individuals, or existence-stipulations, more generally, may be introduced later on (applied logic), but can also be outlined with further notation of pure logic later on.

Categorical propositions

Basics

Syllogstic arguments have traditionally a particular form: Firstly, two categorical propositions are claimed (premises). From them a third categorical proposition is derived (conclusion). Each categorical proposition contains two categorical terms. In doing so in traditional syllogistic for types of judges are differentiated concerning the relation between subject (S) and predicate (P).

code quantifier subject copula predicate type example shorthand
A All S are P universal affirmative All humans are mortal. SaP
E All S are not P universal negative No humans are perfect. SeP
I Some S are P particular affirmative Some humans are healthy. SiP
O Some S are not P particular negative Some humans are not clever. SoP

(The vocals are derived from the Latin words "affirmo" (I affirm) and "nego" (I negate), at which each first vocal stands for a universal, the second for a particular judge.)

From the assumption of a judge it is possible to deduce statements over the performance of the terms. The universal judges claim that all S are P (SaP), respectively ~P (SeP). Therefore, S without P (SaP), respectively ~P (SeP) does not exist. The particular jugdes claim that some S are P (SiP), respectively ~P (SoP). Thus from the judge it is possible to deduce ranges of values which are affirmative (A, for affirmation), negative (N, for negation) or unsettled (u) values. Hence SaP says there is no S without P (negative value), but it does not make a statement whether S and P together exist or P without S. SiP however makes a positiv statement, namely that there are S which are P (S and P appear together), but leaves the question unanswered, whether there are S without P or P without S (therefore unsettled). The adjacent graphic illustrates the situation for all four types of judges (negative values are red, positive are green).

Following tabellarly overview about the settings, which the four types of judges make, results from above:[13]

Categorical propositions
SaP
SeP
SiP
SoP
SaPSePSiPSoP
S, P uNAu
~S, P uuuu
S, ~P NuuA
~S, ~P uuuu

Existential propositions

At the adding of existential propositions (which are made in traditional syllogistic) the specification is done, that every term has to be approchable to be positive, if they should relate to something at all. Looking at SaPs' negative value of its scope, it is possible to draw further implications from this postulate for the terms. Would be SaPs' first value also negative (therefore "S, P"="N"), thus for S no possibility of positive value is left (because "S, P"="N" is already set and the other cases concern "~S"). Therefore, is for this case only a positive value left: There have to be S, which are P, because else there would be no S. One can proceed analogical with the other types of judges and with the second term of the categorical proposition. So it remains uncertain whether there are P which are not S, because a positive value of P is already ensured at "S, P".

The following overview of semantic implications of the categorical propositions result from the above said:

Categorical propositions with existential propositions
SaP
SeP
SiP
SoP
S
P		~S
P		S
~P		~S
~P
SaP AuNA
SeP NAAu
SiP Auuu
SoP uuAu

Direct inferences

It is possible to deduce a slew of direct inferences: subalternation (for example SaP→SiP), conversion, obversion, contraposition, partial inversion, inversion; contradiction, contrariety, subcontrariety.

Among other uses, conversion is needed to result from one syllogistic figure to another; that is why it is colored yellow here.

Initial judge (S•P)a AuNAe NAAui Auuuo uuAu
Conversion (P•S)i ANuAe NAAui Auuu- uAuu
Obversion (S•~P)e NAAua AuNAo uuAui Auuu
Contradiction (~P•~S)a AuNAo uAAN- uuuAo uuAu
Partial inversion (~S•P)o uAANi ANuA- uAuu- uuuA
Inversion (~S•~P)i ANuAo uAAN- uuuA- uAuu

Introduction of a third term for syllogistic reasoning

Within a syllogism overall three different terms are used:

  1. the major term, which appears in the major premise, and on the right side of the conclusion, i. e. as its predicate (P);
  2. the minor term, which appears in the minor premise, and on the left side of the conclusion, i. e. as its subject (S); and
  3. the middle term (M), which appears in the major and minor premise, but not in the conclusion.

So a third term has to be introduced. Therefore, the logical formulas (i. e. the formulas for the categorical propositions) are to be extended for each judge (major premise, minor premise, conclusion) and the values of A and N are written in lower case (a and n) from now on. For this is also possible to say, that they are triadic (i. e. three circumstances are connected with each other) extended dyadic (i. e. two circumstances are connected with each other) formulas. So called par-positions (German: "Gleichstellen") are generated.

S
M
P		~S
M
P		S
~M
P		~S
~M
P		S
M
~P		~S
M
~P		S
~M
~P		~S
~M
~P
MaPaauunnaa
MePnnaaaauu
MiPaauuuuuu
MoPuuuuaauu
SaMaunaauna
SeMnaaunaau
SiMauuuauuu
SoMuuauuuau
SaPauaunana
SePnanaauau
SiPauauuuuu
SoPuuuuauau

Rules for syllogistic reasoning

The [...] syllogistic reasoning can be traced back to the following two rules:

  1. If one of two of the a-par-positions of premises is blocked (through an n of the other premise), the a of the other par-position must go into the conclusion (the par-positions of the conclusion are thus a).
  2. If two par-positions of the conclusion are blocked by the premises through at least one n, they both must be n in the conclusion.
    — Walther Brüning, Grundlagen der strengen Logik

Now it is possible to state two arbitrary categorical judges (synthesis of premisis) and deduce therefrom possible information (analysis of conclusion). First of all, it has to be ensured that the premisis do not contradict each other, by appliance of both rules at the same time. But this is not true for all premisis-pairs in traditional logic.

Examples for strict syllogistic reasoning

Overview about the figures

The middle term can be either the subject or the predicate of each premise where it appears. The differing positions of the major, minor, and middle terms rise a classification of syllogisms known as the figure. Given that in each case the conclusion is S-P, the four figures are:

Figure 1 Figure 2 Figure 3 Figure 4
Major premise M–P P–M M–P P–M
Minor premise S–M S–M M–S M–S
Figure 1
Barbara und Barbari
MaP: a a u u n n a a
SaM: a u n a a u n a
SaP: a u a u n a n a
SiP: a u a u u u u u

The key values are bold and italic. In case that an a is marked, the first rule comes to application. In the case that two n are marked, the second rule comes to application. The rest of the conclusion is unknown (u).

Celarent und Celaront
MeP: n n a a a a u u
SaM: a u n a a u n a
SeP: n a n a a u a u
SoP: u u u u a u a u
Figure 2
Baroco
PaM: a a n n u u a a
SoM: u u a u u u a u
SoP: u u u u a u a u
Overview about the valid syllogism

Figure 1 Figure 2 Figure 3 Figure 4
Barbara Cesare Datisi Calemes
Celarent Camestres Disamis Dimatis
Darii Festino Ferison Fresison
Ferio Baroco Bocardo Calemos
Barbari Cesaro Felapton Fesapo
Celaront Camestros Darapti Bamalip

Terms in syllogism

With Aristotle, we may distinguish singular terms, such as Socrates, and general terms, such as Greeks. Aristotle further distinguished types (a) and (b):

  1. terms that could be the subject of predication; and
  2. terms that could be predicated of others by the use of the copula ("is a").

Such a predication is known as a distributive, as opposed to non-distributive as in Greeks are numerous. It is clear that Aristotle's syllogism works only for distributive predication, since we cannot reason All Greeks are animals, animals are numerous, therefore all Greeks are numerous. In Aristotle's view singular terms were of type (a), and general terms of type (b). Thus, Men can be predicated of Socrates but Socrates cannot be predicated of anything. Therefore, for a term to be interchangeable—to be either in the subject or predicate position of a proposition in a syllogism—the terms must be general terms, or categorical terms as they came to be called. Consequently, the propositions of a syllogism should be categorical propositions (both terms general) and syllogisms that employ only categorical terms came to be called categorical syllogisms.

It is clear that nothing would prevent a singular term occurring in a syllogism—so long as it was always in the subject position—however, such a syllogism, even if valid, is not a categorical syllogism. An example is Socrates is a man, all men are mortal, therefore Socrates is mortal. Intuitively this is as valid as All Greeks are men, all men are mortal therefore all Greeks are mortals. To argue that its validity can be explained by the theory of syllogism would require that we show that Socrates is a man is the equivalent of a categorical proposition. It can be argued Socrates is a man is equivalent to All that are identical to Socrates are men, so our non-categorical syllogism can be justified by use of the equivalence above and then citing BARBARA.

Existential import

If a statement includes a term such that the statement is false if the term has no instances, then the statement is said to have existential import with respect to that term. It is ambiguous whether or not a universal statement of the form All A is B is to be considered as true, false, or even meaningless if there are no As. If it is considered as false in such cases, then the statement All A is B has existential import with respect to A.

It is claimed Aristotle's logic system does not cover cases where there are no instances. Aristotle's goal was to develop "a companion-logic for science. He relegates fictions, such as mermaids and unicorns, to the realms of poetry and literature. In his mind, they exist outside the ambit of science, which is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is 'a phrase signifying a thing's essence.'... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." [14]

However, many logic systems developed since do consider the case where there may be no instances. Medieval logicians were aware of the problem of existential import and maintained that negative propositions do not carry existential import, and that positive propositions with subjects that do not supposit are false.

The following problems arise:

  1. (a) In natural language and normal use, which statements of the forms, All A is B, No A is B, Some A is B, and Some A is not B, have existential import and with respect to which terms?
  2. In the four forms of categorical statements used in syllogism, which statements of the form AaB, AeB, AiB and AoB have existential import and with respect to which terms?
  3. What existential imports must the forms AaB, AeB, AiB and AoB have for the square of opposition to be valid?
  4. What existential imports must the forms AaB, AeB, AiB and AoB have to preserve the validity of the traditionally valid forms of syllogisms?
  5. Are the existential imports required to satisfy (d) above such that the normal uses in natural languages of the forms All A is B, No A is B, Some A is B and Some A is not B are intuitively and fairly reflected by the categorical statements of forms AaB, AeB, AiB and AoB?

For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further, if it is accepted that AiB entails BiA, then AiB and AaB have existential import with respect to B as well. Similarly, if AoB is false if there are no As, and AeB entails AoB, and AeB entails BeA (which in turn entails BoA) then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms. If AaB and AeB is a fair representation of the use of statements in normal natural language of All A is B and No A is B respectively, then the following example consequences arise:

"All flying horses are mythical" is false if there are no flying horses.
If "No men are fire-eating rabbits" is true, then "There are fire-eating rabbits" is true; and so on.

If it is ruled that no universal statement has existential import then the square of opposition fails in several respects (e.g. AaB does not entail AiB) and a number of syllogisms are no longer valid (e.g. BaC,AaB->AiC).

These problems and paradoxes arise in both natural language statements and statements in syllogism form because of ambiguity, in particular ambiguity with respect to All. If "Fred claims all his books were Pulitzer Prize winners", is Fred claiming that he wrote any books? If not, then is what he claims true? Suppose Jane says none of her friends are poor; is that true if she has no friends?

The first-order predicate calculus avoids such ambiguity by using formulae that carry no existential import with respect to universal statements. Existential claims must be explicitly stated. Thus, natural language statements—of the forms All A is B, No A is B, Some A is B, and Some A is not B—can be represented in first order predicate calculus in which any existential import with respect to terms A and/or B is either explicit or not made at all. Consequently, the four forms AaB, AeB, AiB, and AoB can be represented in first order predicate in every combination of existential import—so it can establish which construal, if any, preserves the square of opposition and the validity of the traditionally valid syllogism. Strawson claims such a construal is possible, but the results are such that, in his view, the answer to question (e) above is no.

Syllogistic fallacies

People often make mistakes when reasoning syllogistically.[15]

For instance, from the premises some A are B, some B are C, people tend to come to a definitive conclusion that therefore some A are C.[16][17] However, this does not follow according to the rules of classical logic. For instance, while some cats (A) are black things (B), and some black things (B) are televisions (C), it does not follow from the parameters that some cats (A) are televisions (C). This is because in the structure of the syllogism invoked (i.e. III-1) the middle term is not distributed in either the major premise or in the minor premise, a pattern called the "fallacy of the undistributed middle". Because of this, it can be hard to follow formal logic, and a closer eye is needed in order to ensure that an argument is, in fact, valid.[18]

Determining the validity of a syllogism involves determining the distribution of each term in each statement, meaning whether all members of that term are accounted for.

In simple syllogistic patterns, the fallacies of invalid patterns are:

Other types

See also

References

  1. ↑ Lundberg, Christian (2018). The Essential Guide to Rhetoric. Bedford/St.Martin's. p. 38.
  2. ↑ John Stuart Mill, A System of Logic, Ratiocinative and Inductive, Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation, 3rd ed., vol. 1, chap. 2 (London: John W. Parker, 1851), 190.
  3. 1 2 Frede, Michael. 1975. "Stoic vs. Peripatetic Syllogistic." Archive for the History of Philosophy 56:99–124.
  4. ↑ Hurley, Patrick J. 2011. A Concise Introduction to Logic. Cengage Learning. ISBN 9780840034175
  5. ↑ Zegarelli, Mark. 2010. Logic for Dummies. John Wiley & Sons. ISBN 9781118053072.
  6. ↑ Aristotle, Prior Analytics, 24b18–20
  7. ↑ Bobzien, Susanne. [2006] 2020. "Ancient Logic." Stanford Encyclopedia of Philosophy. § Aristotle.
  8. ↑ Lagerlund, Henrik (2 February 2004). "Medieval Theories of the Syllogism". The Stanford Encyclopedia of Philosophy. Edward N. Zalta. Retrieved 17 February 2014.
  9. 1 2 Bacon, Francis. [1620] 2001. The Great Instauration. – via Constitution Society. Archived from the original on 13 April 2019.
  10. ↑ Boole, George. [1854] 2003. The Laws of Thought, with an introduction by J. Corcoran. Buffalo: Prometheus Books.
  11. ↑ van Evra, James. 2004. "'The Laws of Thought' by George Boole" (review). Philosophy in Review 24:167–69.
  12. 1 2 Corcoran, John. 2003. "Aristotle's 'Prior Analytics' and Boole's 'Laws of Thought'." History and Philosophy of Logic 24:261–88.
  13. ↑ Grundlagen der Strengen Logik. Würzburg 1996.
  14. ↑ "Groarke, Louis F., "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy". Archived from the original on 2017-02-04. Retrieved 2017-03-07.
  15. ↑ See, e.g., Evans, J. St. B. T (1989). Bias in human reasoning. London: LEA.
  16. ↑ Khemlani, S., and P. N. Johnson-Laird. 2012. "Theories of the syllogism: A meta-analysis." Psychological Bulletin 138:427–57.
  17. ↑ Chater, N., and M. Oaksford. 1999. "The Probability Heuristics Model of Syllogistic Reasoning." Cognitive Psychology 38:191–258.
  18. ↑ Lundberg, Christian (2018). The Essential Guide to Rhetoric. Bedford/St. Martin's. p. 39.

Sources

  • Aristotle, [c. 350 BCE] 1989. Prior Analytics, translated by R. Smith. Hackett. ISBN 0-87220-064-7
  • Blackburn, Simon. [1994] 1996. "Syllogism." In The Oxford Dictionary of Philosophy. Oxford University Press. ISBN 0-19-283134-8.
  • Broadie, Alexander. 1993. Introduction to Medieval Logic. Oxford University Press. ISBN 0-19-824026-0.
  • Copi, Irving. 1969. Introduction to Logic (3rd ed.). Macmillan Company.
  • Corcoran, John. 1972. "Completeness of an ancient logic." Journal of Symbolic Logic 37:696–702.
  • — 1994. "The founding of logic: Modern interpretations of Aristotle's logic." Ancient Philosophy 14:9–24.
  • Corcoran, John, and Hassan Masoud. 2015. "Existential Import Today: New Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions." History and Philosophy of Logic 36(1):39–61.
  • Englebretsen, George. 1987. The New Syllogistic. Bern: Peter Lang.
  • Hamblin, Charles Leonard. 1970. Fallacies. London: Methuen. ISBN 0-416-70070-5.
    • Cf. on validity of syllogisms: "A simple set of rules of validity was finally produced in the later Middle Ages, based on the concept of Distribution."
  • Łukasiewicz, Jan. [1957] 1987. Aristotle's Syllogistic from the Standpoint of Modern Formal Logic. New York: Garland Publishers. ISBN 0-8240-6924-2. OCLC 15015545.
  • Malink, Marko. 2013. Aristotle's Modal Syllogistic. Cambridge, MA: Harvard University Press.
  • Patzig, Günter. 1968. Aristotle's theory of the syllogism: a logico-philological study of Book A of the Prior Analytics. Dordrecht: Reidel.
  • Rescher, Nicholas. 1966. Galen and the Syllogism. University of Pittsburgh Press. ISBN 978-0822983958.
  • Smiley, Timothy. 1973. "What is a syllogism?" Journal of Philosophical Logic 2:136–54.
  • Smith, Robin. 1986. "Immediate propositions and Aristotle's proof theory." Ancient Philosophy 6:47–68.
  • Thom, Paul. 1981. "The Syllogism." Philosophia. München. ISBN 3-88405-002-8.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.