The Basics of Philosophy

1. Moscow State Instituteof International Relations (MGIMO-University)School of Government and International Affairs& Alexander ShishkinDepartment of Philosophy The Basics of Philosophy Part IICosmocentric Philosophy Lecture 7Aristotle’s Logic The Tool and Language of Thought

2. Aristotle’s LogicThe Tool and Language of Theoretical Thought • The Principles of Logic (the Laws of Thought) • The Principle of Non-contradiction • The Principle of Excluded Middle • The Forms of Thought • Concepts • Propositions • Inferences • Immediate Inferences • Categorical Syllogism • Rules of Categorical Syllogism • Figures of Categorical Syllogism • Moods of Categorical Syllogism • Hypothetical and Disjunctive Syllogisms • Modal Logic

3. The Organon Categories On Interpretation Prior Analytics Posterior Analytics Topics On Sophistical Refutations Aristotle’s LogicThe Tool and Language of Theoretical Thought Principal Writings Aristotle(384–322B.C.)

4. Aristotle’s LogicThe Tool and Language of Theoretical Thought Logic(Gr. λογική,the science of thinking,from Gr.λόγος,word, concept)is the study of laws, forms and methods of thought,primarily reasoning. The term“logic”is most likely the invention of Stoics.Aristotle’s name for it was “Analytics”. The following account is not limited to Aristotle’s own contribution to the analysis of reasoning, nor does it attempt to reproduce Aristotle’s initial form of presenting his findings; it makes ample use of the subsequent developments in logic, Ancient as well as Medieval, in order to demonstrate the potential of Aristotle’s invention.

5. The Principles of Logic Laws of Thought The Law of Identity The Law of Non-contradiction The Law of Excluded Middle (Third) We may not use the same term in the same discourse while having it signify different senses or meanings. Contradictory statements cannot both be true in the same sense at the same time. For any proposition, either that proposition is true, or its negation is true. A is А and not ~А A and~A are mutually exclusive. EitherA, or~A.

6. The Principles of Logic The Principle of Non-contradiction … the most certain principle of all is that regarding which it is impossible to be mistaken; for such a principle must be both the best known (for all men may be mistaken about things which they do not know), and non-hypothetical. For a principle which every one must have who understands anything that is, is not a hypothesis; and that which every one must know who knows anything, he must already have when he comes to a special study. Evidently then such a principle is the most certain of all; which principle this is, let us proceed to say. It is, that the same attribute cannot at the same time belong and not belong to the same subject and in the same respect;we must presuppose, to guard against dialectical objections, any further qualifications which might be added. This, then, is the most certain of all principles, since it answers to the definition given above. Aristotle.The Metaphysics.

7. The Principles of Logic The Principle of Non-contradiction • The most certain principle of allisthat regarding which it is impossible to be mistaken; • for such a principle must be both thebest known • (for all men may be mistaken about things which they do not know) • and non-hypothetical. • For a principle which every one must have who understands anything that is, is not a hypothesis; • and that which every one must know who knows anything, he must already havewhen he comes to a special study. • Evidently then such a principle isthe most certain of all; which principle this is, let us proceed to say. It is, that • the same attribute cannot • at the same time • belong and not belong • to the same subject • and in the same respect; • we must presuppose, to guard against dialectical objections, any further qualifications which might be added. • This, then, is the most certain of all principles, since it answers to the definition given above.

8. The Principles of Logic The Principle of Non-contradiction The same attribute cannot at the same time belong and not belong to the same subject and in the same respect. Aristotle.The Metaphysics.

9. The Principles of Logic The Principle of Excluded Middle But on the other hand there cannot be an intermediate between contradictories, but of one subject we must either affirm or denyany one predicate. Aristotle.The Metaphysics.

10. The Principles of Logic The Principle of Excluded Middle If that which it is true to affirm is nothing other than that which it is false to deny, it is impossible that all statements should be false;for one side of the contradiction must be true. Aristotle.The Metaphysics.

11. The Forms of Thought Forms of Thought Concepts Propositions(sentences) Inferences Horse Animal Horse animal Every animal mortal Every horse animal Human Mortal Human mortal Every horse mortal

12. The Forms of Thought Conceptis a form of thought that grasps the universal essenceof a species (genus). Propositionis a sentence that affirms or deniesa predicate of a subject. Inference is a logical operation of deriving a new proposition (called conclusion) from one or several other propositions (called premises).

13. The Forms of ThoughtConcepts • A concept is a form of thought that grasps the universal essence of a species (genus). • A concept thus applies (extends) to a lot (a set) of objects. This set is called the extension of the concept in question. • Another aspect of a concept is its intension, i.e properties (attributes, qualities) characteristic of the object in question and implied by its concept. • Intension characterises the content of a notion; extension, its range. • Logically speaking, the extension of a concept is a set of subjects of which the concept in question is predicated; the intension, a set of predicates (essential, necessary, and universal) of the concept in question.

14. The Forms of ThoughtConcepts Intensionis a logical term that indicates the internal content of a term or concept, its connotation apart from what it explicitly names or describes. Extensionis a logical term that indicates the range of applicability of a term or a concept, denotation of things to which it is applicable. Adapted from The Encyclopaedia Britannica (the 2008 Version)and TheMerriam-Webster's Collegiate Dictionary (the 11th edition). Logically speaking, intension can be defined as a set of predicates (essential, necessary, and universal) of the concept in question. Logically speaking, extension can be defined as a set of subjectsof which the concept in question is predicated.

15. The Forms of ThoughtConcepts • A concept is a form of thought that grasps the universal essence of a species (genus). • A concept thus applies (extends) to a lot (a set) of objects. This set is called the extension of the concept in question. • Another aspect of a concept is its intension, i.e properties (attributes, qualities) characteristic of the object in question and implied by its concept. • Intension characterises the content of a notion; extension, its range. • Logically speaking, the extension of a concept is a set of subjects of which the concept in question is predicated; the intension, a set of predicates (essential, necessary, and universal) of the concept in question. • The more properties a concept implies, i.e. the greater its intension, the less its extension, and vice versa. • We can get a wider concept (extending to a larger set of objects) by abstracting from a specific property (set of properties) implied by the original concept; this operation is called generalisation. • The opposite procedure, called specialisation, limits the extension of the original concept, but makes it “richer” (more specific). • Predication marks transition to the form of thought called proposition.

16. The Forms of ThoughtConcepts Logical relations between concepts (terms)depend on the presence or absence of common elements in their extensions. Logical relations betweenconceptsare to be distinguished fromnamesake logical relations betweenpropositions. Even though the intensions(meanings) of the two concepts, viz. logical relation between concepts, on the one hand, and the namesake logical relation betweenpropositions, on the other hand, are not unrelated, their extensionsare different and the concepts are, in fact, incompatible.

17. The Forms of ThoughtConcepts Concepts (terms) Compatibles Incompatibles (disjoints) Cosubalternates Equivalents Incompatible non-cosubalternates Alternates (superaltern and subaltern) Opposites Intersecting Contradictories Intersectingcosubalternates Contraries

18. A B Incompatible terms (incompatibles) are terms that have no common elements in their extensions. Compatible terms A B A B Equivalent terms (equivalents) are terms of which the extensions are identical. A B Intersectingtermsare terms a part of the extension of one of which is identical to a part of the extension of the other. A Alternates(alternate terms)are terms the extension of one of which (subaltern)constitutes a part of, but does not exhaust the extension of the other(superaltern). B A A Cosubalternates (cosubalternate terms)are terms the extensions of which constitute different or intersecting parts of the extension of a common superaltern. B C B C Incompatible terms Opposite terms A ~A Contradictories (contradictoryterms) are a pair of incompatible cosubalternates the extensions of whichexhaust the extension of the superaltern. A B Contraries (contrary terms) are a pair of incompatible cosubalternates the extensions of whichdo not exhaust the extension of the superaltern.

19. A B • Incompatible terms (incompatibles): • a political party • the area of a triangle A B • Equivalent terms (equivalents): • an equilateral triangle • an equiangular triangle A B Intersectingterms: an Englishmana chess player A Alternates(alternate terms): a horse (subaltern)an animal(superaltern) B A A Cosubalternates (cosubalternate terms): a cat a pianist a dog a violinist B C B C A ~A Contradictories (contradictoryterms): whitenon-white A B Contraries (contrary terms): whiteblack

20. The Forms of ThoughtPropositions • Proposition is a sentence that affirms or denies a predicate of a subject. S P is (not) subject copula predicate

21. The Forms of ThoughtPropositions Subjectis that of which a quality, attribute, or relation may be affirmed or denied or in which it may inhere. Predicate is something that is affirmed or denied of the subject in a proposition in logic; a term designating a property or relation. Copula is the connecting link between subject and predicate of a proposition. Adapted from TheMerriam-Webster's Collegiate Dictionary (the 11th edition).

22. The Forms of ThoughtPropositions • Proposition is a sentence that affirms or denies a predicate of a subject. • Proposition is the only form of thought that can be meaningfully characterised as either true, or false. S P is (not) subject copula predicate

23. The Forms of ThoughtPropositions Nouns and verbs, provided nothing is added, are like thoughts without combination or separation; “man” and “white”, as isolated terms, are not yet either true or false. In proof of this, consider the word “goat-stag.” It has significance, but there is no truth or falsity about it, unless “is” or “is not” is added, either in the present or in some other tense. Aristotle.On Interpretation.

24. The Forms of ThoughtPropositions • Proposition is a sentence that affirms or denies a predicate of a subject. • Proposition is the only form of thought that can be meaningfully characterised as either true, or false. • Propositions are divided: • by quality into affirmative (affirmations) and negative (denials), depending on whether the predicate is affirmed or denied of the subject; • by quantity into universal and particular, depending on whether the predicate is affirmed (denied) of every or only some of the elements of the set; • by relation into categorical, hypothetical and disjunctive, depending on whether something is predicated of the subject unconditionally, on condition (specified in the proposition), or alternatively (other alternatives to be listed); • by modality into apodeictic, assertoric, and problematic, depending on whether something is predicated of the subject necessarily, actually, or possibly. S P is (not) subject copula predicate

25. The Forms of ThoughtPropositions Every premise states that something either is or must be or may bethe attribute of something else; of premises of these three kinds some are affirmative, others negative,in respect of each of the three modes of attribution; again some affirmative and negative premises are universal, others particular,others indefinite. Aristotle.Prior Analytics.

26. The Forms of ThoughtPropositions By universalI mean the statement that something belongs to all or none of something else; by particular that it belongs to some or not to someor not to all;by indefinite that it does or does not belong, without any mark to show whether it is universal or particular, e.g. “contraries are subjects of the same science”, or “pleasure is not good”. Aristotle.Prior Analytics.

27. Logical Relations Between Extensions andthe Principal Classes of Propositions P Universal affirmative EverySisP(AllSareP) S P S P S P Universal negative NoS isP (EverySis notP) S P S ParticularaffirmativeSomeS is/areP(There exists/exist S that is/are P) S P SP S P Particular negative SomeSis/are notP(Not everySisP) S P

28. Logical Relations Between Extensions andthe Principal Classes of Propositions P Universal affirmative Sis subaltern, Pis superaltern SandPare equivalents S P S P S P Universal negative SandP are incompatibles S P S Particularaffirmative SandP are intersecting Sis superaltern,Pis subaltern S P SP S P Particular negative SandP are intersecting Sis superaltern, Pis subaltern S P

29. Propositions Logical relations between propositions are those of their truth values. Logical relations betweenpropositionsare to be distinguished fromnamesake logical relations betweenconcepts. Even though the intensions(meanings) of the two concepts, viz. logical relation between propositions, on the one hand, and the namesake logical relation betweenconcepts, on the other hand, are not unrelated, their extensions are different and the concepts are, in fact, incompatible.

30. Propositions Logical relations between propositions are those of their truth values. • A single proposition is eithertrueorfalse. • Propositions are unrelated if the truth value of one of them does in no way affect the truth value of the other. • If truth values of two propositions are related, fivecombinations are possible, viz.: • the two propositions are either bothtrueorbothfalse (equivalence); • they are neither bothtrue,norbothfalse (contradiction); • they are not bothtrue,but can bebothfalse (contrariety); • they can be bothtrue,but are notbothfalse (subcontrariety); • thetruthof one(called “superaltern”) impliesthetruthof theother (called “subaltern”; respectively, thefalsehoodof thesubalternimplies thefalsehoodof thesuperaltern), however, thefalsehoodof thesuperalternimplies neitherthefalsehoodnorthetruthof thesubaltern (respectively, thetruthof thatsubalternimplies neitherthetruth northefalsehoodof thesuperaltern) (subalternation).

31. PropositionsLogical Equivalence Equivalentpropositionsare eitherbothtrue,or bothfalse. An elephantisbigger thana dog A dogis smaller thanan elephant If one of the equivalentpropositions is true,the otheris alsotrue,if one of themis false, the otheris also false. is equivalent to Heavy rainsare the cause offlood Floodis the effect ofheavy rains is equivalent to Philipis the father ofAlexander Alexanderis the son ofPhilip is equivalent to Traditional logic did not consider this relation, however, it should be added for the sake of completeness.

32. PropositionsSquare of Opposition Universal affirmative A E Universal negative contrariety AFFIRMO NEGO subalternation contradiction contradiction subalternation Particular affirmative I O Particular negative subcontrariety The square of opposition is a diagram that represents the relations between the four basic categorical propositions having the same terms, i.e. the same subject and the same predicate, but differing in quantity or quality, or both. It should not, however, be assumed that the four relations represented by this diagram are limited to propositions that have the same terms, i.e.the same subject and the same predicate.

33. PropositionsSquare of Opposition Universal affirmative(A) Everyroseisred Noroseisred Universal negative(E) Contradictoriesare neither bothtrue, nor bothfalse, i.e. one of them is necessarily true, and the other is necessarily false. If one of the contradictories is true, the otheris false, and vice versa: if oneof the two is false,the other is true. contradiction contradiction Somerosesarered Somerosesare notred Particular affirmative(I) Particular negative(O)

34. PropositionsSquare of Opposition Universal affirmative (A) Everyroseisred Noroseisred Universal negative(E) contrariety Contraries cannot be bothtrue,but they can be bothfalse. If one of the contraries is true, the other if false; however if one of the two is false, the other may be either true or false. Somerosesarered Somerosesare notred Particular affirmative(I) Particular negative(O)

35. PropositionsSquare of Opposition Universal affirmative (A) Everyroseisred Noroseisred Universal negative(E) Subcontraries cannot be bothfalse,but they can be bothtrue. If one the subcontraries is false, the other is true; however if one of the two is true, the other may be either true or false. Somerosesarered Somerosesare notred Particular affirmative(I) Particular negative(O) subcontrariety

36. PropositionsSquare of Opposition Universal affirmative (A) Everyroseisred Noroseisred Universal negative(E) If the superalternis true,the subalternis also true; however, if the superaltern is false, the subalternmay be either true or false. If the subalternis false,the superalternis also false; however, if the subalterm is true, the superalternmay be either true or false. subalternation subalternation Somerosesarered Somerosesare notred Particular affirmative(I) Particular negative(O)

37. The Forms of ThoughtInferences • Inferenceis a logical operation of deriving a new proposition (called conclusion) from one or several other propositions (called premises) on the basis of the relations between the extensions of the premises’ terms. • Inferences are divided into inductive (that proceed from the particular to the universal) and deductive (that proceed from the universal to the particular). Since necessary is related to general, it is only through deduction that one arrives at certain (reliable) conclusions, inductive reasoning providing but probable knowledge. • Depending on whether the conclusion is drawn from one or several premises, deductive inferences are divided into immediate inferences and syllogisms. • An example of immediate reference is obversion: the quality is changed and the original predicate replaced with a contradictory. Every man is mortal No man is immortal Caesar is not alive Caesar is dead

38. The Forms of ThoughtInferences • Inferenceis a logical operation of deriving a new proposition (called conclusion) from one or several other propositions (called premises) on the basis of the relations between the extensions of the premises’ terms. • Inferences are divided into inductive (that proceed from the particular to the universal) and deductive (that proceed from the universal to the particular). Since necessary is related to general, it is only through deduction that one arrives at certain (reliable) conclusions, inductive reasoning providing but probable knowledge. • Depending on whether the conclusion is drawn from one or several premises, deductive inferences are divided into immediate inferences and syllogisms. • Another example of immediate reference is conversion which is achieved by switching the subject and predicate terms. No cats are dogs No dogs are cats All cats are animals Some animals are cats

39. InferencesDistribution of Terms • Since deductive inference proceeds from the universal (general) to the particular, its validity depends on the distribution of the terms. • The term is called “distributed”, if all members of the term’s class are affected by the proposition in question.

40. InferencesDistribution of Terms Distributionin syllogistics is the application of a term of a proposition to the entire class that the term denotes.A term is said to be distributed in a given proposition if that proposition implies all other propositions that differ from it only in having, in place of the original term, any other term whose extension is a part of that of the original term – i.e.,if, and only if, the term as it is used in that occurrence covers all the members of the class that it denotes. The Encyclopaedia Britannica (the 1997 Version).

41. InferencesDistribution of Terms • Since deductive inference proceeds from the universal (general) to the particular, its validity depends on the distribution of the terms. • The term is called “distributed”, if all members of the term’s class are affected by the proposition in question. • Subjects are distributed in universal propositions (both affirmative and negative) by definition: in universal propositions the predicate is affirmed (or denied) of every element of the subject’s class. • Predicates are distributed in negative propositions (both universal and particular), since by denying a predicate of a subject we exclude the subject in question from the entire predicate’s class. • Hence distributed are: • in universal denials (E), both subjects and predicates; • in universal affirmations (A), only subjects; • in particular denials (O), only predicates; • in particular affirmations (I), neither subjects, nor predicates.

42. InferencesThe Rule of Distribution A termundistributed in the premisemay not bedistributed in the conclusion. No cats are dogs No dogs are cats All cats are animals Some animals are cats Some cats are pets Some pets are cats Some animals are not cats Some cats are not animals Particular denials do not convert.

43. The Forms of ThoughtInferences • Inferenceis a logical operation of deriving a new proposition (called conclusion) from one or several other propositions (called premises) on the basis of the relations between the extensions of the premises’ terms. • Inferences are divided into inductive (that proceed from the particular to the universal) and deductive (that proceed from the universal to the particular). Since necessary is related to general, it is only through deduction that one arrives at certain (reliable) conclusions, inductive reasoning providing but probable knowledge. • Depending on whether the conclusion is drawn from one or several premises, deductive inferences are divided into immediate inferences and syllogisms. • Still another example of immediate reference is contraposition which combines the obversion with the conversion. Every human is mortal No human is immortal No immortal is human No human is immortal Every human is mortal Some mortals are human

44. The Forms of ThoughtInferences • Inferenceis a logical operation of deriving a new proposition (called conclusion) from one or several other propositions (called premises) on the basis of the relations between the extensions of the premises’ terms. • Inferences are divided into inductive (that proceed from the particular to the universal) and deductive (that proceed from the universal to the particular). Since necessary is related to general, it is only through deduction that one arrives at certain (reliable) conclusions, inductive reasoning providing but probable knowledge. • Depending on whether the conclusion is drawn from one or several premises, deductive inferences are divided into immediate inferences and syllogisms. • Still another example of an immediate reference is counterposition which combines the obversion and the conversion. • Syllogistics is the core of Aristotelian logic. • The most elaborated part of Aristotelian syllogistic is the theory of categorical syllogism.

45. InferencesCategorical Syllogism Сategorical syllogism(Gr. ςυλλογισμός)is a kind of inference (argument)that establishes logical relationbetween two concepts (terms)on the basis of their relations to some third concept (term).

46. InferencesCategorical Syllogism The middleterm M P is The majorpremise is the middle term S M The middleterm The minorpremise is Consequently, The minor term(the subjectof the conclusion) S P The major term(the predicateof theconclusion) is

47. Categorical SyllogismFigures of Categorical Syllogism Figure of syllogismis a pattern of syllogism definedby the arrangement of themiddle term. • The middle term occurs in both premises being either the subject or the predicate in any of the two. • There are, therefore, four possible patterns of its arrangement (2х2). • Consequently, there are four figures of categorical syllogism calledthe first, the second, the third, and the fourth. • Of these Aristotle analysed only the first three.

48. Categorical Syllogism Moods of Categorical Syllogism Figure of syllogism is a pattern of syllogism defined by the arrangement of the middle term. Mood of syllogism is a pattern of syllogism defined by the quantity and quality of the constituent propositions. • Each categorical syllogism has three propositions (two premises and a conclusion) and each proposition may take one of four different forms: universal affirmative (A), particular affirmative (I), universal negative (E), particular negative (O). • There are, therefore, 64 possible combinations of propositions (4х4х4) for each of the four figures of syllogism, yielding, on the whole, 256 patterns. • Of these only 24 do not contradict the rules of categorical syllogism and are accepted as valid. They are called moods. • If we deduct 5 so called subalternate moods (which are of little logical value since they are but “weaker” versions of stronger moods), there will remain 19 valid moods of which Aristotle analysed 14 of the first three figures. • Aristotle also deemed it opportune to reduce moods of the second and third figures to those of the first figure he called “perfect syllogism”.

49. Categorical Syllogism The Axiom of Categorical Syllogism Whatever is affirmed (or denied)about every element of a genus (class)is affirmed (or denied)about every element of any species of that genus (of any subclass of that class),but not vice versa.

50. Categorical SyllogismRules of Categorical Syllogism • Rules of the Terms: • The Rule of Three Terms: The syllogism must have three terms, neither more nor fewer. • The Rule of the Middle Term: The middle term must be distributed in, at least, one premise. • The Rule of the Extremes: Terms that are not distributed in the premises cannot be distributed in the conclusion. • Rules of the Premises: • The Rule of an Affirmative Premise: At least one of the premises must be affirmative; hence nothing follows from two negative propositions. • The Rule of a Negative Premise: If one of the premises is negative, the conclusion is negative, too. • The Rule of a Universal Premise: At least one of the premises must be universal; hence nothing follows from two particular propositions. • The Rule of a Particular Premise: If one of the premises is particular, the conclusion is particular, too.