The Birth of Logic in Ancient Greek.

Modulo CLIL Titolo del modulo: Autore: Massimo Mora Lingua: Inglese Materia: Filosofia The Birth of Logic in Ancient Greek. Contenuti: Aristotelian theory of logic, the difference between truth, falsehood and Validity, the perfect form of reasoning: the Syllogism. Classe: 3 linguistico Durata: 10 ore Prerequisiti: Eraclito, Parmenide, Diagrammi di Venn Obiettivi: comprendere la differenza tra falso, privo di senso e insensato. comprendere la differenza tra validità e verità saper applicare le regole del sillogismo e identificare i ragionamenti validi e quelli non validi saper fare sillogismi validi nelle diverse figure e modi con proposizioni vere e/o false. Modalità di lavoro: Lezione frontale Lettura schede preparate dall insegnante Esercizi alla lavagna e al banco Esercizi in gruppo Materiali: schede in lingua inglese prodotte dall insegnante 1 ; testi di riferimento: -A New Modern History of Western Phylosophy di A. Kenny -A Modern Introduction to Logic di Stebbing Verifica finale: Valutazione: Knowledge of topic (1-4) Ability to synthetize/analyze (1-3) Formal Accuracy (2-1) excellent good ok Could be better 1 Tutto il materiale in lingua inglese è stato prodotto dall insegnate, controllato dal collega di lingua inglese e consegnato agli studenti. 1

Originality (1) Lezioni Fase 1: (1 ora) Content: presentazione progetto CLIL in lingua italiana introduzione generale alle tematiche in lingua inglese Fase 2: (1 ora) Content: Introduction to the notion of syllogism materials: Attached A1 Whole class lesson final exercise Fase 3: (2 ore) content: The elements of the syllogism and the theory of distribution materials: Attached A2 whole class lesson and exercises at the blackboard with Venn diagrams Fase 4: (2 ore) Contents: Terms and figures Materials: A3 Whole class lesson and works in groups Fase 5: (2 ore) Contents: general and particular rules of the syllogism Materials: A4 Whole class lesson and exercices at the blackboard with flow diagram, tables Fase 6: (1 ora) Contents: final reflections on the difference between truth and validity Materials: A5 Modalità: whole class lesson and tables at the blackboard Fase 7: (1 ora) Final test Materials: A6 2

Allegati: A1. Logic and syllogism. The two terms are strongly related because syllogism is, etymologically, a Greek word that means to link together ideas, concepts in other words, reasoning. Logic is, by definition, the study of valid inferences or, to use Kenny s words, Logic is the discipline that sorts out good arguments from bad arguments. 2 We ll see the meaning of validity and the difference between valid and truth in the following lessons. We start with an example of syllogism 1 All soldiers are brave 2 All parachutists are soldiers 3 All parachutists are brave Aristotle defines the syllogism as follows: A syllogism is a discourse in which, certain specific things that have been supposed different from other things result of necessity because these things are so." From certain given things (1 and 2 that we call the premises), something different comes (the point 3, the conclusion) from necessity. This means that given 1 and 2 there are no other choices than 3. Only if the conclusion comes from necessity we can say that the syllogism is a valid one, otherwise we have bad arguments as in the following case: 1.All men are mortal 2.Socrates is mortal 3.Socrates is a man This is a clear example in which, given the premises, the conclusion doesn t follow from necessity because I could have a dog named Socrates that is mortal but not human. As we have seen, a syllogism is composed by three propositions and each of the premises has one term in common with the conclusion: the first two propositions are called premises, the third is the conclusion and there are some triads that work and others that don t. So the question is: What is that allows the realization of a syllogism? Is there a hidden link? Are there rules that govern this form of reasoning? Exercise: Analyze this argumentation, is this a syllogism? Motivate your choice. All priests are saints Some men are not priests Some men are not saints A2 2 A. Kenny A new History of Western Philosophy p.95 3

In the example of the soldiers, the propositions are universals and affirmatives (their logical structure is: All S are P), but there are also particular propositions, so called because they don t involve a class, or a category, in the whole, but they refer to at least one member of the class, as in the case of Students in the example Some students are clever. The logic form, in this case, is Some S are P where S means at least one 3. - Some S are P doesn t exclude All S are P, it just states that at least one S is P and that is enough. On the contrary: - All S are P includes that Some S are P. We can express this by saying that what is true for all is also true for some, but not necessarily what is true for some is true for all. Universal and particular propositions differ in quantity, but they also can differ in quality: 1.No musicians are sensitive 2.Some poets are not sensitive are universal (1) and particular (2), but whereas in the syllogisms above the propositions are affirmatives, these are negatives. We are now able to define four kinds of propositions using a criterion invented by logicians in the middle age. A: all S are P (universal and affirmative) E: No S are P (universal and negative) I: some S are P (particular and affirmative) O: some S are not P (particular and negative) 4 Exercise: each student must write 12 propositions in English, three for every kind (15 minutes). They can consul the Dictionary and at the end, a construction of a list will follow. Doctrine of distribution. In order to understand how a syllogism works, we need to analyze and understand what is called the doctrine of distribution. We shall explain what this term means using a slightly modified version of Venn diagrams Definition: a term is distributed when it refers to all the members of the class for which it stands. The term dog stands for the class of all the dogs, man stands for the class of all men When in a proposition a term refers to the whole class, we say that it is distributed. Let s analyze what happens in our four kinds of propositions: A: All priests (P) are saints (S) This clearly means that all what is included in P is S, every single priest is saint, but not all saints are priests; for example, x. We can say that P is distributed because it refers to all the members of the class for which it stands while S is not. E: No priests are Saints. Here we are referring to all priests and we say that they are not saints, but at the same time we are saying that no saints are priests. Both the terms are distributed. I: some priests are saints. We are talking about some S that are P and some P that are S, so neither S nor P are distributed. 3 For simplicity I don t distinguish between particular and individual propositions so Some S are P and X is P are equivalent. 4 A-I are affirmative from the vowels of the Latin word Adfirmo and E-O are negative form the vowels of Nego. 4

O: some P are not S. Here S is clearly not distributed because we are talking about some S, but P is distributer because we say that there is nothing in P that is also an S. If we suppose that the saints in question are X, Y and W, then no priests are X or Y or W. The general rule for the distribution is the following: In universal propositions the subject is always distributed, in negative propositions the predicate is always distributed A3. Coming back to the syllogism and his structure, we have said that a syllogism is made by three propositions and now we must analyze the structure of the proposition in order to understand how a syllogism works. Propositions express relations between two terms: the subject and the predicate. All Greeks are philosophers All Spartan are Greeks All Spartan are philosophers As we can see, we have got three terms that play in this syllogism: Greeks, philosophers and Spartan, every term occurs twice, but only one occurs in both premisses, Greeks. This in called by Aristotle, the middle term that we indicate with M. So, by definition, the Middle term is the only term that occurs in both premisses. The term that occurs as predicate in the conclusion is called Major term, we indicate it with P and the premise in which it occurs is called first, or major premise. The term that occurs as subject in the conclusion is called Minor term and is indicated with S and the premise in which it occurs is called second or minor premise. If we substitute the terms with the letters, what we obtain is a scheme as follows All M are P All S are M All S are P This is the first figure of the syllogism, the perfect one, that in which the middle term occurs as subject in the first premise and as predicate in the second. Syllogisms may be made using the four kinds of propositions and, also, varying the occurrences as: -subject in both premisses -predicate in both premisses -predicate in the major premise and subject in the minor. On this base, Aristotle identifies 4 figures of syllogism: 1.MP 2. PM 3. MP 4. PM SM SM MS MS SP SP SP SP It s worthy to notice, in order to avoid mistakes and confusion, that S means the subject in the conclusion and not just subject because it occurs as predicate in minor propositions in the third and fourth figure. Similarly, P is the predicate of the conclusion and occurs as subject in the second and fourth figure. Exercise: The students try to make up four syllogisms, one for every figure. 5

1. All Cats are Black All Siamese are cats Sll Siamese are Black 2. All Mice are Gray No Dolphins are Grey No Dolphins are Mice 3. No Humans are Immortal Some Gods are Immortal Some Gods are not Human 4. Nessun onesto è ricco Qualche ricco è generoso Qualche generoso non è onesto. Four figures and four propositions that can be mixed together for the production of 256 possible syllogisms. The table below shows the sixteen combinations for the A-proposition in the first figure. 4 3 =64, combinations of the first figure 64x4=256, combinations of the four figures A A A A A A A A A A A A A A A A Premessa A A A A E E E E I I I I O O O O Premessa A E I O A E I O A E I O A E I O Conclusione Exercise: find the 16 combinations for the E-proposition In the next paragraph We ll see how the rules that govern a syllogism cut off a lot of combinations leaving only 19 valid forms A4. Rules 5 : there are two kinds of rules: the general ones that do apply to every syllogism and the particulars that don t. General rules: A syllogism must be composed by three propositions and three terms The Middle Term must be distributed in at least one premise A term distributed in the conclusion must be distributed in the premise in which it occurs From two negative propositions no conclusion is allowed From two particular propositions no conclusion is allowed If one premise is negative, the conclusion must be negative If one premise is particular, le conclusion must be particular Rules of the first figure: The Minor premise must be affirmative because if it were negative, the conclusion should be negative and the first premise positive, but in this case P would be distributed in the conclusion and not in the premise. 5 We won t consider, neither the fourth figure, nor the syllogisms with universal premises and particular conclusion because it is certainly true that if all Greeks are mortal, also some Greeks are mortal, but this is a weak conclusion that can be drawn only using a general truth: What is true for all is true for some. This could be troublesome. 6

The major premise must be Universal in order to grant the distribution of the middle term. Exercise: find out the combinations at the blackboard. A A E E A I A I A I E O And the corresponding syllogisms are called BARBARA, DARII, CELARENT, FERIO Rules of the second figure One premise must be negative, for the distribution of the middle term The major premise must be universal, for the distribution of P Exercise: to find out the combinations at the blackboard E E A A A I E O E O E O CESARE, FESTINO CAMESTRES, BAROCO Rules of the third figure: The minor premise must be affirmative (see the first figure) The conclusion must be particular, because if it were universal, S would be distributed in the conclusion but not in the minor premise. Exercise: to find out the combinations at the blackboard E I O I A A O I O ferison disamis bocardo A5. Now we are able to understand the difference between truth and validity: truth is a relation between a proposition and the world: if things are as it s stated, the proposition is true, otherwise it s false. Validity is a property that relates to a set of propositions linked together and the way in which they are linked may be valid or not. First comes the proposition that expresses a thought, and then comes the truth. Parmenides talked of the Truth s way, that in which being and thought are the same and we have seen many logical and metaphysical implications. Someone said that Parmenides is the father of the logic because we can directly derive, from his principle, the famous laws of logic A=A (Identity: something is equal only to itself) -(A e -A) (Non-contradiction: it s not possible for something to be and not to be at the same time) A o A (Excluded Middle: something is or is not; a proposition is true or false, there isn t another way) With the Non-contradiction principle, Zeno demonstrated, by absurd, that nothing moves nor changes, but a demonstration is a process that involves propositions and here comes Aristotle with the four kinds of proposition and a famous definition of truth To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true 7

Propositions, whether affirmatives or not, particular or not, may be true or false and we say that a deduction can be valid despite the truth or falsehood of the propositions. We can have the following combinations: First premise V V V V F F F F Second premise V V F F V V F F Conclusion V F V F V F V 6 F Working in group: Students will form eight groups and every group will take one of the eight combinations with the aim to write down three pleasing syllogisms. The theory of validity may have metaphysical implications because truth is stronger (or wicker, it depends from the point of view) than falsity: we can build a valid syllogism with false premises and a true conclusion but we can t have true premises and a false conclusion: there is no way from truth to falsity. (Socrate, Agostino Spinoza: if you know the truth, you can t act wrongly; if you have seen the light; you cannot commit a sin ) As we will see, in propositional logic, the conditional PQ is false only when is the case that P is true and Q is false, and I think that this is the only reason that can justify such a rule. P Q PQ V V V A6. 1 Which is the definition of syllogism? 2 What is a figure and how many figures did Aristotle pinpoint? 3 What does the theory of distribution say and which is the general rule for the four kinds of propositions? 4 Why, in a syllogism of second figure, one premise must be negative? 6 All sailors are philosophers Socrates is a sailor Socrates is a philosopher 8

5 Which is the difference between Truth and Validity? 6 Make a syllogism with false premises and a true conclusion 7 Make a syllogism in CESARE and in FERIO 9