This chapter ‘opens the box’ of propositional logic, and looks further inside the statements that we make when we describe the world. Very often, these statements are about objects and their properties, and we will now show you a first logical system that deals with these. Syllogistics has been a standard of logical reasoning since Greek Antiquity. It deals with quantifiers like ‘All P are Q’ and ‘Some P are Q’, and it can express much of the common sense reasoning that we do about predicates and their Corresponding sets of objects. You will learn a famous graphical method for dealing with this, the so-called ‘Venn Diagrams’, after the British mathematician John Venn (1834–1923), that can tell valid syllogisms from invalid ones. As usual, the chapter ends with some outlook issues, toward logical systems of inference, and again some phenomena in the real world of linguistics and cognition.
A syllogism is a logical argument where a quantified statement of a specific
form (the conclusion) is inferred from two other quantified statements (the premises).
The quantified statements are all of the form “Some/all A are B,” or “Some/all A are not
B,” and each syllogism combines three predicates or properties. Notice that “All A are not
B” can be expressed equivalently in natural language as “No A are B,” and “Some A are
not B” as “Not all A are B.” We can see these quantified statements as describing relations
between predicates, which is well-suited to describing hierarchies of properties. Indeed,
Aristotle was also an early biologist, and his classifications of predicates apply very well
to reasoning about species of animals or plants.
Your already know the following notion. A syllogism is called valid if the conclusion
follows logically from the premises in the sense of Chapter 2: whatever we take the real
predicates and objects to be: if the premises are true, the conclusion must be true. The
syllogism is invalid otherwise.
Directions (28-43): In each of the questions below are given two statements followed by two conclusions. You have to take the given statements to be true. Read all the conclusions and then decide which of the given conclusions logically follow from given statements disregarding commonly known facts.
Give Answer:
a. if only conclusion I follows.
b. if only conclusion II follows.
c. if either conclusion I or II follows.
d. if niether conclusion I not II follows.
e. if both conclusions I and II follow.
28. Statements: I. Some paingings are drawings. II. All sketches are paintings.
Conclusions: I. All sketches are drawings.
II. Some sketches being drawings is a possibility.
Statements(29-30): I. All buildings are houses.
II. No house is an apartment. III. All apartments are flats.
29. Conclusions: I. No flat is a house.
II. No building is an apartment.
30. Conclusions:
I. All buildings being flats is a possibility.
II. All apartments being building is a possibility.
Statements (31-32):
I. Some oceans are seas.
II. All oceans are rivers.
III. No river is a canal.
31. Conclusions:
I. All rivers can never be oceans.
II. All canals being oceans is a possibility.
32. Conclusions:
I. No ocean is a canal.
II. At least some seas are rivers.
Statements (33-34): I. No day is night.
II. All nights are noon.
III. No noon is an evening.
33. Conclusions: I. No day is noon.
II. No day is an evening.
34. Conclusions: I. No evenings are nights.
II. All days being noon is a possibility.
Statements (35-36): I. Some papers are boards.
II. No board is a card.
35. Conclusions: I. No card is a paper.
II. Some papers are cards.
36. Conclusions: I. All cards being papers is a possibility.
II. All boards being papers is a possibility.
37. Statements: I. No gadget is a machine.
II. All machines are computers.
Conclusions: I. No computer is a gadget.
II. All computers being gadgets is a possibility.
38. Statements: I. Some symbols are figures.
II. All symbols are graphics.
III. No graphic is a picture.
Conclusions: I. Some graphics are figures.
II. No symbol is a picture.
39. Statements: I. Some exams are tests.
II. No exam is a question.
Conclusions: I. No question is a test.
II. Some tests are definitely not exams.
Statements (40-41): I. All forces are energies.
II. All energies are powers.
III. No power is heat.
40. Conclusions: I. Some forces are definitely not powers. II. No heat is force.
41. Conclusions: I. Some forces being heat is a possibility.
II. No energy is heat.
Statements (42-43): I. No note is a coin.
II. Some coins are metal.
III. All plastics are notes.
42. Conclusions: I. No coin is plastic.
II. All plastics being metals is a possibility.
43. Conclusions: I. No metal is plastic.
II. All notes are plastics.
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