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# Reasoning (Logic)

Subject: Logic
Professor: Mr. Vincent Navarro

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### Reasoning (Logic)

1. 1. Reasoning: Immediate Inference This chapter takes , as its focus, the third mental operation called Reasoning and the processes it involves.
2. 2.  Reasoning is a mental operation through which the agreement or disagreement of two(2) ideas is inferred from their known relation to a common third idea. A. Definition of Reasoning
3. 3.  There are two methods involved in reasoning : 1. Induction, and 2. Deduction Induction or inductive reasoning is one which proceeds from universal data to particular and individual conclusion, e.g. : Juan is a man. Juan is mortal. All men are mortal. B. Methods of Reasoning
4. 4. On the other hand, deduction or deductive reasoning is one which proceeds from universal data to particular and individual conclusion, e.g. : All animals are mortal. All humans are animals All humans are mortal. We will, however, explain in detail the logical principles governing deduction and induction in the succeeding pages. As of the moment, we will concentrate on inference.
5. 5. In the preceding pages, we made sample explanations that reasoning is expressed through inference. By this we mean that when we speak of reasoning we mean inference or vice-versa. True enough, inference has its distinct meaning (compared to reasoning), but then since inference is the manner through which reasoning , as a mental operation , is expressed, so it would be good to take these two terms as having correlative existence and likewise having synonymous meaning.
6. 6. Definition of Inference Inference refers to any process through which the mind proceeds from one or more propositions to other propositions whose meanings are already implied in the former. Examples: All men are mortal. Julius is a man. Julius is mortal. John is man. John is mortal.
7. 7. In the first example, the first proposition: “All men are mortal” enunciates that all creature who are human beings bound to be mortal. In the second and thirds propositions, respectively: ”Julius is a man,” therefore,” Julius is mortal” the sweeping idea that any human being is bound to be mortal permeates. Thus in as much Julius is a human being therefore, he is bound to be mortal. Simply put in inference the mind proceeds to draw one or more propositions (the first proposition :”All men are mortal”, to other propositions (Julius is a man” therefore, “Julius is mortal”). Evidently, the meaning of both the second and the third propositions are already contained by the first proposition. The same contention can be pursed with regard to the second example: “John is a man. Therefore, john is a mortal”. This is because the idea “mortal” is impliedly contained in the idea of “man” so that if one is a man it follows that he must necessarily be mortal.
8. 8. D. Kinds of inference Basically, there are two (2) kinds of inference, namely : (1) Immediate: and (2) Mediate. Under the immediate inference, we will discuss the following : eduction and logical proposition (or oppositional inference). In our advertent discussion of eduction, we will consider the following conversion, obversion, contraposition, and invension. And in our treatment of logical opposition, we will discuss contradictory, contrary, subcontrary, and subaltern oppositions and the rules governing their oppositions. Under mediate inference, we will discuss only simple categorial syllogism.
9. 9. E. Immediate Inference Immediate inference is a process of reasoning, through which the mind passes directly from one proposition to a new proposition which is nothing else but a reformulation (partial or complete) of the very exact meaning or truth as expressed in the original proposition.
10. 10. It contains only two terms : 1. Subject term 2. Predicate term Example : No metals are stones. (S) (P) No stones are metals. (P) (S)
11. 11. E.1 Eduction This is a kind of immediate inference where a new proposition is being formulated either by interchanging the subjectand predicate terms of the original proposition or by the use or removal of negatives. 4 kinds of Eduction 1. Conversion 2. Obversion 3. Contraposition 4. Inversion
12. 12. E.1.1 CONVERSION This refers to a formulation of a new proposition by way of interchanging the subject and the predicate terms of an original propositions, however, the quality of the original proposition is retained.
13. 13. Two parts of Conversion: 1. Convertend 2. Converse Example: No fish is a mouse. (Convertend) (S) (P) No mouse is a fish. (Converse) (S) (P)
14. 14. TWO KINDS OF CONVERSION 1. Simple Conversion - a kind of conversion where the quantity of the converted is retained in the converse. Examples: No men are mortals. (E) No mortals are men. (E) Some mortals are men. (I) Some men are mortals. (I)
15. 15. 2. Partial Conversion - a kind of conversion where the quantity of the converted is reduced from universal to particular. This implies that partial conversion is applicable only to “A” and “E” propositions. Examples: All computers are gadgets. (A) to (Su) (Pp) Some gadgets are computers. (I) (Sp) (Pp)
16. 16. E.1.2 OBVERSION is kind of education where a new proposition is formulated by retaining the subject term and the quantity of the original proposition; however, the quality of the original proposition is change and the predicate term replaced by its contradictory. simply put in obversion a new propositon (obverse) is being formulated through the following:
17. 17. Obversion is, however, applicable to all kinds of categorical propositions (A, E, I and O) Examples: Obvertend Obverse (1) All men are mortal. (A) Su Pp (1) No men are non-mortal. (E) Su Pp (2) No men are mortal. (E) Su Pu (2) All men are non-mortal. (A) Su Pp (3) Some men are mortal. (I) Sp Pp (3) Some men are not non-mortal. (O) Sp Pu (4) Some men are not mortal. (O) Sp Pu (4) Some men are non-mortal. (I) Sp Pp
18. 18. This is a kind of education which results from a formulation of a new proposition whose subject term is the contradictory of the predicate term in the original proposition. In principle, contraposition is a product of both conversion and obversion . Thus like conversion , contraposition involves an interchange of the subject and predicate terms, and like obversion , it involves either the use or removal of negatives which affects the copula and the terms.
19. 19. We can immediately see that there are two types of Contraposition. They are Partial or Simple and Complete. Partial Contraposition consists the formulation of a new proposition (contraposit) through the following: a. Its (contraposit) subject is the contradictory of the predicate term of the original proposition (contraponend), Ex. (Contraponend) All whales are mammals. (A) to Su Pp (Contraposit) No non-mammals are whales. (E) Su Pu
20. 20. b. The quality of the contraponend is changed in the contraposit; and Ex. (Contraponend) No fishes are dogs. (E) to Su Pu (Contraposit) Some non-dogs are fishes. (I) Sp Pp c. The predicate term in the “A” proposition is changed to “E”, while “E” proposition to “I”, and an “O” proposition to “I”. Ex.(Contraponend) Some students are not studios.(O) to Sp Pu (Contraposit) Some non-studious are not non-students.(E) Sp Pp
21. 21. Complete Contraposition. a new proposition (contraposit) is formulated through the following steps: 1 .The subject term in the contraposit is the contradictory of the predicate term in the contraponend , (Contraponend) All whales are mammals. (A) to Su Pp (Contraposit) All non-mammals are non-whales.(A) Su Pp 2.The quality of the contraponend is not changed in the contraposit ; and 2. (Contraponend) No fish is a dog. (E) to Su Pu (Contraposit) Some non-dog is not a non-fish. (O) Sp Pu
22. 22. 3.The predicate term in the contraposit is the contradictory of the subject term in the contraponend. (Contraponend) Some students are not studious.(O) to Sp Pu (Contraposit) Some non-studious are not students.(O) Sp Pu
23. 23. E.1.4 INVERSION  This is a method of eduction in which the mind, through obversion and conversion, finally arrives at a judgment whose subject and predicate terms are contradictories of the subject and predicate terms in the original proposition.
24. 24. Ii Steps involve in inversion:  (Invertend) All birds are animals. (A) to (Obverse) No birds are non-animals. (E)  (Obverse) No birds are non-animals. (E) to (Converse:Simple) No non-animals are birds. (E)  (Converse:Simple) No non-animals are birds. (E) to (Obverse) All non-animals are non-birds. (A)
25. 25. Ii Steps involve in inversion:  (Obverse) All non-animals are non-birds. (A) to (Converse:Partial) Some non-birds are non-animals. (I)  (Invertend) All birds are animals. (A) to (Inverse) Some non-birds are non-animals. (O)
26. 26. Ii Steps involve in inversion: (Invertend) All birds are animals. (A) to (Obverse) No birds are non-animals. (E) to (Converse:Simple) No non-animals are birds. (E) to (Obverse) All non-animals are non-birds. (A) to (Converse:Partial) Some non-birds are non-animals. (I) (Inverse) Some non-birds are non-animals. (O)
27. 27. E.1.3 Logical Opposition Opposition exists between two propositions when these propositions have the same subject and predicate terms but they differ from each other either in quantity or quality or both in quantity or quality.
28. 28. All Filipinos are Asians. No Germans are Romans. All Filipinos are Asians. No Filipinos are Asians.
29. 29. E.4 TYPES OF OPPOSITION There are four types of opposition. They are as follows: (1) Contradictory; (2) Contrariety; (3) Subcontrariety; and (4) Subalternation or subalternity.
30. 30. LOGICAL SQUARE
31. 31. E.4.1 CONTRADICTORY OPPOSITION This is an opposition existing between a pair of propositions having the same subject and predicate terms. In the foregoing illustration of the square of opposition, we see that the following propositions are contradictories: A and O; and E and I. Examples: “All men are mortal” is the contradictory of “Some men are not mortal,” and vice versa; and “No man is mortal” is the contradictory of “Some men are mortal;” and vice versa. In a contradictory opposition, therefore, one proposition simply denies the other proposition. This means that contradictories cannot be both type or true or false at the same time.
32. 32. Rules of Contradictories. The following are the rules governing contradictories: (a) If one of the two contradictories is true the other is false and vice versa. (b) Contradictories cannot be simultaneously true or false at the same time
33. 33. E.4.2. Contrary Opposition (Contrariety) This is an opposition existing between a pair of universal propositions having the same subject and predicate terms but they differ in quality since A is affirmative while E is negative. So one proposition "All men are mortal", is the contrary of the proposition "No man is mortal" and vice versa. Thus, we can make an immediate inference that if one is true, the other must be false. Rules of Contrariety. The following are the rules governing contrariety. (a) If one of the contraries is true, the other is false. (b) If one of the contraries is false, the other is doubtful.
34. 34. Based on the foregoing rules, we can draw the following conclusions: (a) If A is true, E is false. A: All dogs are animals. (True) E: No dogs are animals. (False) (b) If E is true, A is false. A: All cats are dogs. (False) E: No cats are dogs. (True)
35. 35. (c) If A is false, E is doubtful. A: Every students are rich. (False) E: No students are rich. (Doubtful) (d) If E is false, A is doubtful. A: Every man is white. (Doubtful) E: No man is white. (False)
36. 36. E.4.3. Subcontrary Opposition This is an opposition existing between a pair of particular propositions having the same subject and predicate terms but they differ in quality. In this definition, we can clearly see that “I” and “O” propositions are subcontraries. This is because both “I” and “O” propositions are particular but they differ in quality since “I” is affirmative while “O” is negative. Rules of Subcontraries. The following are the rules governing subcontraries. (a)If one of the subcontraries is true, the other is doubtful. (b)If one of the subcontraries is false, the other is true.
37. 37. Based on the foregoing rules, we can draw the following conclusions: (a)If I is true, O is doubtful. I: Some students are girls. (True) O: Some students are not girls. (Doubtful) (b)If O is true, I is doubtful. O: Some policemen are not honest people. (True) I: Some policemen are honest people. (Doubtful)
38. 38. (c) If I is false, O is true. I: Some books are covered. (False) O: Some books are not covered. (True) (d) If O is false, I is true. O: Some employees are not working. (False) I: Some employees are working. (True)
39. 39. E.4.4. Subaltern Opposition This is an opposition existing between a pair of propositions having the same subject and predicate terms and having the same quality, but different quantity. Rules of Subalternation: The following are the rules governing Subalternation. (a) if the universal is true, the particular is true. (b) if the universal is false, the particular is doubtful. (c) if the particular is true, the universal is doubtful. (d) if the particular is false, the universal is false.
40. 40. Based on the aforestated rules we can draw the following conclusions: (a) if A is true, I is true. (b) if A is false, I is doubtful. (c) if E is true, O is true. (d) if E is false, O is doubtful. (e) if I is true, A is doubtful. (f) if I is false, A is false. (g) if O is true, E is doubtful. (h) if O is false, E is false.