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Inductive and Deductive Reasoning

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Sonarin Cruz
Sonarin Cruz

The document discusses inductive and deductive reasoning. Inductive reasoning involves forming general conclusions from specific observations, while deductive reasoning draws specific conclusions from general statements. Examples are given of inductive arguments building from specific cases to a general rule, and deductive arguments applying a general premise to specific cases. The key features of deductive reasoning, including conditional statements and the five types of if-then logical structures (conditional, converse, counter example, inverse, and contrapositive), are also explained through examples.

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INDUCTIVE AND DEDUCTIVE REASONING
INDUCTIVE REASONING
● The process of formulating and developing a general statement. ● It is sometimes called formulating conclusions from the logical presentations of specific observation of facts. The following are some examples of inductive arguments. 1. Lorenz and Joshua are drummers. They feel pain after rehearsing for three hours straight. Jasper is also a drummer and feels the same pain after rehearsing for three hours straight. All drummers who rehearse for three hours straight feel pain in their arms. INDUCTIVE REASONING
● The process of formulating and developing a general statement. ● It is sometimes called formulating conclusions from the logical presentations of specific observation of facts. The following are some examples of inductive arguments. 2. Eighty percent of the youth of today are computer literate. Glenn is a youth of today. Therefore, the chance that Glenn is computer literate is 80 percent. INDUCTIVE REASONING The given examples show a kind of reasoning building from specific events to general rule. It shows inferences from the actual and factual observations to be able to arrive at a conclusion. Inductive thinking can be done in the following order: observation, analysis, inference, and confirmation.
DEDUCTIVE REASONING
● Is the reverse process of inductive reasoning. ● It is a kind of mental thinking drawn from a general statement down to specific facts or event. The following are some examples of deductive arguments. 1. All Sanglahi Dance Troupe members are great performers. Fatima is a member of the Sanglahi Dance Troupe. Fatima is a great performer. DEDUCTIVE REASONING
● Is the reverse process of inductive reasoning. ● It is a kind of mental thinking drawn from a general statement down to specific facts or event. The following are some examples of deductive arguments. 2. Everyone who accepts Jesus as his Lord and savior will enter the heaven. I accept Jesus as my Lord and savior. I will enter the heaven. DEDUCTIVE REASONING
● Is the reverse process of inductive reasoning. ● It is a kind of mental thinking drawn from a general statement down to specific facts or event. The following are some examples of deductive arguments. 3. If ∠M < 90°, then ∠M is an acute angle. ∠V = 58° Angle V is an acute angle. DEDUCTIVE REASONING The given examples express a premise that is true and each step in the argument follows logically from the previous step.
IF-THEN STATEMENT
● An if-then statement is one of the common features of deductive reasoning. ● It is also called the Law of Detachment. Here are the three basic forms of deductive reasoning. 1. p ⟶ q (Conditional Statement) 2. p (Hypothesis) 3. q (Conclusion) IF-THEN STATEMENT
If-then statements have five different structures: 1. Conditional 2. Converse 3. Counter example 4. Inverse 5. Contrapositive IF-THEN STATEMENT
Statement: If Odie will go to church after work, then he will listen to the preachings. Note that, p (Odie will go to church after work.) q (He will listen to the preachings.) 1. CONDITIONAL ● Combination of two statements p and q and by the words if and then. ● If p, then q.
Identify the hypothesis (p) and the conclusion (q). 1. CONDITIONAL ● If I win the game, then I’ll get the prize. ● If I say bad words, then I will be punished. ● If I study hard, then I will graduate. ● I’ll bring an umbrella, if it rains tomorrow.
Write each statement in if-then form. 1. CONDITIONAL ● The measure of an acute angle is between 0 and 90. If an angle is acute, then its measure is between 0 and 90. ● Equilateral triangles are equiangular. If triangles are equilateral, then it is equiangular. ● All whole numbers are integers. If a number is a whole number, then it is an integer.
Statement: If Odie will listen to preachings, then he will go to church after work. Note that, q (Odie will listen to preachings.) p (He will go to church after work.) 2. CONVERSE From the given example, note that in the converse (q implies p), the hypothesis and the conclusion in the conditional (p implies q) were interchanged. A statement and a converse express different meanings. It does not follow that if the statement is true, its converse is also true. Sometimes, the converse of a certain argument is false.
Statement: Every quadrilateral has at least two parallel sides. Counter Example: Any trapezium is a quadrilateral. A trapezium is a quadrilateral. 3. COUNTER EXAMPLE Notice that the statement is not always true. It means that a counter example is a specific case, which negates the general statement, and it proves that the argument is false.
Statement: If is equal to 25, then x is equal to 5. Inverse: If is not equal to 25, then x is not equal to 5. 4. INVERSE Note that an inverse is an argument, which is simply the conditional form inverted using the word not. Thus, If p, then q. If not p, then not q.
Statement: If is equal to 25, then x is equal to 5. Contrapositive: If x is not equal to 5, then is not equal to 25. 5. CONTRAPOSITIVE Notice that the rules of logic negation shown in the examples are as follows: a. The negation of a true statement is always false. b. The negation of a false statement is always true.
DEDUCTIVE REASONING Type of Statement Words Symbols Conditional If-then form p → q Converse Exchange the hypothesis and conclusion q → p Inverse Negating the hypothesis and conclusion ~p → ~q Contrapositive Negating the converse of the conditional ~q → ~p
EXAMPLE 1: Type of Statement Words Symbols Conditional If animals have stripes, then they are zebras. p → q Converse If the animals are zebras, then they have stripes. q → p Inverse If animals do not have stripes, then they are not zebras. ~p → ~q Contrapositive If the animals are not zebras, then they do not have stripes. ~q → ~p
EXAMPLE 1: Type of Statement Words Symbols Conditional If animals have stripes, then they are zebras. p → q Converse If animals are zebras, then they have stripes. q → p Inverse If animals don’t have stripes, then they are not zebras. ~p → ~q Contrapositive If animals are not zebras, then they don’t have stripes. ~q → ~p
EXAMPLE 2: Type of Statement Words Symbols Conditional If triangles are equilateral, then they are equiangular. p → q Converse If triangles are equiangular, then they are equilateral. q → p Inverse If triangles are not equilateral, then they are not equiangular. ~p → ~q Contrapositive If triangles are not equiangular, then they are not equilateral. ~q → ~p
EXAMPLE 2: Type of Statement Words Symbols Conditional If triangles are equilateral, then they are equiangular. p → q Converse If triangles are equiangular, then they are equilateral. q → p Inverse If triangles are not equilateral, then they are not equiangular. ~p → ~q Contrapositive If triangles are not equiangular, then they are not equilateral. ~q → ~p
EXAMPLE 3: Type of Statement Words Symbols Conditional If it is a whole number, then it is an integer. p → q Converse If it is an integer, then it is a whole number. q → p Inverse If it is not a whole number, then it is not an integer. ~p → ~q Contrapositive If it is not an integer, then it is not a whole number. ~q → ~p
EXAMPLE 3: Type of Statement Words Symbols Conditional If it is a whole number, then it is an integer. p → q Converse If it is an integer, then it is a whole number. q → p Inverse If it is not a whole number, then it is not an integer. ~p → ~q Contrapositive If it is not an integer, then it is not a whole number. ~q → ~p
EXAMPLE 4: Type of Statement Words Symbols Conditional If two lines intersect, then they lie in only one plane. p → q Converse If two lines lie in only one plane, then they intersect. q → p Inverse If two lines do not intersect, then they do not lie in only one plane. ~p → ~q Contrapositive If two lines do not lie in only one plane, then they do not intersect. ~q → ~p
EXAMPLE 4: Type of Statement Words Symbols Conditional If two lines intersect, then they lie in only one plane. p → q Converse If two lines lie in one plane, then they intersect. q → p Inverse If two lines do not intersect, then they do not lie in only one plane. ~p → ~q Contrapositive If two lines do not lie in one plane, then they do not intersect. ~q → ~p

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Inductive and Deductive Reasoning

  • 3. ● The process of formulating and developing a general statement. ● It is sometimes called formulating conclusions from the logical presentations of specific observation of facts. The following are some examples of inductive arguments. 1. Lorenz and Joshua are drummers. They feel pain after rehearsing for three hours straight. Jasper is also a drummer and feels the same pain after rehearsing for three hours straight. All drummers who rehearse for three hours straight feel pain in their arms. INDUCTIVE REASONING
  • 4. ● The process of formulating and developing a general statement. ● It is sometimes called formulating conclusions from the logical presentations of specific observation of facts. The following are some examples of inductive arguments. 2. Eighty percent of the youth of today are computer literate. Glenn is a youth of today. Therefore, the chance that Glenn is computer literate is 80 percent. INDUCTIVE REASONING The given examples show a kind of reasoning building from specific events to general rule. It shows inferences from the actual and factual observations to be able to arrive at a conclusion. Inductive thinking can be done in the following order: observation, analysis, inference, and confirmation.
  • 6. ● Is the reverse process of inductive reasoning. ● It is a kind of mental thinking drawn from a general statement down to specific facts or event. The following are some examples of deductive arguments. 1. All Sanglahi Dance Troupe members are great performers. Fatima is a member of the Sanglahi Dance Troupe. Fatima is a great performer. DEDUCTIVE REASONING
  • 7. ● Is the reverse process of inductive reasoning. ● It is a kind of mental thinking drawn from a general statement down to specific facts or event. The following are some examples of deductive arguments. 2. Everyone who accepts Jesus as his Lord and savior will enter the heaven. I accept Jesus as my Lord and savior. I will enter the heaven. DEDUCTIVE REASONING
  • 8. ● Is the reverse process of inductive reasoning. ● It is a kind of mental thinking drawn from a general statement down to specific facts or event. The following are some examples of deductive arguments. 3. If ∠M < 90°, then ∠M is an acute angle. ∠V = 58° Angle V is an acute angle. DEDUCTIVE REASONING The given examples express a premise that is true and each step in the argument follows logically from the previous step.
  • 10. ● An if-then statement is one of the common features of deductive reasoning. ● It is also called the Law of Detachment. Here are the three basic forms of deductive reasoning. 1. p ⟶ q (Conditional Statement) 2. p (Hypothesis) 3. q (Conclusion) IF-THEN STATEMENT
  • 11. If-then statements have five different structures: 1. Conditional 2. Converse 3. Counter example 4. Inverse 5. Contrapositive IF-THEN STATEMENT
  • 12. Statement: If Odie will go to church after work, then he will listen to the preachings. Note that, p (Odie will go to church after work.) q (He will listen to the preachings.) 1. CONDITIONAL ● Combination of two statements p and q and by the words if and then. ● If p, then q.
  • 13. Identify the hypothesis (p) and the conclusion (q). 1. CONDITIONAL ● If I win the game, then I’ll get the prize. ● If I say bad words, then I will be punished. ● If I study hard, then I will graduate. ● I’ll bring an umbrella, if it rains tomorrow.
  • 14. Write each statement in if-then form. 1. CONDITIONAL ● The measure of an acute angle is between 0 and 90. If an angle is acute, then its measure is between 0 and 90. ● Equilateral triangles are equiangular. If triangles are equilateral, then it is equiangular. ● All whole numbers are integers. If a number is a whole number, then it is an integer.
  • 15. Statement: If Odie will listen to preachings, then he will go to church after work. Note that, q (Odie will listen to preachings.) p (He will go to church after work.) 2. CONVERSE From the given example, note that in the converse (q implies p), the hypothesis and the conclusion in the conditional (p implies q) were interchanged. A statement and a converse express different meanings. It does not follow that if the statement is true, its converse is also true. Sometimes, the converse of a certain argument is false.
  • 16. Statement: Every quadrilateral has at least two parallel sides. Counter Example: Any trapezium is a quadrilateral. A trapezium is a quadrilateral. 3. COUNTER EXAMPLE Notice that the statement is not always true. It means that a counter example is a specific case, which negates the general statement, and it proves that the argument is false.
  • 17. Statement: If is equal to 25, then x is equal to 5. Inverse: If is not equal to 25, then x is not equal to 5. 4. INVERSE Note that an inverse is an argument, which is simply the conditional form inverted using the word not. Thus, If p, then q. If not p, then not q.
  • 18. Statement: If is equal to 25, then x is equal to 5. Contrapositive: If x is not equal to 5, then is not equal to 25. 5. CONTRAPOSITIVE Notice that the rules of logic negation shown in the examples are as follows: a. The negation of a true statement is always false. b. The negation of a false statement is always true.
  • 19. DEDUCTIVE REASONING Type of Statement Words Symbols Conditional If-then form p → q Converse Exchange the hypothesis and conclusion q → p Inverse Negating the hypothesis and conclusion ~p → ~q Contrapositive Negating the converse of the conditional ~q → ~p
  • 20. EXAMPLE 1: Type of Statement Words Symbols Conditional If animals have stripes, then they are zebras. p → q Converse If the animals are zebras, then they have stripes. q → p Inverse If animals do not have stripes, then they are not zebras. ~p → ~q Contrapositive If the animals are not zebras, then they do not have stripes. ~q → ~p
  • 21. EXAMPLE 1: Type of Statement Words Symbols Conditional If animals have stripes, then they are zebras. p → q Converse If animals are zebras, then they have stripes. q → p Inverse If animals don’t have stripes, then they are not zebras. ~p → ~q Contrapositive If animals are not zebras, then they don’t have stripes. ~q → ~p
  • 22. EXAMPLE 2: Type of Statement Words Symbols Conditional If triangles are equilateral, then they are equiangular. p → q Converse If triangles are equiangular, then they are equilateral. q → p Inverse If triangles are not equilateral, then they are not equiangular. ~p → ~q Contrapositive If triangles are not equiangular, then they are not equilateral. ~q → ~p
  • 23. EXAMPLE 2: Type of Statement Words Symbols Conditional If triangles are equilateral, then they are equiangular. p → q Converse If triangles are equiangular, then they are equilateral. q → p Inverse If triangles are not equilateral, then they are not equiangular. ~p → ~q Contrapositive If triangles are not equiangular, then they are not equilateral. ~q → ~p
  • 24. EXAMPLE 3: Type of Statement Words Symbols Conditional If it is a whole number, then it is an integer. p → q Converse If it is an integer, then it is a whole number. q → p Inverse If it is not a whole number, then it is not an integer. ~p → ~q Contrapositive If it is not an integer, then it is not a whole number. ~q → ~p
  • 25. EXAMPLE 3: Type of Statement Words Symbols Conditional If it is a whole number, then it is an integer. p → q Converse If it is an integer, then it is a whole number. q → p Inverse If it is not a whole number, then it is not an integer. ~p → ~q Contrapositive If it is not an integer, then it is not a whole number. ~q → ~p
  • 26. EXAMPLE 4: Type of Statement Words Symbols Conditional If two lines intersect, then they lie in only one plane. p → q Converse If two lines lie in only one plane, then they intersect. q → p Inverse If two lines do not intersect, then they do not lie in only one plane. ~p → ~q Contrapositive If two lines do not lie in only one plane, then they do not intersect. ~q → ~p
  • 27. EXAMPLE 4: Type of Statement Words Symbols Conditional If two lines intersect, then they lie in only one plane. p → q Converse If two lines lie in one plane, then they intersect. q → p Inverse If two lines do not intersect, then they do not lie in only one plane. ~p → ~q Contrapositive If two lines do not lie in one plane, then they do not intersect. ~q → ~p