Deductive Reasoning Valid Arguments Logic Arguments Deductive Reasoning Valid Arguments Resources: HRW Geometry, Lesson 12.1, 2.2
How can you tell if an argument is valid? Introduction Instruction Examples Practice Newspaper reporters, politicians, lawyers, baseball managers, and mathematicians, use logic to convince others of their points of view. Recognizing when an argument is valid or invalid will help you think more clearly in confusing situations. This lesson will help you recognize three important valid arguments, modus ponens, modus tollens, and hypothetical syllogism, and two similar arguments that are invalid. How can you tell if an argument is valid?
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List of Instructional Pages Valid Arguments 10. MT – Truth Table 2. Modus Ponens Fallacy – denying the antecedent 3. MP- Euler Diagram 12. Fallacy – Euler Diagram 4. MP – Truth Table 13. Fallacy – Truth Table Fallacy - affirming the consequent 14. Hypothetical Syllogism 6. Fallacy - Euler Diagram 15. HS – Euler Diagram 7. Fallacy – Truth Table 16. HS – Truth Table 8. Modus tollens 17. Review of Arguments 9. MT – Euler Diagram
Valid Arguments Page list Last Next Introduction Instruction Examples Practice Deductive reasoning is the process of drawing logical, certain conclusions by using an argument. In logic, an argument is a series of statements, known as premises, leading to a final statement called a conclusion. A valid argument guarantees that if the premises are all true, then the conclusion will also be true. Three valid arguments that are often used in logic and mathematics are modus ponens, modus tollens, and hypothetical syllogism. Valid Arguments This is page 1 of 17 Page list Last Next
Modus Ponens Page list Last Next Introduction Instruction Examples Practice Consider this argument: Premise #1: If an animal is an amphibian, then it is a vertebrate. Premise #2: Frogs are amphibians. Conclusion: Therefore, frogs are vertebrates. This is a valid argument called modus ponens. We can demonstrate its validity using an Euler diagram and prove its validity using truth tables and the definition of a valid argument. Modus Ponens Page list This is page 2 of 17 Last Next
Euler Diagram Page list Last Next Therefore, the argument is valid. Introduction Instruction Examples Practice We can represent premise 1, “If an animal is an amphibian, then it is a vertebrate” by drawing a circle labeled “Vertebrates” Vertebrates Amphibians containing a smaller circle labeled “Amphibians”. Frogs Amphibians Vertebrates We can represent premise 2, “Frogs are amphibians” by placing a circle labeled Frogs inside the circle labeled “Amphibians.” In order to place the Frogs circle inside the “Amphibians” circle, it must also be inside the “Vertebrates” circle. Thus, the conclusion “Frogs are vertebrates” must be true because Frogs are inside the “Vertebrates” circle. Euler Diagram This is page 3 of 17 Page list Last Next Therefore, the argument is valid.
Proving Modus Ponens valid with a truth table Introduction Instruction Examples Practice Since every time both premise #1 and premise #2 are true, the conclusion is also true, this is a valid argument. We can also prove the argument modus ponens true using a truth table and the definition of a valid argument (if the premises are true, the conclusion is true) p = an animal is an amphibian q = an animal is a vertebrate Proving Modus Ponens valid with a truth table The symbolic form of modus ponens is: p q p q p q Premise #1: Premise #1 Premise #2 Conclusion p q pq T F Page list If an animal is an amphibian, then it is a vertebrate. Last p Premise #2: Frogs are amphibians This is page 4 of 17 q Conclusion: Therefore, frogs are vertebrates Next
An Invalid Argument Page list Next Last Consider this argument: Introduction Instruction Examples Practice Consider this argument: Premise #1: If an animal is an amphibian, then it is a vertebrate. Premise #2: Birds are vertebrates. Conclusion: Therefore, birds are amphibians. This is a not a valid argument even though it is similar to modus ponens. Logical mistakes such as this are called “fallacies”. We can demonstrate that this is not valid using an Euler diagram and prove it is invalid using a truth table and the definition of a valid argument. This invalid argument is called “affirming the consequent”. An Invalid Argument Page list This is page 5 of 17 Last Next
Euler Diagram Page list Next Last Vertebrates Birds Amphibians Introduction Instruction Examples Practice We can represent premise 1, “If an animal is an amphibian, then it is a vertebrate” by drawing a circle labeled “Amphibians” contained in a larger region labeled “Vertebrates”. We can represent premise 2, “Birds are vertebrates” by placing a circle labeled Birds inside the region labeled “Vertebrates.” Vertebrates Birds Amphibians Notice that placing the “Birds” circle inside the “Vertebrates” region, does not mean it must also be inside the “Amphibians” circle. Thus, the conclusion “Birds are amphibians” is not true because Birds does not have to be inside the “Amphibians” circle to be inside the Vertebrates region. Therefore, the argument is invalid. Euler Diagram This is page 6 of 17 Page list Last Next
Proving an argument invalid with a truth table Introduction Instruction Examples Practice Premise #1 Premise #2 Conclusion p q pq T F The symbolic form of this invalid argument is: p q q p We can also prove the argument false using a truth table and the definition of a valid argument. p = an animal is an amphibian q = an animal is a vertebrate T F Premise #1: p q If an animal is an amphibian, then it is a vertebrate. Proving an argument invalid with a truth table Premise #2: q Birds are vertebrates. Conclusion: p Therefore, birds are amphibians. Page list This is page 7 of 17 Notice that the second time premise #1 and premise #2 are both true, the conclusion is false. Therefore the argument is invalid. This fallacy is called “affirming the consequent”. Last Next
Modus tollens Page list Last Next Consider this argument: Introduction Instruction Examples Practice Consider this argument: Premise #1: If a shirt is a Polo, then it has a horse logo. Premise #2: This shirt does not have a horse logo. Conclusion: Therefore this shirt is not a Polo. This is a valid argument called modus tollens. We can demonstrate its validity using an Euler diagram and prove its validity using truth tables and the definition of a valid argument. Modus tollens This is page 8 of 17 Page list Last Next
Euler Diagram Page list Last Next Introduction Instruction Examples Practice We can represent premise 1, “If a shirt is a Polo, then it has a horse logo” by drawing a region labeled “Polo shirt” contained in a larger circle labeled “Shirts with logo”. We can represent premise 2, “This shirt does not have a horse logo” by drawing a different region labeled “Shirts with no logo”. Shirts with logo Shirts no logo Polo Shirt Since the “Shirts no logo” region cannot be placed in the “Shirts with logo” region, it cannot be in the “Polo Shirt” region. The conclusion, this shirt is not a Polo, is correct. Therefore, the argument is valid. Euler Diagram This is page 9 of 17 Page list Last Next
Proving Modus tollens valid with a truth table Introduction Instruction Examples Practice Premise #1 Premise #2 Conclusion p q pq ~q ~p T F We can prove the argument modus tollens true using a truth table and the definition of a valid argument. p = the shirt is a Polo q = the shirt has a horse logo T T Premise #1: pq T If a shirt is a Polo, then it has a horse logo. T T T Premise #2: ~q The symbolic form of modus tollens is: p q ~q ~p Proving Modus tollens valid with a truth table This shirt does not have a horse logo. Conclusion: ~p Therefore, this shirt is not a Polo. This is page 10 of 17 Page list Since the only time both premise #1 and premise #2 are true, the conclusion is also true, this is a valid argument. Last Next
Another Invalid Argument Introduction Instruction Examples Practice Consider this argument: Premise #1: If Susan overslept, then she is running late. Premise #2: Susan did not oversleep. Conclusion: Therefore, Susan is not running late. This is a not a valid argument even though it is similar to modus tollens. We can demonstrate that it is not valid using an Euler diagram and prove it is invalid using a truth table and the definition of a valid argument. This invalid argument is called “denying the antecedent”. Another Invalid Argument This is page 11 of 17 Page list Last Next
Euler Diagram Page list Last Next Introduction Instruction Examples Practice We can represent premise 1, “If Susan overslept, then she is running late” by drawing a region labeled “Overslept” contained in a larger region labeled “Running Late”. Running Late Overslept Susan Since Susan can be placed in the “Running Late” region without being in the “Overslept” region, the conclusion, Susan is not running late, is not correct. Susan could be running late for other reasons. Euler Diagram This is page 12 of 17 Page list This is not a valid argument. Last Next
Proving an argument invalid with a truth table Introduction Instruction Examples Practice Premise #1 Premise #2 Conclusion p q pq ~p ~q T F F* We can also prove the argument false using a truth table and the definition of a valid argument. p = Susan overslept q = Susan is running late Premise #1: If Susan overslept, then she is running late is represented by pq Premise #2: Susan did not oversleep is an example of ~p Conclusion: Therefore, Susan is not running late is an example of ~q The symbolic form of this invalid argument is: p q ~p ~q Proving an argument invalid with a truth table This is page 13 of 17 Page list Notice that the first time premise #1 and premise #2 are both true, the conclusion is false. Therefore the argument is invalid. Last Next
Hypothetical Syllogism Introduction Instruction Examples Practice Consider this argument: Premise #1: If Alfred is 16, then he can get his driver’s license. Premise #2: If Alfred gets his driver’s license, then he can get a hot car. Conclusion: Therefore, if Alfred is 16, he can get a hot car. This is a valid argument called a “hypothetical syllogism”. It is also called a “chain rule” and can be extended for many steps. It is the argument that is most used in mathematical proofs. Notice the similarity to the transitive property from algebra. Hypothetical Syllogism This is page 14 of 17 Page list Last Next
Euler Diagram Page list Last Next Introduction Instruction Examples Practice We can represent premise 1, “If Alfred is 16, then he can get his driver’s license” by drawing a region labeled “16” contained in a larger region labeled “Driver’s license”. Hot Car Driver’s license 16 We can represent premise 2, “If Alfred gets his driver’s license, then he can get a hot car” by putting the “Driver’s license” region inside a region labeled “Hot Car”. Euler Diagram This is page 15 of 17 Page list If Alfred is in “16”, he is automatically in “Hot Car”. Last Next Therefore, this argument is valid.
The symbolic form of a hypothetical syllogism Proving a hypothetical syllogism valid with a truth table Introduction Instruction Examples Practice The symbolic form of a hypothetical syllogism p q q r pr Note the configuration of a truth table with three variables. If Alfred is 16, then he can get his driver’s license. pq p: Alfred is 16 q: Alfred can get his driver’s license r: Alfred can get a hot car. If Alfred gets his driver’s license, then he can get a hot car. qr Premise #1 Premise #2 Conclusion p q r pq qr pr T F Therefore, if Alfred is 16, then he can get a hot car. pr Every time both premises are true, the conclusion is also true. Therefore this is a valid argument. This is page 16 of 17 Page list Last Next
Page list Last Next p q p q ~q ~p q r p r p q q p ~p ~q Introduction Instruction Examples Practice Modus ponens Modus tollens Hypothetical Syllogism p q p q ~q ~p q r p r Fallacy of “Affirming the Consequent” Fallacy of “Denying the Antecedent” p q q p ~p ~q This is page 17 of 17 Page list Last Next
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Write the argument in symbolic form. Introduction Instruction Examples Practice Examples Write the argument in symbolic form. Draw the proper conclusion using a valid argument. Name that argument. Example 1 Example 2 Example 3
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Practice How can I be sure my arguments are valid arguments? Introduction Instruction Examples Practice Practice HRW Homework Help 12.1 Practice with Logical Chains HRW Homework Help 2.2 How can I be sure my arguments are valid arguments?
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Valid Hypothetical Syllogism Example 1 Back to main example page If you pay your taxes late, then you will pay a late penalty. You do not pay your taxes late. Therefore, you will not pay a late penalty. Write each argument in symbolic form and state whether it is valid or invalid. If it is valid, give the name of the argument. p q ~p Not Valid ~q If you score high on the LSATs, then you will be accepted by the law school of your choice. If you are accepted by the law school of your choice, then you will impress your friends. Therefore, if you score high on the LSATs, then you will impress your friends. Example: If a car has air bags, then it is safe. This car has air bags. Therefore, this car is safe. p q p q Valid Modus ponens Valid Hypothetical Syllogism p q q r p r
Example 1 Back to main example page If Malik buys a satellite dish, then he will get 128 TV channels. Malik does not does not get 128 channels. Therefore, Malik did not buy a satellite dish. Write each argument in symbolic form and state whether it is valid or invalid. If it is valid, give the name of the argument. p q ~q ~p Valid Modus tollens Example: If a car has air bags, then it is safe. This car has air bags. Therefore, this car is safe. If Sarah gets a raise, then she will be able to afford some new clothes. Sarah gets a raise. Therefore, Sarah will be able to afford some new clothes. p q p q p q p q Valid Modus ponens Valid Modus ponens
Example 2 Back to main example page If a car has air bags, then it is safe. This car has air bags. Given the first two statements of an argument, draw the correct conclusion. Name the valid argument you used. If no conclusion is possible, write “No Conclusion” Modus ponens (MP) Modus tollens (MT) Hypothetical syllogism (HS) Therefore, this car is safe. (MP) If a movie is exciting, then it will gross a lot of money. This movie grossed a lot of money. No Conclusion If you do not pay your taxes, then you will not qualify for further government assistance. You qualify for further government assistance. Example: If I have a flat tire, I’ll be late for class. I had a flat tire. Therefore, you did pay your taxes. (MT) Therefore, I will be late for class. (MP)
Example 2 Back to main example page If Smoltz pitches the Braves will win. If the Braves win we will celebrate. Given the first two statements of an argument, draw the correct conclusion. Name the valid argument you used. If no conclusion is possible, write “No Conclusion” Modus ponens (MP) Modus tollens (MT) Hypothetical syllogism (HS) Therefore, if Smoltz pitches, we will celebrate. (HS) If John eats Twinkies he gains weight. John didn’t gain weight. Therefore, John did not eat Twinkies. (MT) Example: If I have a flat tire, I’ll be late for class. I had a flat tire. If Edith earns enough money, then she will visit the Bahamas. Edith did not earn enough money. No Conclusion Therefore, I will be late for class. (MP)
Example 3 Back to main example page r t t s r s Hypothetical syllogism Given the argument in symbolic form, name it as: Modus ponens Modus tollens Hypothetical syllogism Fallacy of affirming the consequent Fallacy of denying the antecedent. a b ~b ~a Modus tollens w k k w Fallacy of affirming the consequent (p m) s (p m) s Modus ponens
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