deductive forms and symbolism instructor: dr. jason sheley...• one way to think of modus tollens...
TRANSCRIPT
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Deductive Forms and SymbolismInstructor: Dr. Jason Sheley
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Opening puzzle: The Experience Machine
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Would you plug into the
machine? Why or why
not?
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If you agree with Nozick, that
gives us reasons to consider Descartes' answer...
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Let's talk about essays
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Inference Rules and Games
• As we shall soon see, there are common patterns that invalid arguments fall into
• But we should also recognize there are common patterns of valid arguments.
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• Here is the first set of patterns.
• (Note: this is a quick introduction. See the textbook for more rules and examples)
• If it helps, you can think of the patterns as rules that we use in a game.
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Disjunctive Syllogism
A Bornot A
B
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• You can either go right or go left.
• You cannot go right.
• Therefore, you must go left.
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• Either the Angels win the World Series, or the Red Sox win the World Series.
• The Angels do not win.
• Therefore, the Red Sox win the World Series.
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Disjunctive Syllogism
• A v B
• ~ B
• Therefore, A
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Modus Ponens
If A, then BA
B
“the mode of placing”
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MODUS PONENS
• A ——-> B
• A
• Therefore, B
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Example
• If the car is parked on the street between 8 am and 10 am, then it is parked illegally.
• The car is parked on the street between 8am and 10am.
• Therefore, it is parked illegally.
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Conditional Statements
• In order to understand Modus Ponens, it helps to understand conditionals better.
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Conditional Statements
• If ______, Then ____
• The “If” part of the statement is known as the antecedent.
• The “Then” part of the statement is known as the consequent.
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• A conditional statement indicates that a relationship holds between two ideas.
• Conditional statements make no claims about the truth of the components alone.
• What a conditional says is IF the antecedent is true, THEN the consequent is true.
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These are all conditional statements
• If there is traffic on the 5, then it is better to take the 91.
• If the Angels win, then the Angels qualify for the playoffs.
• If there is traffic on the 5, then the Angels qualify for the playoffs.
• (The above statements all have the same truth conditions.)
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• Conditional statements are powerful tools of reasoning, and become all the more powerful when we use them in Modus Ponens arguments.
• Let’s get some more practice with conditionals. We are familiar with conditionals from games…
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Modus Tollens
not A
not B
If A, then B
“the mode of taking away”
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Example
• If the car is parked on the street between 8 am and 10 am, then the car is parked illegally.
• The car is not parked illegally.
• Therefore, the car is not parked on the street between 8am and 10am.
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• One way to think of Modus Tollens is to recall the strange truth table that is generated for Modus Ponens.
• We were mainly concerned about the case in which the antecedent is true, and consequent is false.
• Given what the conditional is saying, we can infer that if the consequent is false, the antecedent must have been false.
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Hypothetical Syllogism
If A, then BIf B,
If A, then C
then C
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Example
• If the car is parked on the street between 8 am and 10 am, then the car receives a ticket.
• If the car receives a ticket, then the owner must pay a fine.
• Therefore, if the car is parked on the street between 8am and 10am, then the owner must pay a fine.
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• Now that we have seen the valid forms, we want to also be able to prove that an argument does not work.
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• There are some common mistakes. We can notice these mistakes by looking at deviations from the forms we have seen.
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INVALID
• If A, then B
• B
• Therefore, A
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INVALID
• If A, then B
• If C, then B
• Therefore, if A then C
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INVALID
• If A, then B
• not A
• Therefore, not B
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INVALID
• Either A or B
• A
• Therefore, B
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Why use symbols?
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