2.3 Valid and Invalid Arguments - United States Naval Academy · Valid Arguments De nition 1 An...
Transcript of 2.3 Valid and Invalid Arguments - United States Naval Academy · Valid Arguments De nition 1 An...
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2.3 Valid and Invalid Arguments
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Valid Arguments
Definition1 An argument (argument form) is a sequence of statements
(statement forms).
2 All statements in an argument, except the final one, are calledpremises (or assumptions or hypothesis).
3 The final statement is called the conclusion.
4 An argument form is valid if, no matter what particular statementsare substituted for the statement variables in its premises, wheneverthe resulting premises are all true, the conclusion is also true.
(Hint:If any premises are false, then the argument is vacuously true.)
5 An argument is valid means that its form is valid.
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Valid Arguments
Definition1 An argument (argument form) is a sequence of statements
(statement forms).
2 All statements in an argument, except the final one, are calledpremises (or assumptions or hypothesis).
3 The final statement is called the conclusion.
4 An argument form is valid if, no matter what particular statementsare substituted for the statement variables in its premises, wheneverthe resulting premises are all true, the conclusion is also true.
(Hint:If any premises are false, then the argument is vacuously true.)
5 An argument is valid means that its form is valid.
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Valid Arguments
Definition1 An argument (argument form) is a sequence of statements
(statement forms).
2 All statements in an argument, except the final one, are calledpremises (or assumptions or hypothesis).
3 The final statement is called the conclusion.
4 An argument form is valid if, no matter what particular statementsare substituted for the statement variables in its premises, wheneverthe resulting premises are all true, the conclusion is also true. (Hint:If any premises are false, then the argument is
vacuously true.)
5 An argument is valid means that its form is valid.
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Valid Arguments
Definition1 An argument (argument form) is a sequence of statements
(statement forms).
2 All statements in an argument, except the final one, are calledpremises (or assumptions or hypothesis).
3 The final statement is called the conclusion.
4 An argument form is valid if, no matter what particular statementsare substituted for the statement variables in its premises, wheneverthe resulting premises are all true, the conclusion is also true. (Hint:If any premises are false, then the argument is vacuously true.)
5 An argument is valid means that its form is valid.
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Valid Arguments
Definition1 An argument (argument form) is a sequence of statements
(statement forms).
2 All statements in an argument, except the final one, are calledpremises (or assumptions or hypothesis).
3 The final statement is called the conclusion.
4 An argument form is valid if, no matter what particular statementsare substituted for the statement variables in its premises, wheneverthe resulting premises are all true, the conclusion is also true. (Hint:If any premises are false, then the argument is vacuously true.)
5 An argument is valid means that its form is valid.
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Testing an Argument Form for Validity
Fact1 Identify the premises and conclusion of the argument form.
2 Construct a truth table showing the truth values of all the premisesand the conclusion.
3 A row of the truth table in which all the premises are true is called acritical row.
1 If there is a critical row in which the conclusion is false, then theargument is invalid.
2 If the conclusion in every critical row is true, then the argument form isvalid.
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Testing an Argument Form for Validity
Fact1 Identify the premises and conclusion of the argument form.
2 Construct a truth table showing the truth values of all the premisesand the conclusion.
3 A row of the truth table in which all the premises are true is called acritical row.
1 If there is a critical row in which the conclusion is false, then theargument is invalid.
2 If the conclusion in every critical row is true, then the argument form isvalid.
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Modus Ponens
Fact
An argument form consisting of two premises and a conclusion iscalled a syllogism.
The most famous example is modus ponens (“mood that affirms”):If p → q, p ∴ q
p q p → q p q
T T T TT F F TF T T FF F T F
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Modus Ponens
Fact
An argument form consisting of two premises and a conclusion iscalled a syllogism.
The most famous example is modus ponens (“mood that affirms”):If p → q, p ∴ q
p q p → q p q
T T T T TT F F TF T T FF F T F
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Example
Example
Here is a modus ponens argument: If it snows more than 2” then the NavalAcademy closes. It snowed more than 2”.
∴ The Naval Academy closed.
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Example
Example
Here is a modus ponens argument: If it snows more than 2” then the NavalAcademy closes. It snowed more than 2”. ∴ The Naval Academy closed.
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Modus Tollens
Fact
Modus tollens (“mood that denies”) has the formIf p → q. ∼ q ∴ ∼ p.
Example
If it snows more than 2” then the Naval Academy closes. The NavalAcademy did not close.
∴ It snowed less than 2”.
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Modus Tollens
Fact
Modus tollens (“mood that denies”) has the formIf p → q. ∼ q ∴ ∼ p.
Example
If it snows more than 2” then the Naval Academy closes. The NavalAcademy did not close. ∴ It snowed less than 2”.
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Additional rules of inference
Definition1 Generalization: p ∴ p ∨ q or
q ∴ p ∨ q.
2 Specialization: p ∧ q ∴ p or p ∧ q ∴ q.
3 Elimination: p ∨ q, ∼ q ∴ p or p ∨ q. ∼ p ∴ q.
4 Transitivity: p → q, q → r ∴ p → r .
5 Division into Cases: p ∨ q, p → r , q → r ∴ r .
Example
Give examples of each of the above rules of inference.
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Additional rules of inference
Definition1 Generalization: p ∴ p ∨ q or q ∴ p ∨ q.
2 Specialization: p ∧ q ∴ p or
p ∧ q ∴ q.
3 Elimination: p ∨ q, ∼ q ∴ p or p ∨ q. ∼ p ∴ q.
4 Transitivity: p → q, q → r ∴ p → r .
5 Division into Cases: p ∨ q, p → r , q → r ∴ r .
Example
Give examples of each of the above rules of inference.
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Additional rules of inference
Definition1 Generalization: p ∴ p ∨ q or q ∴ p ∨ q.
2 Specialization: p ∧ q ∴ p or p ∧ q ∴ q.
3 Elimination: p ∨ q, ∼ q ∴ p or
p ∨ q. ∼ p ∴ q.
4 Transitivity: p → q, q → r ∴ p → r .
5 Division into Cases: p ∨ q, p → r , q → r ∴ r .
Example
Give examples of each of the above rules of inference.
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Additional rules of inference
Definition1 Generalization: p ∴ p ∨ q or q ∴ p ∨ q.
2 Specialization: p ∧ q ∴ p or p ∧ q ∴ q.
3 Elimination: p ∨ q, ∼ q ∴ p or p ∨ q. ∼ p ∴ q.
4 Transitivity: p → q, q → r
∴ p → r .
5 Division into Cases: p ∨ q, p → r , q → r ∴ r .
Example
Give examples of each of the above rules of inference.
2.3 Valid and Invalid Arguments 8 / 10
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Additional rules of inference
Definition1 Generalization: p ∴ p ∨ q or q ∴ p ∨ q.
2 Specialization: p ∧ q ∴ p or p ∧ q ∴ q.
3 Elimination: p ∨ q, ∼ q ∴ p or p ∨ q. ∼ p ∴ q.
4 Transitivity: p → q, q → r ∴ p → r .
5 Division into Cases: p ∨ q, p → r , q → r
∴ r .
Example
Give examples of each of the above rules of inference.
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Additional rules of inference
Definition1 Generalization: p ∴ p ∨ q or q ∴ p ∨ q.
2 Specialization: p ∧ q ∴ p or p ∧ q ∴ q.
3 Elimination: p ∨ q, ∼ q ∴ p or p ∨ q. ∼ p ∴ q.
4 Transitivity: p → q, q → r ∴ p → r .
5 Division into Cases: p ∨ q, p → r , q → r ∴ r .
Example
Give examples of each of the above rules of inference.
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Additional rules of inference
Definition1 Generalization: p ∴ p ∨ q or q ∴ p ∨ q.
2 Specialization: p ∧ q ∴ p or p ∧ q ∴ q.
3 Elimination: p ∨ q, ∼ q ∴ p or p ∨ q. ∼ p ∴ q.
4 Transitivity: p → q, q → r ∴ p → r .
5 Division into Cases: p ∨ q, p → r , q → r ∴ r .
Example
Give examples of each of the above rules of inference.
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Fallacies
Fact
We discuss two fallacies:
1 Converse Error
2 Example: If Midshipmen X cheats on the test then Midshipmen X sitsin the last row. Midshipmen X sits in the back row.
∴ Midshipmen Xcheats on the test.
3 Inverse Error
4 Example: If Midshipmen X studies for the test then Midshipmen Xgets a high score on the test. Midshipmen X does not study for thetest. ∴ Midshipmen X gets a low score on the test.
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Fallacies
Fact
We discuss two fallacies:
1 Converse Error
2 Example: If Midshipmen X cheats on the test then Midshipmen X sitsin the last row. Midshipmen X sits in the back row.
∴ Midshipmen Xcheats on the test.
3 Inverse Error
4 Example: If Midshipmen X studies for the test then Midshipmen Xgets a high score on the test. Midshipmen X does not study for thetest. ∴ Midshipmen X gets a low score on the test.
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Fallacies
Fact
We discuss two fallacies:
1 Converse Error
2 Example: If Midshipmen X cheats on the test then Midshipmen X sitsin the last row. Midshipmen X sits in the back row. ∴ Midshipmen Xcheats on the test.
3 Inverse Error
4 Example: If Midshipmen X studies for the test then Midshipmen Xgets a high score on the test. Midshipmen X does not study for thetest.
∴ Midshipmen X gets a low score on the test.
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Fallacies
Fact
We discuss two fallacies:
1 Converse Error
2 Example: If Midshipmen X cheats on the test then Midshipmen X sitsin the last row. Midshipmen X sits in the back row. ∴ Midshipmen Xcheats on the test.
3 Inverse Error
4 Example: If Midshipmen X studies for the test then Midshipmen Xgets a high score on the test. Midshipmen X does not study for thetest.
∴ Midshipmen X gets a low score on the test.
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Fallacies
Fact
We discuss two fallacies:
1 Converse Error
2 Example: If Midshipmen X cheats on the test then Midshipmen X sitsin the last row. Midshipmen X sits in the back row. ∴ Midshipmen Xcheats on the test.
3 Inverse Error
4 Example: If Midshipmen X studies for the test then Midshipmen Xgets a high score on the test. Midshipmen X does not study for thetest. ∴ Midshipmen X gets a low score on the test.
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Contradiction Rule
Definition
If you can show that the supposition that statement p is false leadslogically to a contradiction, then you can conclude that p is true.Symbolically
∼ p → c ∴ p.
Example (Knights and Knaves)
The logician Raymond Smullyan describes an island containing two typesof people: knights who always tell the truth and knaves who always lie.You visit the island and are approached by two natives who speak to youas follows: A says: B is a knight. B says: A and I are of opposite type.What are A and B?
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Contradiction Rule
Definition
If you can show that the supposition that statement p is false leadslogically to a contradiction, then you can conclude that p is true.Symbolically ∼ p → c
∴ p.
Example (Knights and Knaves)
The logician Raymond Smullyan describes an island containing two typesof people: knights who always tell the truth and knaves who always lie.You visit the island and are approached by two natives who speak to youas follows: A says: B is a knight. B says: A and I are of opposite type.What are A and B?
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Contradiction Rule
Definition
If you can show that the supposition that statement p is false leadslogically to a contradiction, then you can conclude that p is true.Symbolically ∼ p → c ∴ p.
Example (Knights and Knaves)
The logician Raymond Smullyan describes an island containing two typesof people: knights who always tell the truth and knaves who always lie.You visit the island and are approached by two natives who speak to youas follows: A says: B is a knight. B says: A and I are of opposite type.What are A and B?
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Contradiction Rule
Definition
If you can show that the supposition that statement p is false leadslogically to a contradiction, then you can conclude that p is true.Symbolically ∼ p → c ∴ p.
Example (Knights and Knaves)
The logician Raymond Smullyan describes an island containing two typesof people: knights who always tell the truth and knaves who always lie.You visit the island and are approached by two natives who speak to youas follows: A says: B is a knight. B says: A and I are of opposite type.What are A and B?
2.3 Valid and Invalid Arguments 10 / 10