Valid arguments

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Valid arguments A valid argument has the following property: It is impossible for the conclusion to be false, if we assume that every premise is true. In a valid argument, “the truth of the premises forces the truth of the conclusion.”

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Valid arguments. A valid argument has the following property: It is impossible for the conclusion to be false, if we assume that every premise is true. In a valid argument, “the truth of the premises forces the truth of the conclusion.”. Use of truth tables to analyze arguments. - PowerPoint PPT Presentation

Transcript of Valid arguments

Page 1: Valid arguments

Valid arguments

A valid argument has the following property:It is impossible for the conclusion to be false, if we assume that every premise is true.

In a valid argument, “the truth of the premises forces the truth of the conclusion.”

Page 2: Valid arguments

Use of truth tables to analyze arguments

1. Symbolize (consistently) all of the premises and the conclusion.

2. Make a truth table having a column for each premise and for the conclusion.

3. If there is a row in the truth table where every premise column is true but the conclusion column is false (a counterexample row) then the argument is invalid. If there are no counterexample rows, then the argument is valid.

Page 3: Valid arguments

Why does this method work?

When we have filled in the truth table, we are checking to see if there is a row where the conclusion is false while every premise is true.

If there is a row where the conclusion is false while every premise is true, then the truth table has shown that it is possible for the conclusion to be false at the same time that every premise is assumed to be true: this is exactly the definition of an invalid argument.

Page 4: Valid arguments

Exercise

Use a truth table to test the validity of the following argument.

If I enter the poodle den, then I will carry my electric poodle prod or my can of mace.

I am carrying my electric poodle prod but not my can of mace.

Therefore, I will enter the poodle den.

A. Valid B. Invalid

Page 5: Valid arguments

Solution

If I enter the poodle den, then I will carry my electric poodle prod or my can of mace.

I am carrying my electric poodle prod but not my can of mace.Therefore, I will enter the poodle den. Step 1: SymbolizeLet p: I enter the poodle denq: I carry my electric poodle prodr: I carry my can of macep(qr) q ~r p

Page 6: Valid arguments

Solution, page 2

Step 2: Make a truth table having a column for each premise and a column for the conclusion.

p(qr)

q ~r p

p

q r

~r

q∨r

Premise p→ (q∨r)

Premise q∧ ~r

Conclusion p

T T T F T T F T T T F T T T T T T F T F T T F T T F F T F F F T F T T F T T F F F T F T T T T F F F T F T T F F F F F T F T F F

Page 7: Valid arguments

Solution, page 3

Step 3: Is there a row where the conclusion is false while both premises are true?Yes, the sixth row shows that the argument is INVALID.

p

q

r

~r

q∨r

Premise p→ (q∨r)

Premise q∧ ~r

Conclusion p

T T T F T T F T T T F T T T T T T F T F T T F T T F F T F F F T F T T F T T F F F T F T T T T F F F T F T T F F F F F T F T F F

Page 8: Valid arguments

Exercise

Test the validity of the argument.

I don’t like muskrats.

If I own a badger or I don’t own a wolverine, then I like muskrats.

Therefore, I own a wolverine and I don’t own a badger.

A. Valid B. Invalid

Page 9: Valid arguments

Solution

I don’t like muskrats.

If I own a badger or I don’t own a wolverine, then I like muskrats.

Therefore, I own a wolverine and I don’t own a badger.

First, reduce the argument to symbols.

Let p: I like muskrats q: I own a badger r: I own a wolverineWith this choice of symbols, the argument as the following form:

~p (q~r) p r ~q

Page 10: Valid arguments

Solution, page 2

Step 3: There is NO ROW where the conclusion is false while both premises are true, so the argument is VALID.

Step 2: Make a truth table having a column for each premise and a column for the conclusion.

~p (q~r) p r ~q

Page 11: Valid arguments

Exercise

The Argue-mentor, Part 3http://www.math.fsu.edu/~wooland/arg3/pats.html

Test the validity of this argument.~p~q q ~p

A. ValidB. Invalid

Page 12: Valid arguments

Solution

Below is a truth table for the argument. (There may be some redundant

columns.)

The three columns in bold are the important columns. An argument is INVALID if it is possible for the conclusion to be false while every premise is true.

There is no row where the conclusion is false while every premise is true, so the argument is valid.

p q ~p ~q PREMISE ~p ∨ ~q

PREMISE q

CONCLUSION ~p

T T F F F T F T F F T T F F F T T F T T T F F T T T F T


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