Valid arguments 1. Write the following argument in symbolic form, and determine
ID: 3144574 • Letter: V
Question
Valid arguments
1. Write the following argument in symbolic form, and determine whether it is valid. Give a brief explanation. Define your statements precisely. Do not use negatives in your statement, or compound statements. Use the rules of inference. Carefully number every step in your argument, and justify each step.
• (1) If this computer program is correct, then it produces the correct output with the test data.
• (2) This computer program produces the correct output.
• Conclusion: Therefore the computer program is correct.
2. Fill in the blank so that the argument is valid, and write the complete argument in symbolic form.
• (1) If they were unsure of the address, then they would have telephoned.
• (2) — ??? —
• Conclusion: They were sure of the address.
Explanation / Answer
1. Let p : This computer program is correct and q : It products the correct output (with the test data)
(1) If this computer program is correct, then it produces the correct output with the test data.
This can be written in symbolic form as
p -> q
(2) This computer program produces the correct output
q
Conclusion: Therefore the computer program is correct
p
The argument has a not a valid conclusion:
Reason: The two statements can be combined as
(p -> q) ^ q.
When p is false and q is true the statement evaluates to
(false -> true) ^ true
= true ^ true
= true.
Thus p could be false even if p -> q and q are true.
Thus the conclusion is invalid.
2. Let p : They were unsure of the address and q : They telephoned.
(1) If they were unsure of the address, then they would have telephoned.
This can be written in symbolic form as
p -> q
Conclusion: They were sure of the address.
~p
Using the modus tollens rule,
(p -> q) ^ ~q => ~p
(2) is ~q
i.e They did not telephone.
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