a) Using long or short truth tables (symbolize as necessary and show tables) determine and state whether the following arguments are valid, briefly explaining your table; and b) if valid, demonstrate validity by deductions using Group I rules.
Arg
1. P-->Q
2. ~P ^:~Q
To determine the validity of the argument, we will first construct the truth tables for the given premises and the conclusion.
a) Constructing a long truth table:
1. P → Q
2. ~P ^ ~Q
|--
∴ ~Q (conclusion)
We will consider all possible combinations of truth values for P and Q.
| P | Q | ~P | ~Q | P → Q | ~P ^ ~Q | ~Q |
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| T | T | F | F | T | F | F |
| T | F | F | T | F | T | T |
| F | T | T | F | T | F | F |
| F | F | T | T | T | F | T |
In the table, we can see that in every row where the premises (columns 1 and 2) are true, the conclusion (column 7) is also true. Therefore, the argument is valid.
b) Demonstrating validity by deductions using Group I rules:
1. P → Q (premise)
2. ~P ^ ~Q (premise)
3. ~P (simplification, from premise 2)
4. Q (modus ponens, from premises 1 and 3)
5. ~Q (simplification, from premise 2)
From step 4 and step 5, we obtain both Q and ~Q, which is a contradiction. Therefore, the assumption that there is an invalid argument is incorrect. Hence, the argument is valid.