Validity Of Arguments

Validity of an argument is determined as follows:

• An argument $P_1, P_2, P_3 \dots P_n \therefore Q$ is valid if $Q$ is true whenever all the premises $P_1, P_2, P_3 \dots P_n$ are true.
• Validity of an argument is determined if and only if the proposition $(P_1 \land P_2 \land P_3 \land \dots \land P_n) \to Q$ is a tautology.

Prove whether the following argument is valid or not: $P, P \to Q \therefore Q$

Solution

Draw a truth table for $P \to Q$:

1. Since in row 1 the conclusion is true and all the premises are true, then the argument is valid.
2. Since column 5 is a tautology, then the argument is valid.

Use the truth table to show whether the given argument is valid or not:

$P \to Q, Q \to R \therefore P \to R$

Symbolize the given argument and then test its validity:

If I like mathematics, then I will study, either I study or I fail. But I failed, therefore I do not like mathematics.

Solution

The given argument is symbolized as follows:

• Let $p$: I like mathematics
• Let $q$: I will study
• Let $r$: I fail

Then the given argument is as follows:

$p \to q, q \lor r, r \therefore \lnot p$

Testing the validity

$[(p \to q) \land (q \lor r) \land r] \to \lnot p$

Since column 8 is not a tautology, the given argument is not valid.

1. Translate the following arguments in symbolic form and then test its validity:

   • If London is not in Denmark, then Paris is not in France. But Paris is in France, therefore London is in Denmark.
   • If I work I cannot study. Either I work or I pass mathematics. I passed mathematics, therefore I studied.
   • If I buy books, I lose money. I bought books, therefore I lost money.

2. Determine the validity of:

   • $p \to q, \lnot q \therefore \lnot p$
   • $\lnot p \to q, p \therefore \lnot q$
   • $p \to \lnot q, r \to q, r \therefore \lnot p$