Proposition: tautology, contradiction, truth tables

Question

Classify each of the following propositions as a tautology, a contradiction, or neither. Note that if you claim that a proposition is a tautology, then you must argue( by using truth tables or otherwise) that it is true for every assignment of truth values to the propositional variables; if you claim that it is false for every assignment of truth values to the propositional variables; and if you claim that it is neither a tautology nor a contradiction, then you must find an assignment of truth values to the propositional variables that makes it true and another assigns that makes it false.

A- P → ¬P

B- P → P

C- ( P ∧ Q) → ( P ∨ Q)

D- ¬P ∨ ( P → Q)

E- P ∧ ¬( P ∨ Q)

F- ( P ∨ Q) → P

G- ( P ∨ Q) ∧ ( ¬P ∨ ¬ Q)

H- ( P ∧ Q) ∨ ( ¬P ∨ ¬Q) I know how to do the truth tables but I don't understand what they're saying you must argue.

Answer 1

If you know how to make a truth-table, great: you're almost there!

For every statement that you work out on a truth-table, there are three possible outcomes:

1. The statement is True in all rows. This means that the statement is a tautology

2. The statement is False in all rows. This means that the statement is a contradiction

3. The statement is True in at least one row, and False in at least one other row. Then the statement is a contingency. And you can look at the reference columns on the left to see under what conditions it is True, and under what conditions it is False.

I'll just do the first one as an example:

\begin{array}{|c|c|} \hline P & P \to \neg P \\ \hline T & F\\ F & T\\ \hline \end{array}

We are dealing with case 3 here, so this is a contingency. The statement is True when $P$ is False, and the statement is False when $P$ is True.