Practice Problems: Truth Tables
The following are some practice problems on truth tables; i.e., they cover Part III of forall x: Calgary. For each problem, you can use the "Check" button to see if you have filled in the truth table correctly. You will get a message that says "Success!" if so, and otherwise one that says "Something's not quite right!".
Building Complete Truth Tables
Fill in the complete truth table for each of the following sentences. These all have just one logical operator, so this is just testing if you remember the characteristic truth tables for the connectives of TFL.
Validity of arguments
An argument is valid in TFL if every valuation (row of the complete truth table) makes one of the premises false or makes the conclusion true. A counterexample is a valuation that shows that an argument is not valid, i.e., a valuation where the premises are all true and the conclusion is false. Here are two example truth tables to test your understanding of this concept. First, fill in the table to compute the truth value of each sentence on each valuation. The conclusion is the sentence on the far right; it is separated from the premises by a ⊢ symbol, which basically here means the same as the therefore symbol ∴ in forall x. Underneath the ⊢ symbol, select ✓ if the valuation is "good" (makes a premise false or the conclusion true), and select ✗ if the valuation is "bad" (makes all premises true and the conclusion false). If the argument is invalid, you should indicate this by clicking the "Invalid!" button, and entering a combination of truth values that shows that the argument is invalid (i.e., the T/F combination underneath G and H in a line which you think shows that the argument is invalid. You should see a "Success!" message when you've picked a truth value combination which actually shows that the argument is invalid, and "something's not quite right" if you've not given a valuation that shows the argument invalid if it is in fact invalid.
Here is a complex truth table to test for the validity of an argument. We'll fill in the truth values of the sentence letters for you in this one.
Equivalence in TFL
Two sentences of TFL are equivalent if they agree in truth value on every valuation. To test for equivalence, compute the truth values of both sentences, and then compare them row-wise. If their truth values match in every valuation, the two sentences are quivalent; if there is a valuation in which they have different (opposite) truth values, they are not. If you think the are not equivalent, click the "Inequivalent!" button and enter the truth-value combination of the valuation that shows that they are not.
You should have managed to figure out that the two sentences in III.9 are not equivalent, but the two sentences in III.10 are. The latter equivalence is one of De Morgan's Laws.
Some sentences of TFL are such that they end up being true whichever truth values the sentence letters have. We call these tautologies. For instance, P ∨ ¬P is true on every valuation. Give truth tables and determine if the following sentences are tautologies. If they aren't, click the "non-tautology" button and enter the valuation that shows they are not tautologies.
There are interesting connections between valid arguments and tautologies. For instance, modus ponens is the argument P, P \to Q \therefore Q$. Now do the truth table for the "corresponding conditional", (P ∧ (P → Q)) → Q:
You'll notice this is a tautology. This is no accident. Can you explain why?
Here's another example, requiring a larger truth table.
For some applications, it is important to know if there is a way some sentences can all be made true at the same time. We say in such a case that the sentences are jointly satisfiable. If you want to find all such ways, you have to do a complete truth table. You should click the "consistent" button and enter a valuation that makes all sentences true, if they are jointly satisfiable.
However, if all you want is a single valuation on which all sentences are true, it is enough to construct a partial truth table (see Chapter 13). For instance, to find a valuation on which the sentences of the previous exercise are all true, complete the following partial truth table: