Lecture Notes 2: Truth Tables

Review of Truth-Functional Operators

So far we have seen five truth-functional operators below.

𝒜	~𝒜
T	F
F	T

𝒜	ℬ	𝒜 & ℬ
T	T	T
T	F	F
F	T	F
F	F	F

𝒜	ℬ	𝒜 ∨ ℬ
T	T	T
T	F	T
F	T	T
F	F	F

𝒜	ℬ	𝒜 → ℬ
T	T	T
T	F	F
F	T	T
F	F	T

𝒜	ℬ	𝒜 ↔ ℬ
T	T	T
T	F	F
F	T	F
F	F	T

Truth Tables For Formulas

Suppose you don’t know the truth values of P and Q.

What can you know about P → (Q → P)? A lot!

There are four possibilities for P and Q: both are true, P is true and Q is false, or vice versa, or both are false.

Writing the possibilities under the atomic statements makes a table.

(Here we repeat the same possibilities under both occurrences of “P”, since they abbreviate the same statement.)

By comparing the highlighted columns we can determine the truth value for the conditional sub-statement on the right.

We can now use the column we just calculated, along with the column under the first P to calculate the truth value of the whole.

(I’ll change the colors to indicate which columns we’re now looking at.)

Let us now highlight this final column we just filled in.

The column under the main operator (here highlighted) is very important.

It tells you the truth value of the whole statement.

In this case, it tells you that this statement cannot be false.

Tautologies and Self-Contradictions

A tautology is a statement that is true for every possible assignment of truth values to its atomic parts.

P → (Q → P) is a tautology.

A self-contradiction is a statement that is false for every possible assignment of truth values to its atomic parts.

P & ~P is a self-contradiction.

A contingent statement is a statement that is true for some (one or more) possible assignments of truth or falsity to its atomic parts, and false for others.

P ↔ (P → Q) is a contingent statement.

How to Draw a Truth Table

1. Count the number of distinct atomic statements.
2. For n atomics, we 2ⁿ rows. (Doubles with each new one.)
   • For 1 atomic, we need 2 rows.
   • For 2 atomics, we need 4 rows.
   • For 3 atomics, we need 8 rows.
   • For 4 atomics, we need 16 rows, etc.
3. For the first atomic, make the first half true, second half false.
4. For the next, do half as many trues consecutively as the previous, then the same number of Fs, and repeat.
5. Last one should alternate T, F, T, F, etc.

Suppose a statement has P, Q and R as distinct letters. You need eight rows. For P make the first four rows T, and the second four F. For Q, do two Ts, then two Fs, repeat. For R, alternate T/F.

The Process in Action

On the computer you can click a square to toggle it between T, F and blank. You can also press T or F when the mouse cursor is hovering over a square. You can highlight rows or columns with the checkboxes on the side and bottom. Highlighting is not checked as part of the answer; you may use it however you wish.

Try it yourself:

Remember:

1. Count the letters.
2. First letter: half Ts, half Fs.
3. Cut each half in half for next letter; repeat.
4. Last letter alternates Ts and Fs.
5. Rest of table, inside parentheses to out.
6. Check final column.

Ready for 4 atomics (16 rows)?

Try it yourself:

Some additional examples

Logical Equivalence Truth Tables

Logically equivalent statements are those that necessarily have the same truth value (the same for every possible truth value assignments to their atomic parts).

We can test for logical equivalence with a combined truth table for two statements. If they are equivalent, their final columns should match exactly.

Do ~(P & Q) and ~P & ~Q mean the same?

Let us make a table comparing them.

Try it yourself:

Another example.

Important: This is done like one table, not two, so P and Q are treated the same in the two statements.

Try it yourself:

These are true on precisely the same rows.

In other words, they are logically equivalent.

Some additional examples

Truth Tables for Arguments

We can also test the validity of an argument with a combined truth table for the premises and conclusion.

We do a combined table for the argument.

If there is any possibility that (even one row where) all the premises are true, and the conclusion false, the argument is invalid. Otherwise the argument is valid.

Try it yourself:

Here there are two rows—the second and fifth—where the premises are all true and the conclusion false.

However, if not a single row has all true premises and a false conclusion, the argument is valid.

This argument is valid. (We don’t know whether or not it is sound.)

A Table for a Real Argument

If there is a God (G), then God created everything in the universe (C). If God created everything in the Universe, then everything in the universe is good (E). It’s not the case that everything in the universe is good. Therefore, there is not a God.

Try it yourself:

This argument is valid. I’ll let you think about whether or not it’s sound.

Some additional examples

Some Review

Review: Truth Tables for Statements

Review: Truth Tables for Arguments