1
Only some rows count
A row can refute an argument only if every premise is true in it. Rows in which a premise is false tell you nothing about the argument, because the argument promises nothing there.
Lesson · Arguments and Validity · Propositional logic
Validity by Truth Table
An argument is valid when no row of its table makes every premise true and the conclusion false. The lab finds such rows, and names four forms on sight: two valid, modus ponens and modus tollens, and two invalid, affirming the consequent and denying the antecedent.
invalid: a row with premises T, conclusion F
p → q, p ∴ q modus ponens: valid
p → q, ¬q ∴ ¬p modus tollens: valid
p → q, q ∴ p affirming the consequent
p → q, ¬p ∴ ¬q denying the antecedent
One test, four familiar forms
Core ideas
What this lesson establishes
The test is the same for every argument over a few simple sentences: write every row, and look at the rows where all premises hold.
2
One row is enough
A single counterexample row makes the argument invalid. Validity needs every row examined; invalidity needs one.
3
Validity belongs to the form
Replace p and q with any sentences you like and the verdict does not change. That is why the four forms have names.
The material
The counterexample row
A definition from the last two lessons, put to work on four arguments.
A counterexample is a row, and only a row. It is an assignment of truth values to the simple sentences under which every premise comes out true and the conclusion comes out false; nothing about the world is being claimed, only that such a case is describable. So an argument is valid when no such row exists, and invalid when one does, and the second verdict needs exactly one row to establish it. For p → q, q ∴ p the lab finds the row p = F, q = T: the premises hold, the conclusion fails, and no argument about the other three rows can rescue the form — which is why “if God exists life has meaning; life has meaning; so God exists” is settled by a table and not by theology.
p q | p → q p q
--------------------------
T T | T T T premises T, conclusion T
T F | F T F a premise is F
F T | T F T a premise is F
F F | T F F a premise is F
That is modus ponens: premises p → q and p, conclusion q. The columns shown are the two premises and the conclusion. Only the first row has every premise true, and there the conclusion is true. The other rows have a false premise and are irrelevant. No counterexample, so valid.
p q | p → q q p
--------------------------
T T | T T T no counterexample
T F | F F T a premise is F
F T | T T F COUNTEREXAMPLE
F F | T F F a premise is F
That is the same table for affirming the consequent, p → q and q, concluding p. Two rows have both premises true, and in the second of them the conclusion is false. One row is enough.
Theorem · The four forms
Modus ponens (p → q, p, so q) and modus tollens (p → q, ¬q, so ¬p) are valid. Affirming the consequent (p → q, q, so p) and denying the antecedent (p → q, ¬p, so ¬q) are invalid, and each is refuted by the row p = F, q = T.
The pair that look alike are the pair that differ. Modus tollens denies the consequent and concludes against the antecedent; denying the antecedent starts from the wrong end. If you cannot tell which you are looking at, build the table.
In the lab, choose each of the four presets. It lists the rows, marks any counterexample row, and names the form from a catalogue. The name is looked up from the shape of the formulas, not from what the sentences mean, so an argument with the same shape and different subject matter is named the same way.
Interactive
Validity by cases
Every row, every counterexample, and the form
Method
Testing an argument by table
Five steps; the fourth is the one that does the work.
1
Translate the argument
Give each simple sentence a letter, write each premise and the conclusion as a formula, and keep the same letters throughout.
2
List every row
n letters give 2ⁿ rows. List them in a fixed order, and add a column for each premise and for the conclusion.
3
Fill the columns
Evaluate each premise and the conclusion in every row, from the connective definitions.
4
Look only at rows where every premise is true
In each such row, read the conclusion. A row where it is false is a counterexample; the argument is invalid and you can stop.
5
If no row survives, the argument is valid
Every row where the premises hold has a true conclusion, or no row has the premises all true. Both count as valid.
Counterexample rows
Score 0 / 0Choose the answer you can justify.
Life has meaning
Let p be “God exists” and q be “life has meaning”. If God exists, life has meaning; life has meaning; so God exists.
premises: p → q , q conclusion: p
row p q | p → q q | p
1 T T | T T | T
2 T F | F F | T
3 F T | T T | F counterexample
4 F F | T F | F
Row 3 makes both premises true and the conclusion false: there is no God, and life has meaning anyway. The argument is invalid, affirming the consequent. This does not show that God does not exist, or that the premises are false; it shows that these premises do not settle the conclusion.
Common mistakes
1
Counting a row with false premises and a false conclusion as a counterexampleA counterexample needs every premise true and the conclusion false. In the table for affirming the consequent the row p false, q false has a false conclusion, but its second premise q is false too, so the argument promises nothing there. Only the row p false, q true refutes it.
2
Treating the lab's name for a form as the verdictThe name is looked up from the shape of the formulas. The verdict is computed from the rows. They agree, and if they ever seemed not to, trust the rows: the table is the definition.
3
Concluding the conclusion is false because the argument is invalidA counterexample row shows that the premises do not guarantee the conclusion. The conclusion may be true anyway. “God exists” is not refuted by a bad argument for it, any more than it is established by a good one.
Check your understanding
✓
Finish when you can produce the row, not just the verdict.
Given an argument over two or three letters, write its table, mark the rows where every premise is true, name the counterexample row if there is one, and name the form if it is one of the four.
Note: A table has 2ⁿ rows, so this method is practical for a handful of sentences and hopeless for a hundred. Later lessons keep the number of letters small, and the lab refuses more than six.