Unit 3: Proving Invalidity (Shorter Truth-Table Method)

Shorter Truth-Table Method (or Indirect Method)

The full truth-table method is effective but very slow, especially with 3 or more variables (8+ rows). The Shorter Truth-Table Method is a much faster way to check for validity.

It is also known as the "Indirect Method" or "Reductio ad Absurdum" (reduction to absurdity). It works by trying to *prove* the argument is invalid. If you succeed, it's invalid. If you fail (by finding a contradiction), it must be valid.

The Core Strategy:

1. Assume the argument is INVALID.
2. To do this, we try to find a "counterexample row." We assign FALSE (F) to the conclusion and TRUE (T) to all the premises.
3. Work backward from these assigned values to find the truth values of the simple variables (p, q, r).
4. Check for Consistency:
   • If you can assign consistent values (e.g., 'p' is always T, 'q' is always F) that make all premises T and the conclusion F, your assumption was correct. The argument is INVALID.
   • If you are forced into a contradiction (e.g., 'p' must be T in one premise but F in another), your assumption was impossible. The argument is VALID.

Step-by-Step Example 1 (Proving Invalidity)

Argument: "If I study (S), I will pass (P). I passed (P). Therefore, I studied (S)."

Symbolized: (S ⊃ P), P / ∴ S

Step 1: Assume Invalid.

We set up the argument and assign T to premises and F to the conclusion.

(S ⊃ P)   /   P   //   ∴ S
    T      /   T   //   ∴ F

Step 2: Assign values from the "easiest" parts.

The easiest parts are the conclusion (S) and Premise 2 (P), which are just simple variables.

• From the conclusion: S = F
• From Premise 2: P = T

Step 3: Check for consistency.

Now we plug these values (S=F, P=T) into Premise 1 to see if it can still be T.

(S ⊃ P)
(F ⊃ T)
    T

This works! A (F ⊃ T) statement is TRUE.

Step 4: Conclusion.

We successfully found a consistent assignment of truth values (S=F, P=T) that makes both premises TRUE and the conclusion FALSE.
Therefore, the argument is INVALID.

Step-by-Step Example 2 (Proving Validity)

Argument: "If it's raining (R), the ground is wet (W). The ground is not wet (~W). Therefore, it is not raining (~R)."

Symbolized: (R ⊃ W), ~W / ∴ ~R

Step 1: Assume Invalid.

Assign T to premises and F to the conclusion.

(R ⊃ W)   /   ~W   //   ∴ ~R
    T      /   T   //   ∴ F

Step 2: Assign values from the conclusion and simple premises.

• From the conclusion: If ~R is F, then R = T.
• From Premise 2: If ~W is T, then W = F.

Step 3: Check for consistency.

Now we plug these values (R=T, W=F) into Premise 1 to see if it can be T.

(R ⊃ W)
(T ⊃ F)
    F

Step 4: Conclusion (Reductio ad Absurdum).

Here we have a CONTRADICTION!

Our assumption in Step 1 requires Premise 1 to be T.
But the values derived from the other premises and conclusion force Premise 1 to be F.
It is impossible to make all premises T and the conclusion F at the same time. The initial assumption was absurd.

Therefore, the argument is VALID.