Partial Truth Tables: Find a Countervaluation and Repair a Failed Choice

Make every premise true and the exact conclusion false

A countervaluation is one complete assignment that makes all premises true together and the conclusion false. Start by imposing those targets, then use the truth rules to force letter values or split genuine alternatives. Reject a contradictory branch, but keep searching any remaining branches.

This is a method for classical truth-functional arguments, with material implication and inclusive disjunction. The examples are constructed, not reports about real people or events. A failed attempt at a countervaluation does not by itself prove validity; it may only show that your particular choice was wrong.

Formal background: [2] [1].

We will inspect the inference from P→(Q∨R) and P to Q. A validity worksheet explains what a counterexample is. This guide shows how to record forced values, recognize a choice, backtrack when it fails, and check the successful assignment.

Write the target before making convenient choices

Premise 1: P → (Q ∨ R)   target T
Premise 2: P             target T
Conclusion: Q           target F

Premise 2 forces P=T. The conclusion target forces Q=F. Since the first premise must be T and its antecedent P is T, its consequent Q∨R must be T. With Q already F, this forces R=T.

You could start with a tentative R value before propagating, but a forced-value-first record avoids needless guessing. It also makes clear which entries follow from the target and which were choices. Never silently erase a forced assignment when a later requirement conflicts with it.

Repair a failed trial without changing the task

On a narrow screen, scroll inside this table. Keyboard users can focus the region and use the arrow keys.

In the trial, Q∨R is F∨F=F. Since P=T, the conditional is T→F=F. The trial gives the desired false conclusion but sacrifices a premise, so it is not a countervaluation.

The correction changes only the tentative R choice. P=T and Q=F remain fixed by the original targets. Now Q∨R is F∨T=T, the conditional is T→T=T, and premise P is T. Conclusion Q stays F. The inference is therefore invalid.

Download the complete target, rejection and successful assignment

This is a particularly small search: after propagation there is actually no remaining choice for R. The failed trial illustrates what happens if you guess too early. In larger formulas, several alternatives may remain after all currently forced values are entered.

Separate a forced assignment from a genuine alternative

A disjunction A∨B required T allows A=T, or B=T, or both. A conditional A→B required T allows A=F, or B=T, or both. These requirements do not always force a unique letter value. Record a branch when other constraints do not decide the alternative.

By contrast, A∨B required F forces A=F and B=F. A→B required F forces A=T and B=F. A conjunction required T forces both conjuncts T. A conjunction required F leaves alternatives until other constraints settle which conjunct fails.

On a narrow screen, scroll inside this table. Keyboard users can focus the region and use the arrow keys.

If you require the same letter both T and F on one branch, that branch closes. Separate branches can assign different values to a letter because they represent different candidate valuations. Combining their favorable entries into one supposed answer would lose the requirement of a single coherent assignment.

For a search with a genuine remaining choice, use the single premise (P∨Q)∧¬(P∧Q) and conclusion R. A countervaluation needs the premise T and R=F. Propagation requires P∨Q=T and P∧Q=F, but does not yet decide P or Q. Choose P=T as one branch; then Q must be F. The alternative branch P=F forces Q=T. Both completed branches work.

On a narrow screen, scroll inside this table. Keyboard users can focus the region and use the arrow keys.

If a P=T branch tentatively sets Q=T, reject that Q choice because P∧Q becomes true. Backtrack to Q=F while retaining P=T and R=F. Unlike the first example, P itself was not initially forced. To refute validity, either successful branch suffices; you do not need both. To claim a search has no success, every remaining alternative must be accounted for.

See what makes a complete no-countervaluation argument

Premises: P → (Q ∨ R), P, ¬Q
Conclusion: R

For the modified inference, require R=F. Premise ¬Q=T forces Q=F, and premise P=T forces P=T. To keep P→(Q∨R) true, Q∨R must be true. Yet the forced Q=F and R=F make it false. The targets are incompatible.

Here the rejection is decisive because the relevant values are all forced by the target; there is no unexplored alternative that could rescue them. It is not the unsupported claim “I tried one row and failed”. The argument is valid under these classical truth rules.

If a larger search branches, every alternative must be ruled out before this method establishes that no countervaluation exists. You can instead use a complete truth table or a valid formal proof. A partial search is useful only when you can account for its remaining possibilities.

Formal background: [2].

Adding ¬Q also changes the conclusion from Q to R in this comparison. Keep each argument’s premise list and conclusion visible. A correct valuation for one inference does not automatically apply to a similar-looking inference.

Report the assignment and all truth checks

When the search succeeds, supply the full assignment of every distinct letter and evaluate the original formulas. “Set Q false” is not a completed countervaluation: it leaves the other letters and the premises unchecked. If a letter is genuinely unconstrained, choose an explicit value and show that the completed assignment still works.

One verified countervaluation refutes validity. It does not show that every assignment makes the conclusion false, nor that the premises are actually true in the world. Validity concerns the impossibility of true premises with a false conclusion; soundness adds premise truth to a valid argument.

A compact search record should preserve four things: original targets, forced entries, rejected choices with reasons, and the final whole-formula check. These make a result auditable and help locate the first wrong step when your search stalls.

Try six countervaluation decisions

Keep the same assignment across all formulas in each argument.