Joint Satisfiability: Why Three Possible Premises Can Fail Together

Require one valuation for the entire premise set

Premises are jointly satisfiable when at least one valuation makes all of them true together. Separate valuations showing that each premise can be true do not establish this. Even a different successful valuation for every pair need not work for the whole set.

Our constructed classical example has three premises: P∨Q, ¬P and ¬Q. Each premise is satisfiable by itself. Every pair is satisfiable too. Nevertheless, the complete triple is inconsistent: no assignment makes all three true.

Formal background: [1].

This matters when interpreting an argument with inconsistent premises. Under classical validity, such premises cannot provide a row with all premises true and the conclusion false. That makes the inference valid in the semantic sense; it does not make an arbitrary conclusion a true statement about the world.

Distinguish individual, pairwise and joint questions

The first two questions permit the witness valuation to vary as you move to another premise or pair. The third requires a single common witness. The fourth adds a further constraint on the conclusion. These are different jobs, so keep their outputs separate.

In the triple here, ¬P asks for P=F and ¬Q asks for Q=F. Those two values make P∨Q false. There is already a direct conflict between the conjunction of the requirements, even though no printed premise is itself an outright contradiction.

Check the whole triple in all four rows

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

Download all rows and the pairwise witness records

In the first row, both negated premises fail. In the next two rows, one negated premise fails. In the last row, both negations are true but the disjunction fails. Thus every possible P,Q assignment fails at least one premise.

The table is complete because there are two distinct letters and all four combinations appear exactly once. Its final column is a check of the whole set, not another premise secretly added to it. You can equivalently inspect the column for (P∨Q)∧¬P∧¬Q: it is always F.

No extra sentence letter can repair this inconsistency. Adding R=T or R=F leaves the existing P,Q contradiction unchanged. It simply duplicates each failed row with two R values.

Show why pairwise success does not solve the triple

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

The witnesses are deliberately different. Taking Q=T from the first row and Q=F from the third would not create a stronger common witness; it would assign incompatible values to Q. Similarly, taking each pair’s successful result without its assignment hides the incompatibility.

This triple is minimally inconsistent: removing any one premise leaves a satisfiable pair. “Minimal” here describes removal of premises from this particular set. It is not a claim about shortest notation, the importance of a premise, or every inconsistent set.

For a larger set, a failed pair is enough to refute joint satisfiability, because all members of the full set would include that pair. Successful pairs, however, do not settle the whole-set question. The three-premise counterexample already disproves that proposed shortcut.

Understand validity with inconsistent premises

Premises: P ∨ Q, ¬P, ¬Q
Proposed conclusion: R

To refute this inference semantically, we would need a valuation with all three premises T and R=F. No such valuation exists because the premise requirements already conflict. The inference is therefore classically valid regardless of R’s assigned value.

Do not infer that R is a tautology. R is false on every valuation assigning it F, and true on those assigning it T. The absence of a true-premise countervaluation is a property of this argument, not a change to R’s own truth table.

The argument cannot be sound under one classical valuation: its premises cannot all be true together. If the letters formalize real claims and you accept them all, revisiting those claims or their formalization is the useful next step. Formal validity does not certify a contradictory factual account.

Formal background: [2].

The cited proof package also has explosion, which allows a conclusion after a contradiction has been derived. That classical rule is not shared by every nonclassical logic. This page is confined to the stated two-valued semantic setting; do not transfer its result to a different logic without checking that logic’s rules.

Compare a consistent subset without erasing the original result

Remove ¬Q and keep P∨Q with ¬P. The witness P=F,Q=T makes both remaining premises true. They also entail Q: once P is false, the disjunction can be true only through Q. This is a satisfiable, informative inference rather than the inconsistent triple.

Deleting a premise changes the question. It can demonstrate which requirements conflict, but it does not show that the original set was satisfiable. For factual reasoning, the choice of which claim to revise needs evidence about those claims; the truth table alone cannot tell you which report is reliable.

A practical audit records the whole set, the common-witness search, any smaller inconsistent subset, and what happens when a specific premise is removed. Keep the original premises visible so that “repair” does not become an unnoticed substitution of a different argument.

Try six common-witness checks

A satisfiability answer needs the appropriate scope of witness.