# Identification of conflcting consequents within a set of logical statements

Let's say I have a set of three logical statements:

1. If it is sunny AND it is the weekend then I play football (S * W -> F)
2. If it is NOT sunny OR it is NOT the weekend then I do NOT play football (!S + !W -> !F)
3. If it is NOT sunny AND it is NOT the weekend then I play football (!S * !W -> F)

I can show that the antecedents are complete, i.e. at least one antecedent is true for all combinations in a truth table. I can show that there is at least one combination that is not unique, i.e. more than one antecedent is true for the same combination in a truth table.

You may notice that in the case of Statement 2 the consequent is 'I do NOT play football when S=0 and W=0' which conflicts with the consequent of Statement 3 which is I do play football.

Whilst grounded in predicate or propositional logic is it possible to test whether the consequent of a set of logical statements are conflicting with one another or not?

Thanks

• There is Robinson's resolution algorithm that decides in finitely many steps whether a set of given propositional formulas is satisfiable (non-contradictory) or not. There are algorithms for some special cases of predicate formulas also, like those including 1-place predicates only, but no such algorithm for general predicate formulas. Commented Jan 5, 2023 at 21:47
• In its generality you're dealing with the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) which is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. SAT is the first problem that was proved to be NP-complete by Cook and Levin, while heuristic SAT-algorithms are able to solve problem instances involving tens of thousands of variables and formulas consisting of millions of symbols... Commented Jan 9, 2023 at 1:18

A piece of terminology first. The words 'premise' and 'conclusion' apply to arguments, not sentences. What you have are three sentences in conditional form. The left side of a conditional is the antecedent and the right side is the consequent.

Your sentences are consistent, in the sense that there are some valuations under which all of them are true. If you construct a truth table for them, they hold true under the following three circumstances: It is sunny and the weekend and I play football; it is not sunny and the weekend and I don't play football; it is sunny and not the weekend and I don't play football.

However, if it is not sunny and not the weekend then there is no way for all the sentences to be true, whether I play football or not. This would have the consequence that if the sentences were all true, then there are no rainy weekdays. This would be strange because we might reasonably take it for granted that all combinations of sunny/not-sunny and weekend/not-weekend days are possible.

So your question becomes: if we include the constraint that all combinations of the truth values of S and W are possible, how do we detect that the sentences are inconsistent for some valuation of S and W? You could look at the truth table for the conjunction of all three sentences and find rows which come out false for a given valuation for S and W, irrespective of F.

Or you could take the conjunction of the three sentences and conjoin this with each in turn of the four possibilities: S ∧ W, S ∧ ¬W, ¬S ∧ W, ¬S ∧ ¬W. In the last case, you will get inconsistency, i.e. a value false in every row.

• Thanks for clarifying the terminology, I'll update my OP to reflect that. My second statement is actually 'it is not sunny OR the weekend and I don't play football', not 'it is not sunny AND the weekend and I don't play football'. When you say 'take the conjunction of the three sentences' does that apply to the whole statement or just the antecedent of the statement? Commented Jan 8, 2023 at 19:50
• You need to take the conjunction of all the sentences, and then conjoin each of the various combinations of S ∧ W, S ∧ ¬W, ¬S ∧ W, ¬S ∧ ¬W in turn. The result will show that ¬S ∧ ¬W is inconsistent with your given sentences. Commented Jan 9, 2023 at 16:01

is it possible to test whether the conclusions of a set of logical statements are conflicting with one another or not?

In mathematical logic, there is no known general algorithm to decide that a set of statements are consistent with each other or not.

We can sometimes find the answer using our native logical intuition, but there is no deterministic procedure.

This is the equivalent of finding the exit of a labyrinth without a map.

Theoretical problems may be simple enough for the answer to be intuitively evident or easy enough. Real life conundrums may not be.

• I know of a test for a contradiction which says that if all combinations of a truth table provide an output of False then the statement is therefore a contradiction, take P * !P as an example. Commented Jan 8, 2023 at 19:42
• @cptnemo Even if that we true, this does not answer the question. Commented Jan 9, 2023 at 16:19