# Logic Question: Can a logically valid argument have a logically false conclusion? [duplicate]

Are there any examples of such an argument? What would its premises be like? How could such an argument be possible?

Where an argument is logically valid "if and only if it is not possible for all the premises to e true and the conclusion false."

And logically false is defined as "if and only if it is not possible for the sentence to be true." (e.g. 'June will pass Chemistry 101 and she will not pass Chemistry 101').

1. Premise: all horses are brown
2. Premise: X is a horse
3. Conclusion: from premise 1 and 2 follows that X is brown

However, suppose that X in fact isn't brown, but white (meaning that either X isn't a horse - premise 2 is false -, or not all horses are brown - premise 1 is false).

The reasoning leading to the conclusion is logically valid, it's a valid reasoning, but the conclusion is not true, because we started with false premises.

• But surely the conclusion is not logically false. After all it's logically possible that X is brown. So you haven't given a valid argument, whose conclusion is logically false. – sequitur Jan 10 '15 at 16:42
• @sequitur a formula (e.g. a conclusion) is said to be logically false when it's not true under every possible interpretation. A reasoning however is valid when the conclusion follows from the premises. Don't mix them up. See en.wikipedia.org/wiki/Validity – user2953 Jan 10 '15 at 16:55
• I'm quite clear about the difference. The point is that the OP wanted someone to provide a logically valid argument whose conclusion is logically false. Your argument is valid (in any structure where the premises get a designated value the conclusion get such a value) but the conclusion is not logically false (since its negation is satisfiable). By the way being non-true in every structure is no adequate general definition of logical falsity, since it presupposes bivalence and not all logics having logical falsities are bivalent. – sequitur Jan 11 '15 at 17:10
• My bad, @sequitur. I didn't see OP edited his question. In the first version I interpreted it as whether the conclusion could be logically invalid, not "false in every possible interpretation" which is the current version. – user2953 Jan 11 '15 at 17:19

Any argument with necessarily false premises is valid, regardless of the conclusion. Therefore as long at least one of your premises is always false, you can have a false conclusion and still have a valid argument.

This is a counterintuitive fact. However, we need to remember that the only thing logical validity guarantees us is that we will NEVER have a situation where the premises are true and the conclusion is false. Outside of that single guarantee, validity tells us nothing.

This may not seem useful, but in fact, it is very useful because it preserves the quality of truth (where truth is as defined within the system of logic).

This depends on what kind of logic we're talking about and on how logical consequence is spelled out. For simplicity let's settle the second issue by assuming that logical consequence amounts to model preservation in every structure. Different logics will differ on the notion of structure and the notion of a structure being a model of a sentence.

Now, the case of interest indeed arises in classical logic as well as in intuitonistic logic, since both logics are explosive: From a contradiction every sentence follows and so every logically false sentence follows as well. For a trivial example consider (A & ~A) ⊨ (A & ~A), for some sentence A.

In a formal system, if you can prove a statement is valid, it is valid for that formal system (yay for tautologies!)

However, part of the formal system is the axioms one is assuming. If you disagree with the axioms, you can disagree with the conclusion.

Consider:

• Assume: All cats are human is a human.
• Assume: Pixel is a cat
• Conclusion: Pixel is a human.

The logic is sound, but the first assumption is suspect.

• Validity applies to arguments not statements as normally defined. – virmaior Jan 9 '15 at 4:42

Yes, you can have a logically valid argument with a false conclusion. An example is below.

1. Every US president is a white male.
2. Barack Obama is a US president.
3. Therefore, Barack Obama is a white male.

Valid argument because the conclusion follows logically from the premises. Premise 1 is false, however, so the argument is said to be valid and unsound.