# Could an argument with false Premises and a true Conclusion be logically valid?

Am unsure (leaning more towards no): true P and true C is valid, false P and false C is valid, True P and false C is invalid, how about false P and true C?

P= Premises; C= Conclusion.

People who says.. Yes, it is valid, I would like to hear some answers for

How can we derive the truth out of falsehood?

If we can, then True P - False C is valid too (tell me why it is invalid?).

• Yes; an argument is not valid when we have TRUE premises and FALSE conclusion. Aug 8, 2019 at 12:52
• No what I asked, I am interested in false P and true C. Aug 8, 2019 at 13:15
• You must understand valid does NOT mean TRUE or indicate TRUTH in reality. Mathematicans typically rant that logic is about validity. Old school philosophy professors ranted about the SOUNDNESS of an argument--not about just VALIDITY. So YES you can have a valid argument with false premises & a true conclusion. Does this mean all valid arguments apply to reality? No. There is a difference between soundness & validity. A sound argument must be valid & must also have true premises that apply to reality.Valid arguments alone dont have this impact. So shoot for sound arguments not just validity. Aug 8, 2019 at 14:36
• "You must understand valid does NOT mean TRUE", I understood that, an argument can be valid or invalid.. I didn't talk about TRUE in any place, not to mention the premises can be true or false. Aug 8, 2019 at 15:03
• The point I was making is that the strongest arguments one can make are SOUND arguments secondly, many people think that if an argument is valid then they have won an argument. This is false.. You can use less reliable types of argument if you desire but why would you choose that? Why would someone in reality argue using false premises? This is a sign the person doesn't know what they are doing. Rational arguments have true premises to begin with in reality. Do not confuse reality arguments with mathematical arguments designed for only a classroom. Aug 8, 2019 at 15:32

Yes, an argument with false premises and a true conclusion can be valid. For example:

All cats are human

Socrates is a cat

Therefore, Socrates is human

The argument has false premises and a true conclusion. But the argument is valid since it's impossible for the premises to be true and the conclusion false. In other words, if the premises are true the conclusion is guaranteed to be true, which is how validity is defined.

• Great example, thank you ! +1 Aug 9, 2019 at 9:17
• "it's impossible for the premises to be true and the conclusion false" ....I can say the same thing to your example, it is impossible to derive the truth out of falsehood. so that's why I am not happy about saying: false P and true C as valid. Aug 9, 2019 at 11:51
• @Explorer_N An argument is valid if there is no way for the premises to be true and the conclusion false at the same time. That’s the standard definition of validity. In my example imagine that the premises are true: in a world where they’re true the conclusion is also true. That’s validity.
– E...
Aug 9, 2019 at 13:06
• @RaGa__M "it is impossible to derive the truth out of falsehood." No, it's not. The conclusion of an argument does not need to contain all the information of the premises: we can "lose strength" along the way (think about proving a special case from a general result). The point is that losing strength can only make something more true. May 7, 2021 at 18:53

A valid argument guarantees that the conclusion shall be true whenever all premises are true.

This guarantee is broken only when the conclusion may be false when all premises are true.

So a valid argument does allow for a case where the conclusion is true while some (or all) of the premises are false.   Its guarantee is not broken by that.

Could an argument with false Premises and a true Conclusion be logically valid?

Validity is assessed on form only. Whether the premises are actually true or actually false is irrelevant.

For example,

Donald Trump is a martian;

All martians are Presidents of the United States of America;

Therefore, Donald Trump is President of the United States of America.

Valid argument, false premises, true conclusion. QED.

Note 1 on logical validity

The truth of the conclusion is not derived from the truth of the premises since the premises are (presumably) false. And it is also clearly not derived from the falsehood of the premises.

The truth of the conclusion is derived from the form of the argument, and by assuming that the premises are true.

If you understand the argument, then you should be certain, once you assume the premises, that the conclusion is true.

There is nothing else to it.

Aristotle didn't provide any more details as to how we arrive at the certainty that the argument is valid. And, so far, nobody else did, even though many great thinkers since Aristotle pondered the issue.

Note 2 on logical validity

Many logicians accept as valid arguments which are not formally valid. For example:

Everyone is female.

So, any siblings are sisters.

This argument will be accepted on the semantic ground that, first, the definition of the word "sister" in English makes any sister female by definition and, second, the definition of the word "sibling" in English makes any sibling either male or female.

However, semantic is a murky issue and admitting validity on semantic grounds can only lead to endless debates about the meaning of the words used in the argument which are not logical terms (i.e. not "or", "imply", etc.).

Further, any definition accepted as relevant to justify validity on semantic ground, is de facto an assumption, i.e. an implicit premise.

Whenever an argument is admitted as valid on semantic ground, it should be possible to make it formally valid by making explicit all relevant definitions by incorporating them as additional premises of the argument.

Thus, the argument above could be made formally valid by making it "formal", as follows:

For all x, Brother(x) implies not Female(x);

For all y, Sibling(y) implies either Sister(y) or Brother(y);

For all z, Female(z);

Therefore, for any a, Sibling(a) implies Sister(a).

Here, we can ignore the semantic of the non-logical terms. The validity of the argument is now entirely a function of the form of the argument.

To qualify an informal argument as valid, without any qualification, is therefore seriously misleading. An informal argument is valid to you only because you admit, if only implicitly, all relevant definitions.

Try to get anyone who doesn't know the definition of the English words used in the argument to agree that the argument is valid! Good luck. And would you yourself sign a document written in any language you don't understand on being told that the document is valid?

All formally valid arguments are also informally valid. However, informally valid arguments are not necessarily formally valid.

Thus, it is never misleading to use the word "valid" to refer to formally valid arguments, but it is misleading to use it to refer to informal arguments. When talking about the validity of informal arguments, we should use the expression "informally valid".

• In a proof by contradiction p is not a premise. If it was, that would mean you have a valid argument with p as a premise and ~p as a conclusion. But any such argument is of course invalid.
– E...
Aug 9, 2019 at 2:49
• @Eliran Alright, changed. I hope you're gonna like this one. Aug 9, 2019 at 8:32
• @Speakpigeon that's another great example, thank you ! +1 Aug 9, 2019 at 9:22
• How it becomes valid? no one is answering that part. you can't derive the truth out of falsehood. Aug 9, 2019 at 11:52
• For reference, this article (plato.stanford.edu/entries/logical-consequence) mentioned a whole bunch of people apart from Aristotle who have detailed "how we arrive at the certainty that the argument is valid."
– jhch
Aug 9, 2019 at 20:46

I will answer your question first by talking about the definition of 'validity' (which I think is necessary to consider very precisely) and then explaining the reasoning behind this definition.

As already mentioned here, validity is a property not of a concrete (single) argument, but rather of the form of an argument.

The argument

(1) If it is raining, the street is wet. (2) It is raining. Conclusion: The street is wet.

has the following form: (1) If p, then q. (2) p. Conclusion: q.

Now the definition of validity says: An argument form is valid if and only if it is not possible that all premises are true and the conclusion is false.

That means, in order to prove that an argument form is valid, we have to prove that whenever we insert true propositions for its variables (here in the example: p and q), the conclusion must be guaranteed to be true.

I will show the general steps for this prove before considering your question regarding false premises. (Possibly you are already familiar with these steps.)

In the easiest case (like ours), a truth table can be made. A truth table shows all possible combinations of the truth values of the premises of our argument form.

The truth table for the above argument form would look as follows:

``````p  |   q   |   p -> q (1. premise) |   p (2. premise)   |  q (conclusion)
-------------------------------------------------------------------------
F      F         T                     F                   F
F      T         T                     F                   T
T      F         F                     T                   F
T      T         T                     T                   T
``````

(The '->' is the common symbol for 'if... then' in logic; an explanation of how the truth value for this logical operator is calculated can be found here.)

Since the definition of validity only talks about the case of true premises, all other lines of the truth table can be completely ignored. In this simple example, it is therefore only the last line that is relevant. Since the last line of the truth table yields a true conclusion, we know that this argument form is valid.

Before we have this proof, we can make no inference regarding validity: An argument with false premises could either be an instance of a valid argument form as well as of an invalid one.

One last illustration: If we again take our simple argument form above, we could really construct an instance with false premises and a true conclusion:

Let 'p' stand for 'You are a cat.', and q for 'You are a human.' (inspired by the answer by Eliran). The our concrete argument would look as follows: (1) If you are a cat, you are a human. (2) You are a cat. conclusion: You are a human.

Both premises are false, and the conclusion is true. From the reasoning above we know that the argument form of which this argument is an instance is valid.

The reasoning behind the concept of validity:

Maybe these illustrations can also make clear why 'validity' is defined precisely in this way. Validity can (metaphorically spoken) be seen as a quality criterion of argument forms. Argument forms can be seen as the logical construct that lies behind a concrete argument. If only we insert true premises into this construct, we are guaranteed to come up with a true conclusion. If this were not the case, we would have made a logical mistake. Correct logical reasoning guarantees that truth is preserved! If we however insert false premises... Well, since the logical construct is still the same, we have not made a logical mistake. We have rather made some other mistake (we have false beliefs regarding reality etc.) So our argument, from the logical point of view, still deserves this quality criterion. That's what validity is about.

Of course, validity is not everything. We indeed want to have true premises. That's why there is also the notion of 'soundness', as also already mentioned here. A sound argument is a valid argument + true premises. That is, a sound argument does not only involve correct logical reasoning but more: E. g. correct beliefs about our world.

• so if I apply the same argument form: if you are a human, you are a cat....you are a human... does this argument valid? @Andreas Schütz Aug 9, 2019 at 12:52
• @Explorer_N: I am not sure if I understand correctly, but: Yes, no matter which propositions you insert into a valid argument form, the validity won‘t be affected. And sure we might call an argument that has a valid argument form a ‚valid argument‘. Aug 9, 2019 at 15:06
• @Explorer_N: I think I see now what you mean. Yes, the following argument with the same argument form as above is still valid: (1) If you are a human, you are a cat. (2) You are a human. conclusion: You are a cat. In this case, we have one false, one true premise, and a false conclusion. Aug 10, 2019 at 8:23
• you are a act is a conclusion not premises. "you are a cat" is a consequent which I am using as a conclusion, you are a human is a true premises. Aug 10, 2019 at 13:41

Trivially, the argument could be:

1. If it is raining and I go outside with my umbrella, I will not get wet.
2. It is raining.

Conclusion:

• If I go outside with my umbrella, I will not get wet.

This argument is perfectly valid. The fact that both premises are wrong: umbrellas do not keep you dry on windy days, and it is not currently raining, does not change the fact that the valid argument led to a true conclusion.

• The condition given by the OP directly were that all the premises were FALSE, While the conclusion was TRUE. You seem to use an example of possibly one false premise. This does not meet the criteria posed by the original question. Aug 9, 2019 at 0:27
• @Logikal "The condition given by the OP directly were that all the premises were FALSE" Uhm... there's exactly one revision of edit... that just adds the epistemology tag. The title literally reads "an argument with false Premises". The body and multiple comments just seem to contain the phrases "false P" and "true P" multiple times. Collectively taken, the only way I can fathom your deriving that the OP is giving the condition that all premises were false would be if you're overinterpreting the plurality in "Premises" in the title. Aug 9, 2019 at 4:44
• No I am Not taking a literal sentence reading. The definition of an argument in philosophy requires there to be at least two premises followed by a conclusion. Perhaps you are a math person & don't realize some terminology does NOT carry over between Mathematical logic & Philosophy. That means all of the premises must be false with the conditions set by the OP while the argument is still valid with a true conclusion. It is typical math people turn everyt original premise into a conditional. In reality this method is not always true & can mislead you. Aug 9, 2019 at 9:20
• ...and if we want true conclusions: 1. Socrates is a chicken. 2. All chickens are mortal. C: Socrates is mortal. ...has one false premise. 1. Socrates is a cow. 2. All cows have two legs. C: Socrates has two legs. ...has all false premises. Both arguments here are valid; neither is sound. There's nothing special here about all premises being false versus some of them being false. Aug 9, 2019 at 14:01
• As to ill-formed modus ponens? This is not modus ponens. It is a different argument form: premises "(A and B) implies C" and "A" leads to the conclusion "B implies C". That is a valid argument. Aug 9, 2019 at 14:41

The Material Conditional doesn’t always cleanly match our expectations of a relation of inference. This is well enough known that there is a Wikipedia article on Paradoxes of Material Implication.

One particular concern is the thought that assuming this as a blanket rule potentially allows not just valid arguments but definitionally Sound arguments of arbitrary statements as long as the conclusions are in fact true.

Intuitively, we may wish to demand that truth on its own is not quite enough on which to hang implication. This is a common theme in Relevance Logic, which is a topic that might interest you.

• You did not define sound argument correctly. You might be a math person which is why you think the way you think, but you should know that math terminology and philosophy can have different context; I know for sure they are not identical. You can tell a tree by its fruit with the use of some terms for alleged topic “LOGIC”. Aug 9, 2019 at 0:14
• @PaulRoss We would need to have some example to make clear what you mean by "definitionally Sound arguments". I never heard of this expression and it draws a blank on a search engine. Also, Relevance logic I'm sure is interesting but the notion that validity is really a relation between premises and conclusion isn't specific to it and in fact dates back to Aristotle. Are we not in danger of forgetting the basics if the debate turns around material implication v. relevance logic? Aug 9, 2019 at 8:51

Yes.

The definition of validity is extremely narrow. From IEP:

A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

The definition does not mention what happens when the premise is logically false (that is, a contradiction). If the premise is logically false, the argument is trivially valid (more precisely, we call this "vacuously valid" but trivial works as well).

This is because if the premise is a contradiction, then it can never be true, and so it is impossible for the premise to be true while the conclusion is false.

• You link to the IEP article, which is fine, but the comment you then make after the IEP quote concerning false premises is not supported by the article, and this makes your answer misleading. Aug 9, 2019 at 9:01
• Fair concern, I edited to make more clear.
– jhch
Aug 9, 2019 at 14:27
• @JohnHugues The problem I had in mind is still there. Your comment suggests that the IEP's article doesn't mention contradictory premises "because if the premise is logically false, the argument is trivially valid". This falsely suggests that the IEP article substantiates the notion that logical arguments may be "trivially valid". Aug 9, 2019 at 18:50
• Well... it does substantiate that notion, in that 1) it defines validity and 2) arguments may be trivially valid. I'll expand further to make myself more clear.
– jhch
Aug 9, 2019 at 19:21
• The IEP article talk about validity. It doesn't talk about arguments being "trivially", or "vacuously" valid. It doesn't even touch upon contradictory premises. You are simply assuming too much. Aug 9, 2019 at 20:11

This truth functional situation of false premises making a conditional true is my favorite example of how formal languages do track natural languages but with different aesthetics. It seems the aesthetics of all languages formal and natural in the end may be the same and thus imply there may be foundational knowledge there somewhere.

The concept of material implication is universally known in human reasoning: “if Socrates is human, then he is mortal” written logically as P⊃Q. Though it is universally used by all, as your question brings out, there are serious intuitive problems with such reasoning such as its truth conditions. If the antecedent P is false or if both antecedent and consequent are false then the compound implication in its entirety is true. “If life exists on the moon, then life exists on earth” and “If the moon is made of green cheese, then life exists on other planets” are both considered valid reasoning by most of modern logic because it is impossible for their truth conditions to be: the antecedent is true and the consequence is false. Seems absurd in our natural language but it makes perfect sense and the rationality is easier to see written aesthetically in natural language: “If congress passes serious immigration reform, then I am a monkey’s uncle”. This statement expresses what I consider to be a true proposition: “congress will not pass serious immigration reform”. Yet, I express it in the form of a false antecedent that by being false is taken even in natural language to prove anything; also, both antecedent and consequent are false but I use this conditional to state a truth.

Surprisedly, this is as common in mathematical reasoning as it is in natural language. The statement “if n is a perfect square, then n is not a prime number” is true throughout. But, “if 3 is a perfect square, then 3 is not prime” is also a true conditional though both its antecedent and consequent are false. So, as much as I hate to admit it, the nonsensical aesthetics of non-analytical or continental philosophy writers such as Sontag, Foucault, Derrida et al may actually contain something with which I agree but is simply not worth the effort of getting to it. Aesthetics trumps intuition and perhaps much more.

• Regarding your "if 3 is a perfect square, then 3 is not prime", a math deductive system won't proceed like this which is like a sophist's technique to assume some thing as an axiom or self-claimed given condition. Since we also know 3 is not a perfect square (can be proved easily from Piano Arithmetic), then you arrive at a contradictory from 2 statements, per principle of explosion of classical logic or intuitionistic logic (non-paraconsistent logic) you can get any conclusion u want. Same for natural language, seems no aesthetics involved just sophistry using false premise to confuse people. May 8, 2021 at 0:24
• Calling a valid argument "sophistry" because you do not understand satire is the worst kind of sophistry. I used the statement “if n is a perfect square, ...” and inserted "3" to state “if 3 is a perfect square, ...” as examples of why false premises in a conditional have a truth value of "T". This is why I chose "3", because it leads to a contradiction. That's the point. This is called "satire". The antecedent in “If congress passes serious immigration reform, ...” is also false and leads to contradiction. That is the aesthetics involved: using a contradiction to state a truth. Jul 10, 2021 at 20:28
• Thx for you feedback! I'm not the one who downvoted your answer and I'm not currently active on PSE... I remember my comment above re your answer was based on your last conclusion. This is a purely classic logic very basic Q/A, I don't see how aesthetics or continental philosophy involved and arrived at your conclusion in the last paragraph. I fully understand material implication and closely related logic JTB knowledge. Jul 11, 2021 at 0:13
• ... My sohphistry comment above is not about your answer, it's about the way used in above contrived example. It's just "irrelevant" non-indicative" material condition, that's why we also need other logics such as relevance logic, indicative condition, or Lewis' counterfactual conditions... They're applicable for different use cases. Aesthetics and especially continental style philosophy may also employ such logics if they want, and it's moot from this purely very basic logic question... Jul 11, 2021 at 0:13
• Though logic, as much of mathematics, has become an end-in-itself game, this is not its foundation. It is supposed to be descriptive of how to preserve truth. Natural language employs sarcasm to preserve truth. A formal language must be able to do so or is no improvement on natural language. Truth-functional logic can be sarcastic through its acceptance that false premises can state a truth. The false premises are not irrelevant or however you dismiss them. My "If 3 ..." example is as valid as a statement using the Ideal Gas Law (a false premise, there is no such thing as an ideal gas). Jul 13, 2021 at 17:30