# In formal logic, how is it possible for an argument with a contradictory conclusion to be valid?

I've recently been trying to introduce myself to formal logic, and I've hit a stumbling point:

In the text I'm reading, a valid argument is defined as "an argument in which it is impossible for the premises to be true and at the same time the conclusion false."

The text then later prompts me to declare whether it's possible for an argument to be valid with a contradiction for a conclusion. The answer key then claims that yes, it is in fact possible, but offers no further explanation. I'm also supposed to create an example.

I just can't wrap my head around this one.

Here's the example I've worked up, is this a valid argument?

• If grass is green then it is raining.
• If water is wet then it is not raining.
• Therefore: It is raining and not raining.

• Just brainstorming: A contradiction within the logic has a side effect of "proving" everything, so it is usually considered "bad." However, there are several forms of arguments where one proves something by showing that its negation forms a contradiction (I.e. "assume pepsi and coke are the same. If you arrive at a contradiction, then you know there must have been a flaw in your premise, thus proving pepsi and coke are different"). Your text may be aluding to this sort of technique. Apr 7 '15 at 16:21
• Hey I appreciate you jumping to my assistance! Unfortunately, I'm finding it difficult to consider that the author intended me to be pondering "techniques" in any form because this is a part of the first problem set of the first chapter of an introductory text. Here is the problem set if it helps: Which of the following is possible? If it is possible, give an example. If it is not possible, explain why. 1. A valid argument that has one false premise and one true premise 2. A valid argument that has a false conclusion 3. A valid argument, the conclusion of which is a contradiction Apr 7 '15 at 16:25
• Would it be valid if I created an argument in which the premises can never be true? For example, making all the premises contradictions? Example: Bats are mammals and are not mammals. Bats exist and don't exist. Therefore bats exist and don't exist. Apr 7 '15 at 16:30
• Yes. A common phrasing of that concept is, "valid argument, but faulty premise." Apr 7 '15 at 16:35
• Blackwood, you might be using a bad textbook, for it seems that the author has a reductio ad absurdum in mind without explaining it properly: If you assume that the premises are true and then arrive at a contradiction, then you have shown that the premises are false. That's correct, because in classical logic from "p implies q" it follows that "not-q implies not-p", hence form "p implies q" and "not q" you can conclude "not p". Since the premises are false, the definition is still correct, but the author should have explained that better... Apr 9 '15 at 11:17

An argument is valid if and only if the truth of its premises entails the truth of its conclusion and each step, sub-argument, or logical operation in the argument is valid. – Wikipedia

This definition only talks about concrete arguments1 with true premises. It basically does not tell you anything of concrete arguments with false premises. From this follows that the definition does not forbid a valid argument (with false premises) to have a false conclusion.

Validity of arguments is best checked with abstraction. Consider for example2:

P → Q.
P.
∴ Q.

This argument is valid; it follows from the Modus Ponens. If the premises are true, we know for sure that the conclusion is true (this is the definition of validity of an argument; see above).

However, now consider:

If it's raining, I eat the cat.
It's raining.
∴ I eat the cat.

This argument is valid since it has the same form as the abstract argument above. However, as it turns out, it's not raining (the second premise is false). Moreover, I am not eating a cat at this moment. Hence, the conclusion is false while the argument is valid3.

Validity is often confused with soundness:

An argument is sound if and only if

1. The argument is valid.
2. All of its premises are true.

Your argument is not a valid argument. The conclusion does not logically follow from the premises. For example, if grass is not green and water is not wet, then this does not tells us anything about whether it's raining or not. However, if you also have as premises that grass is green and water is wet, then, yes.

1: By concrete argument I mean an argument with concrete premises, i.e. without variables. So some 'P' can only be part of a concrete argument if we know what is meant by 'P'. I will use abstract argument for an argument with variables.
2: The symbol ∴ means 'therefore' and is used to indicate the conclusion.
3: Whether or not the first premise is true is out of the scope of this answer.

Another way of thinking about it is this: A valid argument is "an argument in which it is impossible for the premises to be true and at the same time the conclusion false." If it's impossible for the premises to be true, then it's clearly also impossible for "the premises to be true and at the same time the conclusion false." So, an argument with premises that can't be true is valid.

• What you say is correct, although I think you don't realise the implications of your last sentence, "So, an argument with premises that can't be true is valid.". From this would follow that any argument with false premises is valid.
– user2953
Apr 7 '15 at 18:38
• @Keelan Any argument with contradictory premises would be valid. My understanding is that this is in fact the case, even if there seems to be no connection between premises and conclusion, because a contradiction implies everything. Apr 7 '15 at 18:46
• I see what you mean, but I'm afraid that's not the case. See for example this or my answer above, but also your definition: "an argument in which it is impossible for the premises to be true and at the same time the conclusion false." - that the premises are not true is irrelevant for the validity of the argument: if the premises being true entails truth of the conclusion, the argument is valid. This entailing is not an implication, but a logical consequence (for which it doesn't hold that everything follows from a contradiction)
– user2953
Apr 7 '15 at 18:50
• @Keelan This certainly falls within the definition in the question: if the premise is a contradiction (not necessarily if it's just false), it's impossible for the premise to be true and the conclusion false. The definition in the question is not "premises are taken to the conclusion by a generally valid structure," it's merely that it is impossible for all premises to be true and the conclusion false (and if premises are contradictory, this is the case under any definition of "impossible"). The argument "P and ~P therefore Q" is valid under this definition. Apr 7 '15 at 19:17

If your conclusion is a contradiction then your argument can only be valid if the truth of the conclusion is entailed by the truth of the premises.

An argument is invalid if it takes a form where the premises are true whilst the conclusion is false.

Take Modus Ponens as an example:

``````If p then q
p
Therefore q
``````

Is it possible to construct an argument which follows the above logical form, but has true premises whilst the conclusion is false?

If it is possible, then Modus Ponens becomes an invalid form of argument. If it isn't possible, then Modus Ponens is a valid argument form.

Take the following argument:

``````P: X is a square

C: X is a rectangle
``````

If the premise were true, then the conclusion would be true and the argument would be valid.

But what if our premise was actually false?

For example, what if X was actually a triangle?

If `X is a triangle` is true, then the conclusion is also false. We would therefore have a false premise which leads to a false conclusion.

We would need the premise to be true and the conclusion to be false for the argument to be invalid. But, this can't happen. If X is a triangle, then the conclusion is false. If X is a square, then the conclusion is true. Either way, we are stuck in a situation where the truth of the premise entails the truth of the conclusion. The argument is therefore valid.

Let's define two terms, of which only one term is directly relevant to the question at hand:

Tautology: something that is true in all possible worlds.

Contradiction : something that is false in all possible worlds.

So, if we have a conclusion that is false in all possible worlds, the argument would only be valid if we have premises that entail the truth of the conclusion if the premises were true. Here is an example (Modus Ponens):

If the grass is green then I will be happy and not happy.

The grass is green.

Therefore, I will be happy and not happy.

This conclusion is clearly contradictory. The argument is valid however, since it follows a valid form of reasoning (MP). If the premises were true, the conclusion would be true. Remember that validity isn't concerned with the actual truth of the premises.

Here's the example I've worked up, is this a valid argument?

If grass is green then it is raining. If water is wet then it is not raining. Therefore: It is raining and not raining.

The general principle here is that if an assumption leads to a contradiction, then that assumption is false.

Define the following propositions:

G = Grass is green

R = It is raining

W = Water is wet

Suppose [G => R] & [W => ~R] & G & W.

We can then infer R & ~R obtaining a contradiction as you suggest. A valid argument so far, but we can continue.

We can conclude that the initial assumption is false: ~[[G => R] & [W => ~R] & G & W]

(Can also be verified with a truth table.)

• The premise: "If grass is green, then it is raining" = G -> R. In the OP's argument, neither G nor W are supposed as stated. Apr 8 '15 at 11:37
• Unstated assumptions, it is true: Grass is green and water is wet. Apr 8 '15 at 13:54
• In formal logic, you don't get unstated assumptions. If you don't state it, you don't get it. Apr 8 '15 at 22:27
• I like to be helpful. Sometimes, in these postings, you have read between the lines to be helpful. Sometimes, you have to exercise some judgement. It seemed obvious to me that the OP was assuming that grass is green and water is wet. It was the only way he could have gotten the contradiction, "It is raining and not raining." Apr 9 '15 at 2:43
• I'm sorry for the terseness of my comments. I agree that to get there you need to posit G & W . What I think would make your answer much better is to explain that how the OP did it is missing a vital and important step, viz., the adding of these two premises before you can even get to the contradiction. The biggest errors students make in formal logic is jumping past important minutae like failing to state all of their premises or leaving out important bits of definitions. Consequently I don't think this is a good answer for this question.. Apr 9 '15 at 9:40

Sure, just use modus ponens:

If water is liquid, then fire is a substance and fire is not a substance. Water is liquid. Therefore, fire is a substance and fire is not a substance.

This argument has the right form, so it is valid, but has a contradictory conclusion ( feel free to pick your own contradiction ).

I also struggled with this..it’s from the open source logic textbook forallx..after contemplating it

An argument is invalid if

• it can have premises that are all true
• and a conclusion that is false

A contradiction is any sentence that must be logically false. Eg

It is raining AND it is not raining

This sentence is made up of 2 sentences

A: it is raining

B: it is not raining

One has to be false.

Since a contradiction has to be made up by at least one false premise, it can’t be made up of premises that are all true. Therefore it can’t be invalid, so it must be a valid argument.

I would appreciate wiser minds commenting on this.

• I like forallx. Good reference. Welcome to this SE! Nov 16 '18 at 14:54
• Thank you :) I’m using it as a companion text to Steve Gimbels formal logic lecture series on The Great Courses Nov 17 '18 at 6:06

The OP has been given the following challenge:

The text then later prompts me to declare whether it's possible for an argument to be valid with a contradiction for a conclusion. The answer key then claims that yes, it is in fact possible, but offers no further explanation. I'm also supposed to create an example.

Here is an example using Klement's proof checker.

Note that the conclusion, "P∧¬P", is a contradiction but the inference rule, conjunction introduction (∧I) is used correctly. Since the argument from the premises is valid, this would be a simple example of a valid argument with a contradiction as a conclusion.

The OP also provides an example:

Here's the example I've worked up, is this a valid argument?

• If grass is green then it is raining.

• If water is wet then it is not raining.

• Therefore: It is raining and not raining.

Since "grass is green" is true and "water is wet" is true, but it can be either raining or not raining, one of the conditionals in the two premises is false. One might symbolize this as follows. Let "G" be "grass is green". Let "W" be "water is wet". Let "R" be "It is raining". Since "G" is true and "W" is true, we have the following argument:

So this example would also work as a valid argument with a contradiction as a conclusion. To make this argument clearer, one could add to the premises two assumptions: (1) "grass is green" and (2) "water is wet".

The proofs (derivations) above are considered "syntactically valid" according to forallx, page 147. If I used truth tables I could also show they are "semantically valid". Since the truth-functional proof system is "sound" (see Chapter 20 of forallx), these derivations of arguments are not invalid by truth tables.

To illustrate this using the second derivation, the following truth table shows that this sentence, "((G ∧ (W ∧ ((G → R) ∧ (W → ¬R)))) → (R ∧ ¬R))", is a tautology, that is, it is true for all valuations of the atomic sentences, "G", "R" and "W". Note that the premises of the derivation above are conjuncts of the antecedent of this conditional sentence.

Here is a truth table generated by the Stanford Truth Table tool:

References

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

Stanford Truth Table Tool, http://web.stanford.edu/class/cs103/tools/truth-table-tool/

• A small point regarding your first example: you’ve shown that the argument is provable, not that it’s valid. These are different notions. Of course, given a soundness theorem, if it’s provable it’s also valid, but that’s a further and non-trivial step. Nov 16 '18 at 18:06
• @Eliran I suspect the correct term for the deductions is to say they are "syntactically valid*. If I used a truth table, I would be able to show they are also "semantically valid". In forallx, page 147, they write about soundness: "A proof system is sound if there are no derivations of arguments that can be shown invalid by truth tables." I agree there is a distinction between the two approaches, using derivations or using truth tables. Nov 16 '18 at 18:38
• Even so, it's not just a matter of two different approaches; there is a notion of syntactic validity and a notion of semantic validity, and these are different notions. It seems to me that the OP is referring to the latter, given their wording. Nov 16 '18 at 18:43
• @Eliran Just to make sure, I edited my answer to include a truth table. This would show that the OP's example would likely work assuming "G" and "W" are true. Nov 16 '18 at 19:14

I can think of an example that shows the case:

(P ^ not P). Therefore not (P => P).

The conclusion is always false (not (P => P)), therefore a contradiction.

And if the premises were true (which they are not, but that is ok - we are checking for validity, not soundness), the conclusion had to be true.

If you replace the second P for Q in both places and check the truth table you would see that they are logically equivalent and if P=1 and Q=0 shows that the premises are true and the conclusion has to be true, therefore a valid argument.

New contributor
Anderson Goulart is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.