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

Question

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.

My head hurts.

Answer 1

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.

At this point, please rethink your answer to the textbook question. When you've made your mind up, hover your mouse here:

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.}

Answer 2

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.

Answer 3

I'm going to start with a short answer.

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.

Longer answer:

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.

Answer 4

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.)

Answer 5

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 ).

Answer 6

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.

Answer 7

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 following section addresses a concern raised in the comments.

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/

Answer 8

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.

Answer 9

JDisTheBest seems to have the best understanding of logic.

I do not see anyone else applying the same common sense. I say common sense because supposedly everyone commenting on the formal system is somewhat educated on the subject matter. Leaving really just intuitive judgements to differentiate between the views.

From Knowledge, Reality, and Value by Michael Huemer:

An argument would be valid when "it would be impossible (in the sense of contradictory) for all the premises to be true and the conclusion to be false."

For example:

P1. An orange is a fruit

P2. An orange is not a fruit

C. Therefore, A is not A.

Is valid.

Simply because it is not possible for the premises to be true and the conclusion be false. Of course, that is obviously because P1 and P2 cannot both be true.

When people say "if p, then q", they are simply making a statement. Not an argument. More specifically, they are making a conditional statement.