Validity and Soundness (WritePhilosophy Guide)

We saw that a philosophy paper consists of an argument for a thesis. A paper is evaluated based on whether the argument persuaded them of the truth of the thesis. We laid out two criteria for a persuasive argument:

  1. That the conclusion follows from the premises.
  2. That the premises are all true.

But what does it mean for a conclusion to “follow from” the premises? If this is going to be the central requirement of a philosophical argument, we better be more precise.

All arguments consist of a set of premises and a conclusion, all of which are declarative sentences. But there are many different logical structures for arguments. One of them is especially important to philosophers: deductive inferences. A deductive inference is a special kind of argument in which the truth of the premises guarantees the truth of the conclusion. Or to put it another way, in a deductive inference, its impossible for the conclusion to be false if the premises are all true.

We use two technical terms to assess deductive arguments: validity and soundness. We want to know whether an argument is valid and whether it is sound.


Validity

Validity is about truth-transmission. Set aside for now the question of whether the premises are true. Validity captures that unique property of deductive inference that if the premises are true, then the conclusion must be too.

We define it as follows:

Validity:

An argument is deductively valid if and only if it is impossible for all the premises to be true while the conclusion is false.

Note the term “if and only if”. We’ll encounter this in more detail later – but this term gives us a complete definition of validity. For every valid argument, its impossible for the premises to be all true while the conclusion is false. And, for every argument, if it’s impossible for the premises to all be true while the conclusion is false, it is valid.

Consider this argument:

1. Eddie Vedder is the greatest lead singer in rock music.
2. Eddie Vedder is the lead singer of Pearl Jam.

Therefore, Pearl Jam are the greatest band in rock music.

Set aside the truth of the premises – that doesn’t matter to deductive validity. We want to know if it’s possible that the conclusion is false but the premises are both true. Could there be a band who had the greatest lead singer in rock music, but that band weren’t the greatest band in rock music? Yes. It takes more than a great lead singer to make a great rock band. If Eddie Vedder goes and joins your high school rock group, they’ll be a better band than they were, but they won’t be the best band in rock. So this argument is invalid. It remains invalid even if Pearl Jam actually are the greatest band in rock music. It doesn’t matter that the conclusion is true, because its truth was not guaranteed by the truth of the premises.

Take another example:

1. All philosophers read Plato.
2. Plato is a philosopher.

Therefore, Plato reads Plato.

Is it possible for both premises to be true, while the conclusion is false? Suppose the conclusion is false, Plato doesn’t read Plato. Either Plato is a philosopher or he’s not. If he is a philosopher, then there’s a philosopher who doesn’t read Plato, so Premise 1 would be false. If he’s not a philosopher, that contradicts Premise 2, so that’s false. There’s no possible way for both premises to be true yet the conclusion false. So this argument is logically valid. It doesn’t matter whether Plato read his own work or not – it’s valid, and that’s that.

Another way to think of this is by reformulating the argument as a single conditional sentence. We take the conjunction of all the premises as the antecedent of the conditional, and the conclusion as the consequent, like so:

If all philosophers read Plato and Plato is a philosopher, then Plato reads Plato.

If that sentence is true, then the argument is deductively valid.

This works even when the content of the premises and conclusion are plainly false.

1. Mars is made of hummus.
2. Everything that is made of hummus is revolting.

Therefore, Mars is revolting.

Setting aside your possible astronomical revulsion, it’s clear that Premise 1 is false. But it doesn’t matter. This argument is valid. There’s no way Mars could not be revolting if it was true that it was made of hummus and that everything that’s hummus is revolting.

It should be obvious, then, that just because an argument is valid, doesn’t mean it’s any good! The Martian hummus argument is not a good argument and it won’t convince anyone to loathe Mars. Validity alone won’t cut it. Validity corresponds to the first of the two things that an argument needs to be persuasive: if the argument is valid, then the conclusion follows from the premises.

We know that we need to show that the conclusion follows from the premises in order for our argument to convince our reader. We know that if the argument is valid, the conclusion follows from the premises. So, if we show that our argument is valid, we’ve proven that the conclusion follows from the premises. In fact, that is, itself, a deductively valid inference:

1. If an argument is valid, then the conclusion follows from the premises.
2. My argument is valid.

Therefore, my argument’s conclusion follows from its premises.

But now we need to turn our attention to the second element: showing that every premise is true.


Soundness

When considering soundness, we bring the truthfulness of the premises back into the picture. A deductively sound argument has both of the two properties we need – the conclusion follows from the premises, and the premises are true. So:

Soundness:

An argument is deductively sound if and only if the argument is valid and all of the premises are true.

Let’s go back to our arguments from before.

1. Eddie Vedder is the greatest lead singer in rock music.
2. Eddie Vedder is the lead singer of Pearl Jam.

Therefore, Pearl Jam are the greatest band in rock music.

Is this argument deductively sound? No. Before you start asking questions about whether Eddie Vedder is the greatest lead singer in rock (he isn’t – Chris Cornell is), notice that it doesn’t matter whether the premises are true in this argument. Why? Because it is not deductively valid. Validity is a prerequisite for soundness. We can’t have an argument which is sound but invalid. So, this argument is unsound.

What about our second argument:

1. All philosophers read Plato.
2. Plato is a philosopher.

Therefore, Plato reads Plato.

We already saw that this argument is valid, so we can tick the first box in the pair of soundness tests. Plato probably read his own stuff. But that doesn’t matter: we’re interested only in the truth of the premises. Plato certainly was a philosopher, so Premise 2 is false. But there were philosophers who died before Plato was even born, so there are some philosophers who didn’t read Plato. Therefore, Premise 1 is false, and the argument is unsound.

Let’s think of a sound argument. It’s harder than you might expect. To give a clear example, we need premises that are uncontroversially true. Try this one:

1. George Orwell is Eric Blair
2. Eric Blair is dead.

Therefore, George Orwell is dead.

This argument is valid as there is no possible way George Orwell could be Eric Blair, who is dead, and yet still be alive. George Orwell indeed was Eric Blair (Orwell was his pen-name), and he died in 1950. So the argument is sound.

Crucially, when we have a sound argument, we know that the conclusion is true, even if we didn’t know it beforehand. If we didn’t know that Orwell was dead, but we did know that Eric Blair was dead and that Blair and Orwell were the same person, then this argument would allow us to discover, for sure, that Orwell was indeed dead. When the conclusion follows from the premises in a deductively valid inference, and those premises are true, then the conclusion is true. This is why our two criteria were enough for us to be satisfied that our reader would be persuaded by our argument (assuming we stated it clearly enough to be understood).

This also works the other way around. If a conclusion is false, then we know the argument must be unsound. So, if you have a valid argument with a false conclusion, then at least one of the premises must be false. This realisation underpins the philosophical method of reductio ad absurdum (of which more later, ad nauseum) in which we prove that a premise (or one amongst a set of premises, at least) must be false by showing that a false conclusion can be deduced from it. For example:

1. All birds can fly.
2. A penguin is a bird.

Therefore, a penguin can fly.

This is a logically valid argument, but the conclusion is known to be false. So, the argument cannot be sound. Because it is valid but unsound, we know at least one of the premises must be false. We can’t tell which one it is from this alone, but we at least learn that either penguins aren’t bird or not all birds can fly.

If we don’t know whether or not a premise is true, it could be tempting to say that an argument is neither sound nor unsound. This is not the case. Every argument is either sound or unsound. We might not know whether it is sound or not, but that doesn’t mean it is actually in some nether-realm betwixt the two.


Tautologies and Contradictions

Our definition of validity throws up a few anomalies. We said that an argument is valid if and only if it is impossible for the conclusion to be false while the premises are true. Let’s consider two special cases: those where it’s impossible for the conclusion to ever be false, and those where it’s impossible for the premises to ever be true.

For an argument to be invalid, we have to be able to describe a case where the premises would be true while the conclusion is false. But what if the conclusion is never false? A statement which is never false is called a tautology. Tautologies are almost always entirely obvious, redundant statements, like:

All bears are bears.
George Orwell is George Orwell.
Either Mars is made of hummus or Mars is not made of hummus

And so on. There are a few basic formulas for making tautologies: “X or not X” is one. “X is X” is another. Replace X with pretty much any statement, and you’ll get a tautology. Later, we’ll be able to define these formally. There are some slightly more interesting statements which we might agree are tautologies. These might include mathematical truths and some basic definitions. For instance, can any of the following sentences ever be false?

2 + 3 = 5
A bachelor is unmarried.
If all philosophers read Plato and Plato is a philosopher, then Plato reads Plato.

We could dispute the first two if we really wanted to (and philosophers are exactly the people who would!), but we might accept that there is no case where either are false. The third example is the conditional version of one of our valid arguments. We can now see another, more complex, way to define validity: if the conditional formed by the conjunction of the premises as antecedent and the conclusion as consequent is a tautology, then the argument is valid.

If the conclusion of your argument is itself a tautology, then your argument must be valid. A tautology can never be false, so there can never be a case where the conclusion is false while the premises are true. This is not surprising to philosophers. A tautology can be proven without any premises! Consider the follow argument:

Therefore, all bears are bears.

There is no way the conclusion can be false, so the argument is valid. There are no premises, so it cannot have any false ones. So, it is a sound argument! Any tautology can be demonstrated without any premises. Adding premises only complicates the picture. Adding some false premises would render the argument unsound, but it would remain valid. (It can be proven that adding new premises to a valid argument cannot make the argument become invalid – we’ll prove this later). We can therefore conclude that all arguments with tautologous conclusions are valid.

On the other end of the spectrum, a contradiction is a statement which is cannot ever be true. For example:

2 + 3 = 6
Mars is made of hummus and Mars is not made of hummus.
George Orwell is not George Orwell

As before, we have some simple formulae that generate contradictions, like “X is not X” and “X and not X” for any sentence X. The negation of any tautology is a contradiction. So these are all contradictions, too:

Not all bears are bears.
2 +3 5
It’s not the case that if all philosophers read Plato and Plato is a philosopher, then Plato reads Plato.

We have to be a little careful here, though, as some things which look like contradictions at face value actually aren’t. For example, the sentence: “All unicorns are red and no unicorns are red.” This looks like a contradiction because it looks like “X and not X”. But the sentence is true just when there are no unicorns – you can find me a red unicorn to disprove ‘All unicorns are red’, nor a non-red one to disprove ‘no unicorns are red’, so the sentence is true and thus not a contradiction. Be wary of universal generalisations (‘All Xs are Ys’): a sentence like ‘All unicorns are red’ is trivially true if there are no unicorns.

Now, suppose we have an argument which has a contradiction amongst its premises. For instance:

1. All bears are cuddly.
2. Not all bears are bears.

Therefore, Santa Claus is coming to town.

“Not all bears are bears” is a contradiction, so can never be true. This means we know the argument is unsound, as the premises are not true. But the argument must be valid! This is because the premises can never all be true while the conclusion is false, since the premises can never all be true. Again, philosophers are not particularly surprised by this consequence of our definition of validity. We say that from a contradiction, anything follows (or if you like a bit of Latin, ex falso quodlibet). This is sometimes excitingly named the Principle of Explosion. Once there’s a contradiction amongst my premises, literally any conclusion (every conclusion!) validly follows.

This also works without a single contradictory premise, if two or more premises contradict one another. We’ll say that a group of premises are consistent if and only if it is possible for them all to be true together. If a group of premises are inconsistent, then the Principle of Explosion applies, and any conclusion will validly follow. For example, from this group of inconsistent premises, any conclusion can be validly deduced:

1. Anthea is taller than Benedict.
2. Benedict is taller than Carla.
3. Carla is taller than Anthea.

That any conclusion validly follows from a contradiction serves as a vivid reminder that it’s not enough to have a deductively valid inference – we need sound inferences.


Test your understanding

Analysing a set of premises and a conclusion to determine validity is a skill which all philosophers must acquire. You can try out the WritePhilosophy logical validity quiz to test your understanding.

Latest edit: 02/03/2021 by CJ Blunt