# Validity stemming from contradictory premisses

I'm a student new to the study of logic, and having had my first tutorial on it yesterday, while I generally understand the characterisation of logical validity, there were a couple of examples my tutor used which I can't quite get my head around.

The argument:

Birds can fly
Birds cannot fly
Therefore there are no birds

Is apparently valid, while:

Fish can swim
Fish cannot swim
Therefore there are no animals

Is invalid.

My textbook (Halbach's Logic Manual) states that an argument is "logically valid if and only if there is no interpretation under which the premisses are all true and the conclusion is false," but I have no idea how the conclusions in both of these situations is related to the premisses, and then how one is logically valid and the other isn't.

My tutor said that it was something to do with the first conclusion specifying about birds, but I still don't really get where he's coming from. Any help explaining the two arguments would be much appreciated.

• I would assume the tutor is wrong or there's a miscommunication here before anything else. From how I'm reading the arguments they are both valid. – Casey May 1 '15 at 17:34
• @Casey: You are coming at this propositionally, where (A and not A) => False => AnythingYouWant. But (A and not A) is not always false! It is equally true that all unicorns are white and that no unicorn is white, in a universe without unicorns. – user9166 May 2 '15 at 15:38
• @jobermark yeah I see where I went wrong now. – Casey May 2 '15 at 20:38

There are animals other than fish.

You want "no interpretation", "no X does Y" is easiest to address by contradiction. So let us imagine an interpretation as a counterexample.

We only need one counterexample, to contradict "no interpretation".

It has to be one with animals. So let's choose the simplest easy interpretation with animals: one animal -- me. So the conclusion is false.

But fish are animals, too, and I am not a fish. So we have ruled out fish entirely.

Then both premises can be true. The nonexistent, necessarily swimming fish still don't swim.

• This is what my tutor was trying to get at, but again, I still don't understand your explanation. Why does that contradict it? – user14497 May 2 '15 at 16:32
• Because a world where I am the only animal is one where the conclusion is false, but the premises are all true. – user9166 May 2 '15 at 19:24

The simplest form of modern symbolic logic is propositional logic. This is what I would expect to start with for a beginning logic student. If we translate these sentences into propositional logic, both have the form:

Premise: A
Premise: NOT A
Conclusion: THEREFORE B

Since the premises are contradictory, they can never be both true, so any conclusion of any truth value can be inserted in for "B" and the argument will still be valid (by the definition in your text).

To be charitable to your tutor, let us assume he is starting with First Order Logic, which is more complex then propositional logic. In first order logic, more of the fine detail of the arguments is preserved, and we would expect more of a translation like this.

Premise: For ALL BIRDs, CanFly(BIRD) and NOT CanFly(BIRD)
Conclusion: NOT EXISTS BIRD

And the other argument would be

Premise: For ALL FISH, CanSwim(FISH) and NOT CanSwim(FISH)
Conclusion: NOT EXISTS ANIMAL

If we made them more general, we would have:

P: For ALL X, f(X) and NOT f(X)
C: NOT EXISTS X

And

P: For ALL X, f(X) and NOT f(X)
C: NOT EXISTS Y

As you can see, these two arguments no longer have the same form. They also have to be evaluated for validity in a more complex manner involving a domain of discourse --see here for more.

The biggest question is why your tutor is skipping propositional logic and going straight to FOL, which seems like an approach guaranteed to confuse a beginner.

• This does explain why these two different forms could have different meanings. It does not indicate why they do. We also don't know anything got skipped. If they have just added quantification, and therefore changed the definition of valid, it could be just as confusing. – user9166 May 3 '15 at 15:21

I think I see what your tutor is getting at.

The trick here is the work being done by "interpretations" in your working theory of validity. With an interpretation, we're building up a model according to which a particular statement or set of statements might be true. In formal logic we are often very mathematically precise about what sorts of things count as models (we usually need to give an account in terms of algebra or set theory), but we can usually speak informally about conceivable pictures of how things might be.

Consider a conceivable model in which all birds can fly. Well, one reasonably like our own, where we have a number of bird species (pigeons, eagles etc.) capable of flight. There is another conceivable model in which all birds cannot fly - all of the possible species of bird are ones that don't fly (penguins, Ostriches etc.).

This suggests one way of looking at a conceivable model in which both of those things are the case - look for the intersection of the list of species satisfying both criteria. Since the two criteria are in this case exclusive, though, the only models are those in which the set of bird species is just empty; i.e. there are no birds. In this model, it's absolutely the case that all birds can fly, and that all birds can't fly, because there aren't any birds!

What this points to is a curious feature about the "interpretations" theory of validity, which demonstrates that a number of things that look like they would be deductively contradictory turn out to not necessarily be so. On the face of it, you would think that "fish can swim" and "fish cannot swim" would look like they're in logical contrast with each other. But actually, in constructing a model in which there are lots of other animals but unfortunately no fish, we demonstrate logically coherent possibilities in which both of the premises of the second argument can be deemed admissable, while the conclusion would turn out simply false.

• One thing to note carefully: while I think you've got the right answer, it's also important to note that there is another interpretation that both arguments are not self consistent. If, instead of forming sentences as CanFly(x) and not CanFly(x), the sentences become BirdsCanFly and not BirdsCanFly, the latter is a logical inconsistency that allows absolutely any conclusion to be drawn using logical rules (even unrelated ones). From that perspective, one could argue that the argument is "valid, but not consistent" – Cort Ammon May 1 '15 at 20:14
• Oh, sure, there can be no model satisfying simultaneously the conditions that "everything that is a bird can fly" and "there exists something that is a bird and that cannot fly", for instance. Thus, according to the definition of validity, of course, the inference would be valid, since every interpretation of the premises (i.e. all zero of them) is also an interpretation of the conclusion. =) – Paul Ross May 1 '15 at 21:45
• @CortAmmon Extra definitions of 'sound', 'consistent', 'valid' are not going to help. The question is whether A is a fact, or a formula. If A is a fact, then A or not A is false. If A has variables within it, it is simply not true that A or not A is always false. Whatever notation and vocabulary you choose, the point is that the definition intended by the English quantifies over a set, and that makes it not have a fixed truth value. The empty collection of facts implies true, not false. – user9166 May 3 '15 at 15:17
• @jobermark "The definition intended by the English..." is often not sufficient, because English phrases are often ambiguous. This is why first order logic is studied using its own grammar, rather than English. The version I used appears in fallacious arguments quite often, where one sneaks a contradiction into the logic which allows the speaker to logically claim any assumption. It's a limitation/perk of English. If the textbook was kind enough to convert the English into more traditional first order logic grammar, it would be clear which of the interpretations was intended. – Cort Ammon May 3 '15 at 17:25
• @CortAmmon The faith that notation actually helps is baseless. It only helps once you understand the concepts in your own way. That generally means learning the hidden implications and omitted details of your own language. The ambiguous (and pointlessly numerous) ways English marks quantification are something that anyone whose basic representation of formulas is in English already will have to learn, in order to use real quantification with ordinary notation well, anyway. So this is not help. – user9166 May 3 '15 at 17:40

In the birds argument, the premises establish that 'bird' is a contradictory predicate in your model. So no entity can satisfy the predicate. Therefore the interpretation will assign the empty set to that predicate. So there is no interpretation on which both the premises can be true and the conclusion false.

In the fish argument, the premises establish that the predicate 'fish' is a contradictory predicate in your model. So no entity can satisfy the predicate. Therefore the interpretation will assign the empty set to that predicate. But for all the premises establish, there can be an interpretation in which 'fish' is empty but 'animals' has members. So there is a model in which the premises are true and the conclusion false. So the argument is invalid.

The set of premises in each argument are actually consistent. A sentence like "birds can fly" reads "for all x, if x is a bird, then x can fly." Equivalently this reads, "either x isn't a bird, or x can fly." "Birds cannot fly" reads "there doesn't exist some x such that x is a bird and x can fly." Now the only interpretations for which both premises are true in the first argument is when nothing is a bird. Since this is also what the conclusion asserts it will be true in those interpretations. This leaves us with no interpretation where every premise is true and conclusion false, hence it's a valid argument. The second argument, however, has interpretations where the premises are true and conclusion false. The structure of the premises is the same as in the first argument. The only interpretations where both premises were true in the first argument is when there were no birds; similarly the only interpretations for which is the case in the second argument is when there are no fish. The difference here is that the conclusion here doesn't assert that there are no fish, but rather that there are no animals. But there are interpretations where something isn't a fish, but where that something is an animal. Hence there is an interpretation where every premise is true yet the conclusion is false, making the argument invalid.

So I imagine you thought the premises were inconsistent hence you could derive anything from them. But they're not. In the first argument the premises are linked to the conclusion in a way which makes it valid. In the second argument that link is missing.