Is formal logic unsuitable for philosophical reasoning?

Question

I'm taking a course at university about philosophical reasoning / argumentation. The professor came up with an example where formal logic was wrong:

1. If Dave is in London, then he is in England. (p: D. is in London, q: D. is in England; p -> q)

2. If Dave is in Paris, then he is in France. (r: D. is in Paris, s: D. is in France; r -> s)

Furthermore the following tautology:

3. ((if p then q) and (if r then s)) then ((if p then s) or (if r then q))

He argues that this allows for the following sentence to be true:

Therefore Dave is in England, when he is in Paris, or he is in France, when he is in London.

From this he follows, that formal logic should not be used for arguments (since the conclusion is wrong).

While I accept the example argument I do not believe that the conclusion is wrong. I do agree on the validity of the conclusion regarding formal logic - the tautology used definitely is a tautology and thus the conclusion is correct according to the premises.

I believe the problem is that our understanding of the conclusion states that it should be wrong, since we know something about London, England, Paris, and France. But it isn't. As to my understanding, the premises simply lack the information that someone is not in England when he is in Paris and not in France, when he is in London.

Thus I do not accept my professors critique on formal logic as suitable for philosophical reasoning.

Is formal logic really unsuitable for philosophical reasoning?

I'm asking since this argument undermines my conviction of formal logic being the one reliable tool for reasoning!

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For those of you who don't believe that (3) is a tautology: It is a tautology given that any configuration for p, q, r, and s will yield True.

For those of you who are into programming, here is a Haskell Program that checks the claim:

class Stmt a where
    isTautology :: a -> Bool
instance Stmt Bool where
    isTautology = id
instance (Enum a, Bounded a, Stmt b) => Stmt (a -> b) where
    isTautology f = all (isTautology . f) [minBound .. maxBound]
a ==> b = not (a && not b)
claim p q r s = ( ( p ==> q ) && ( r ==> s ) ) ==> ( ( p ==> s ) || ( r ==> q ) )
checkClaim = isTautology claim

Answer 1

It's a tautology, but the mapping between logic and the English-language description is wrong.

(p ==> s) || (r ==> q)

is equivalent to

(!s || p || !q || r)

which is:

Either we're in London, or we're in Paris, or we're not in England, or we're not in France.

Quite so--in fact, for geography, we could drop the whole London and Paris thing, and indeed--we would either be not in England or not in France at any given time.

So what went wrong?

The problem is the description

either when in London I'm in France, or when in Paris I'm in England

is that it makes it sound like either one of these rules is a true rule quantified over all locations, or the other rule is a true rule quantified over all locations. That's not what's going on at all. Only the whole statement is quantified over all locations--for any location you pick, at least one of these things is true.

So, how do we manage that? For example, suppose we're not in Paris. We choose the "when in Paris" rule; the premise is false, so the rule tells us nothing. Now, suppose we enter Paris. We choose the "when in London" rule; again the premise is false, so the rule tells us nothing.

In particular, what we cannot say is that

∀x(P(x)||Q(x)) => ∀xP(x) || ∀xQ(x)

which is what was sneakily implied by the English language phrasing.

Logic works fine on argumentation. Just don't trip yourself up with moving your existential quantifiers around and it's all good.

Answer 2

There are a few issues conflated here.

First, let's take the example given. The proposition fails, because the "tautology" listed is not actually valid.

((if p then q) and (if r then s)) then ((if p then s) or (if r then q))

...is not true. (If p then q) and (if r then s) does not in any way allow us to claim that one of (if p then s) or (if r then q) is true. So, we're not dealing with valid formal reasoning here, but fallacious reasoning.

Now, as to the larger issue of whether or not formal logic is appropriate for arguments. The short answer is: yes, it should be used, as far as it can be-- but the meat of most arguments lies in the definitions, and the implications to be drawn. It is rare that you will see a philosophical argument fall due to an error in formal reasoning; much more likely, it is attacked because of some unquestioned assumption implied in the structure of the claims.

Finally, it is dangerous to think that formal logic is the "one reliable tool for reasoning", because it raises the question: by what means do you reliably know that formal logic is reliable? Logic is (famously) not self-grounding; there is no logical proof that the axioms of logic are valid. For a nice (and canonical) example, see Lewis Carroll's What the Tortoise Said to Achilles.

EDIT:

I just want to clarify that I know that the "tautology" listed above is, indeed, tautologous according to the truth table. However, in order to be able to apply the formal logic encased within to something less abstract than the truth table, we need to construe the material implication operator as having some real-world significance. In this case (and in many other cases), this translation fails to obtain. This does not mean that formal logic is wrong, per se, as much as it is not applicable to the domain (of facts, and not the domain of truth tables.)

I suspect that this is the very point the OP's professor is making, and his warning should be heeded.

Answer 3

This must be a common example, as Wikipedia has the example same example in the article on Paradoxes of material implication. The article explains the paradox as:

The first might be read "If John is in London then he is in England, and if he is in Paris then he is in France. Therefore, it is either true that (a) if John is in London then he is in France, or (b) that if he is in Paris then he is in England." Using material implication, if John really is in London, then (since he is not in Paris) (b) is true; whereas if he is in Paris, then (a) is true. Since he cannot be in both places, the conclusion that at least one of (a) or (b) is true is valid.

But this does not match how "if ... then ..." is used in natural language: the most likely scenario in which one would say "If John is in London then he is in England" is if one does not know where John is, but nonetheless knows that if he is in London, he is in England. Under this interpretation, both premises are true, but both clauses of the conclusion are false.

But you ask if formal logic is suitable for philosophical reasoning. I would say that formal logic is a tool, but a dangerous tool if used the wrong way. You should only count on it when you have mastered it. Seems that many of the answers here were by people who weren't aware of the paradoxes of material implication, so this shows you how difficult formal logic really can be especially when you actually try to use it to solve real problems.

It is unfortunate that we even use material implication, that we have a special logical connective and symbol for it, and that we are all taught that "if then translates as material implication in formal logic". You can do just fine in formal logic without material implication, and you should only translate if/then as material implication when you are sure that material implication has all of, and only, the logical properties that you need. This example would be much better off if you use first-order relations instead of material implication. For instance, you could express explicitly what sort of logical properties that one thing being located in something else imply, e.g., that if Dave is in London, and London is in England, then Dave is in England.

It helps to break material implication down when you use. For instance, Dave being in London material implies that Dave is in England breaks down into either Dave is not in London or Dave is in England, for instance. Is that really the break down that you were expecting?

Answer 4

The literature on the analysis and meaning of conditionals is enormous. I have studied this a fair bit, and I can tell you there are at least 20 books and over 20,000 papers on the subject. Try going to JSTOR and searching for conditionals to get an idea.

Conditionals in natural languages cannot simply be captured by the material conditional. Frege introduced the material conditional, but his main concern was to describe logical, mathematical and scientific relationships. Natural language conditionals do not generally make a good fit, which is why the paradoxes of material implication exist. This is not to say that logic is unhelpful in reasoning, just that you need to be careful how to use it when analysing natural language expressions.

When we say "if p then q" in ordinary usage, we assert something much stronger than a truth function. We are typically saying that we are willing to infer q from p. Inferences can be of different kinds, e.g. deductive, inductive or abductive, but any can form the grounds for asserting a conditional.

To change your example slightly, it seems reasonable to say A:"If Dave is in London then he is in England" but not B:"If Dave is in London then he is in France". If the material conditional were used here, both statements would be true in the event that Dave is not in London. It might seem possible to rescue the logic by replacing Dave with a universally quantified variable. A':"If anyone is in London then he is England" and B':"If anyone is in London then he is France" but even this doesn't work. The point is more subtle, but in the event that London was uninhabited, both A' and B' would still be true. The reason we find them different is because we are willing to draw an inference from "Dave is in London" to "Dave is in England", but not from "Dave is in London" to "Dave is in France".

Another general reason for the difference between the material conditional and ordinary language conditionals is that in the real world we lack certainty and have to reason with defaults that allow exceptions. C:"If the switch is in the down position then the current flows". Except it doesn't if there's been a power outage to the building that houses the circuit. This is why the logical rule of strengthening generally doesn't work in natural language. Strengthing has it that "p => q" entails "p and d => q" which is true for the material conditional but easy to find counterexamples for with real world conditionals.

Speaking of uncertainty, very commonly when we make ordinary conditional assertions we are really claiming only that the relationship is probable. In other words, much of the time "if p then q" is better represented as "Pr(q | p) is high". This is why, for example, the rule of hypothetical syllogism doesn't always work with real conditionals. Hypothetical syllogism has it that "if p then q" and "if q then r" entails "if p then r". Again, true for the material conditional, but easy to find counterexamples for with real world conditionals.

There are many other differences between the material conditional and natural language conditionals, but my response has become very long already. You'll need to dig into some of the literature to understand it better. Jonathan Bennett's book A Philosophical Guide to Conditionals" is a good start.

Answer 5

I have no idea why it is voted up, but Rex Kerr's answer begs the question. I'll explain why in a minute. Your logic teacher was probably trying to show you the limits of propositional logic (P), not quantificational logic (Q). The original argument was in P, not Q. What he's trying to show is a specific limitation of P, i.e. P seems to have problems with capturing predicates and quantifiers. This also extends to the other logical languages. There are valid arguments that cannot be captured by both P and Q, but can be captured by languages like Modal Logic. This is also why there are a plethora of logical languages out there beyond P and Q, because those two languages cannot capture the richness, nor the validity, of all natural language arguments. You can explore this topic more by reading up on philosophical logic, which talks about the problems the various languages have with trying to capture natural language arguments.

As for Rex's answer, as I can't comment on his answer, I'll comment here. The original argument was in P, and the OP asked about logic's failings in that language. Rex shifted it to Q without addressing the original limitations that you asked in your question. He is right that Q captures aspects of the argument that P cannot, but he is begging the question, because the original issue was that of P.

Answer 6

You can't use truth tables or sentences like these ones as it is about situations. If Dave is in London at time t, then he is in England at time t. If he is in Paris at time u, t differs from u significantly, he is in France at time u. What your prof showed was that you have to be carefull about using formal reasoning as a tool that always gives the right answer.