## Is formal logic unsuitable for philosophical reasoning?

14

11

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:

1. ((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!

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


How come no one has mentioned the paradoxes of implication? I would argue that, yes, formal logic shouldn't be used in philosophical reasoning because they still haven't found a system of formal logic that doesn't contain these paradoxes. – Kevin Holmes – 2014-07-21T15:53:30.590

Can I ask about the "when" in the 4th quote? I'm not sure that "Dave is in England when Dave is in London" is a fair paraphrase of "If Dave is in London then Dave is in England". The "when" sounds counterfactual - in any situation in which Dave is in London, Dave is in England. But "if" sounds more like a declarative point - Dave might or might not be in London (right now, say); if he is, he's in England. – Paul Ross – 2014-07-21T23:16:26.367

6considering your and between ((if p then q) AND (if r then s)) you implicate that Dave is able to be at two places at the same time, therefore this doesn't represent anything realistic anyways. What would happen if you say ((if p then q) XOR (if r then s)) meaning, if Dave is in London then he is in england OR if he is in Paris then he is in France but he cannot be at both places, this would seem more realistic and therefore could rather be used as an argument. – Sim – 2011-12-03T20:15:38.720

1"considering [...] you implicate that Dave is able to be at two places at the same time" -- no, he does not. – None – 2011-12-04T00:27:41.960

if he is saying "(if Dave is in London he is in England) AND (if Dave is in Paris he is in France") there is at least no rule which states that he must not be at both place at the same time, you might be right that he didn't state the opposite as I first thought. But my argument, that this doesn't picture anything realistic and therefore will never have a realistic result, might still be valid. – Sim – 2011-12-04T00:59:33.593

## Answers

15

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.

But logic alone can be insufficient, if not complemented by critical interpretation. Even so Gödel's incompleteness theorem is valid, some critical interpretation was appropriate to clarify what it really "means". However, some newer "no-free lunch" theorems have similar issues, but have not been complemented by much critical interpretation. The resulting conclusions have problematic consequences, from my point of view. – Thomas Klimpel – 2011-12-04T14:25:35.667

1@ThomasKlimpel - The question was not "does logic prove all statements" but rather "shall I use logic in argumentation". But I agree that understanding incompleteness theorems is interesting, and that mapping reality into and out of a logical framework is a critical and nontrivial task. – Rex Kerr – 2011-12-05T05:44:47.970

5

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.

5(3) is a Tautology, you can create yourself a truth-table and fill in every value for p, q, r, and s. It will always be true. Can you give a configuration for p, q, r, and s which yields False? – scravy – 2011-12-03T23:20:15.767

Sure. We'll take P to be true, and say that if P is true, then Q is also true. We'll leave R and S undefined, as they are irrelevant to my purposes. Now, the fact that P implies Q does not allow us to make any claims about S (such as "P implies S") nor does it allow us to make any claims about R (such as "R implies Q"). – Michael Dorfman – 2011-12-04T13:00:04.607

You can't just ignore truth-tables and assert the opposite; you have to identify the mismatch between the nonintuitive interpretation and the truth-table. Or you have to accept that "formal logic should not be used in argumentation". – Rex Kerr – 2011-12-04T15:14:26.383

OK, I'll spell this out in more detail. We'll set P to be "It is raining outside", and Q to be "The ground is wet." If P is true, then Q is true. Similarly, we'll set R to be "It is sunny outside", and S to be "The ground is dry". If R is true, then S is true. Now, both "if P then Q" and "if R then S" are true-- however, we cannot derive from this that either "if P then S" or "if R then Q" is true-- in fact, both are trivially false. – Michael Dorfman – 2011-12-04T15:47:26.423

You are either still confused, or you are being too imprecise with your language. It looks to me like you've made the same mistake as the lecturer three times in a row, and then argued that the truth-table isn't what it in fact is in order to avoid his conclusion. What exactly do you think (if p then s) or (if r then q) means as a logical statement? – Rex Kerr – 2011-12-05T05:48:43.937

@RexKerr: The truth-table is irrelevant to philosophical discourse; that's my point. We can play around with logical abstractions like "(if p then s) or (if r then q)" all we want to, but if we want to be able to apply this (outside of the level of pure abstraction), we need to be able to substitute propositions in for the variables. And once you do this, tautologies like the one listed are no longer necessarily true; in fact, they rarely are. Now, if you want to argue that formal logic would apply to my set of 4 propositions, you're welcome to attempt to make the language more precise. – Michael Dorfman – 2011-12-05T07:39:10.373

3I'm really not interested in giving a tutorial on first order predicate logic. ∀x(((p(x)⇒q(x)) ∧ (r(x)⇒s(x))) ⇒ ((p(x)⇒s(x)) ∨ (r(x)⇒q(x)))) is a tautology, but throwing universal quantifiers ∀x around on the pieces is not a valid transformation of the tautology, no matter how tempting it is to interpret the English language analog that way. Put another way, truth tables are completely relevant to philosophical discourse, but only if you do your logic right. Incorrect application of logic is just as irrelevant as other forms of confusion are to philosophical discourse. – Rex Kerr – 2011-12-05T08:14:28.893

I have studied FOPC, believe it or not, and I do understand quantification; I intentionally used statements of the type "S is P" in my examples as they are (as Heidegger points out) metaphysical statements par excellence. You say "truth tables are completely relevant to philosophical discourse, but only if you do your logic right", but I would argue that this is itself tautological; the notion of "doing your logic right" begs the question. I'm all for formalization, but we have to recognize that it is a process of abstraction, and there is a danger of abstracting away the essential. – Michael Dorfman – 2011-12-05T08:37:18.873

1If there are essentials being abstracted away here, you haven't pointed them out. As such, this answer and ensuing comments are either in danger of misleading people about what is or is not a tautology, or provide no help in answering the original question since you do not specify the system of logic that you're using. If someone says, "I have two green cups--two green things plus two cups is four, four things!", the appropriate thing to do is correct the usage of abstraction. This is no different. – Rex Kerr – 2011-12-05T15:01:48.353

3

The question here comes down to the "translation" of the material implication operator from the domain of truth tables to the domain of actual propositions. Wikipedia has a few words on the subject here: http://en.wikipedia.org/wiki/Material_conditional, concluding that the material implication operator does not translate to an "if...then" construction. If such a "translation" is impossible, then the domains of formal logic and of philosophical argumentation are divergent, and one must tread with caution when one wishes to apply one to the other (as one may stray outside of the overlap.)

– Michael Dorfman – 2011-12-05T15:14:09.050

The clarification is helpful. – Rex Kerr – 2011-12-05T19:03:39.833

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?

It is certainly surprising that nobody had mentioned the phrase "paradox of material implication" prior to this answer, because that's exactly what's at play in the OP's test case - the mistranslation of "if" as a disjunction. – Paul Ross – 2014-07-21T22:53:46.840

3

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.

2

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.

1If the logic teacher was trying to show the limits of propositional or sentential logic, then he did a really bad job given that he left the OP with the impression that "formal logic is unsuitable for philosophical reasoning". All that was done is to demonstrate that the English-language interpretation of q | s | !p | !r is misleading when translated into implications. If you don't want to end up confused about what the sentence actually means, then you use first-order / quantificational logic where you see the problem explicitly. – Rex Kerr – 2011-12-11T04:36:37.087

-1

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.