Logic in many ways comes from philosophy. Many of the fathers of logic, such as Aristotle, who developed and early form of deductive logic, saw logic as codifying the laws of reasoning. And so, to be absolutely sure of our arguments, it would seem that we should argue exclusively using formal logic. However, philosophical arguments are rarely made using formal logic, and in many cases when formal logic is invoked its use is questionable.
At first glance this appears quite contradictory. Philosophy, more than any other discipline, relies on reason and argument. If the laws of reasoning really are captured by logic then it is hard to see how philosophers could have a good reason not to use it. Of course it is easy to come up with bad reasons. Perhaps philosophers simply aren’t smart enough to use logic for all their arguments, or perhaps they are too lazy (since constructing an argument using logic usually has the effect of sacrificing clarity for precision). Perhaps this is true for some philosophers (and I know a few who treat formal logic like a poisonous snake), but it certainly can’t be true for all of them. Bertrand Russell, for example, wrote the Principia Mathematica (along with Alfred Whitehead) in which much of mathematics was derived in a completely formal way. And yet Russell didn’t develop his philosophy using such a formal system. Thus we are still left with the question: if logic truly captures the laws of reasoning why didn’t great logicians such as Russell and Frege write philosophy using it?
Of course some may be tempted to reject logic as reflecting the proper way to reason. But that is a position that is hard to defend. Certainly logic seems to work well enough where it is applied, and so the claim that logic is somehow in error cannot be reasonably defended. And so if we are to deny that logic can capture reason it must be because logic necessarily omits valid patterns of reasoning. If these patterns are necessarily left out that must mean that there is no way to consistently formalize them, which is to say that we can’t explicitly state when and how to use these patterns of reasoning in a way that doesn’t lead to error. But if that is the case it is hard to see how we can claim that they are valid patterns of reasoning at all. If they can’t be described in a consistent way then it would seem like they are a kind of intuitive leap we are making which is inexpressible (the leap itself, not where it leads us). And if that is the case how can we defend the correctness of such a leap? Certainly it is possible to reason incorrectly, and so, given that we cannot defend our argument on the basis of it following a pattern that is generally known to be correct, there is no good reason for anyone to be convinced by such arguments.
However, just because a valid pattern of argument could be formalized doesn’t mean that it can be formalized by logic as we know it (at least, not without great effort). This is the reason that best explains why philosophy is not done in a formal system. Although we would like to be able to do philosophy in a purely formal manner, and thus reduce the number of mistakes we make, the formal systems that we have simply aren’t suited for doing philosophy. And one day, when our formal systems are superior to those we have now, perhaps we will do philosophy using them.
Allow me to point out a few ways in which formal logic is unable to capture certain philosophical arguments. One such area is any argument involving truth or provability itself. It is well known that if formal logic is extended by adding the ability to say of a sentence “this sentence is true (or can be proved)” then the system becomes inconsistent. This does not mean that it is impossible to talk about truth or provability of a formal system from within a formal system. What it does mean is that such a formal system must be able to handle inconsistency (so that it is “contained”, meaning that inconsistency doesn’t imply absurd conclusions), or the rules of deduction must be weakened (meaning that they must prove less), so that inconsistency doesn’t arise. Which illustrates my point quite nicely, we might be able to formally reason about truth and probability, we just can’t do it properly with the formal systems we currently have.
Formal systems also have problems expressing the underlying ideas that justify many inferences. For example, consider the following argument: a society that tolerates injustice will not flourish. Therefore most societies punish the unjust. And so if the unjust man wishes to avoid punishment they must appear just. Putting on the appearance of being just makes unjust men unhappy since it is often contrary to their desire to be unjust. Conclusion: the unjust man is generally unhappy. We can formalize this argument, but such a formalization captures only the superficial features of the claims. For example we could write the argument (roughly) as follows:
P(Pun(s)|s∈S) > .9
P(Cares(j)|j∈P) > .99
∀x∈P∀m∈S(Unj(x) → (PretJ(x) ∨ ∃y∈P(L(y,m) ∧ ApU(x,y))))
∀j∈P∀m∈S(Pun(m) → (L(j,m) ∧ Unj(j) ∧ ∃x∈P(L(x,m) ∧ ApU(j,x)) ∧ Cares(j)) → ~H(j))
P(~H(j)|PretJ(j) ∧ Unj(j) ∧ j∈P) > .9
∴ P(~H(j)| Unj(j) ∧ j∈P) > .7
Where: S is the set of societies, P is the set of people, P(a|b) is the probability of a given b, Pun(s) asserts that the society s punishes the unjust, Cares(j) asserts that person j cares about punishment, Unj(j) asserts that j is unjust, PretJ(j) asserts that person j pretends to be just, L(y,m) asserts that person y lives in society m, ApU(x,y) asserts that person x appears unjust to person y, H(j) asserts that person j is happy.
So the argument can be formalized, with a great deal of effort, but the formalization clearly doesn’t add much of value in this case. Every step except the last must be accepted as an axiom, none follows from other principles. And that is the problem; our formal system allows us to capture the large scale features of the argument, to say that one claim implies another. But it doesn’t allow us to easily dig into the content of the claims, and derive those connections instead of just stating them. So here formalization doesn’t really add anything because the premises in the formalization are what we might want to challenge, and what we want our formal system to validate. An analogy might be made to attempting to use a formal system that lacks quantification. In such a system we can still formalize sentences that claim that a proposition holds all the time or some of the time, for example by assigning each sentence to its own logical variable. However, this loses the underlying structure of the claim, and so many inferences using them must simply be accepted as axioms. For example, if we formalize the sentence “all men are cruel” with the logical variable c we can formally deduce that c ∨ k, that all men are cruel or all men are not cruel. But we can’t deduce that c → ~k, which we should be able to, because we can’t unpack the content of those sentences using our formal logic. Similarly, our formal logic isn’t helping us unpack our ideas about society, happiness, actions, and appearance involved in our argument, and so formalization doesn’t add much.
But just because philosophy currently can’t be done in formal logic doesn’t mean that philosophers can ignore it. There are many patterns of reasoning that are captured quite satisfactorily by formal logic. Thus an understanding of formal logic can help us in our philosophical reasoning, not because we can do philosophy in terms of it, but because it trains us to reason correctly in many simple cases, which we run across all the time. And so no one who cares about philosophy should avoid an understanding of logic.