Logic also and its philosophy

When one mentions “logic”, I’d wager for most people what comes to mind are those “fun little logic problems”, that are usually word problems where one has to “use logic” to solve. And here, “use logic” usually denotes that no special knowledge or acquaintance with special theories of special fields are required - and then, upon grasping the problem, usually stated in our good old ordinary language, proceed by “pure reasoning” to solve it.

Also, most “educated people” would have an idea that, well, you could formalize the procedure, exchange words for symbols and then perhaps automate at least some parts of the ordeal, but all that with no fundamental change in understanding - that is, those symbols are supposed to only formalize our natural languge anyway, so that is no really big difference.

And then with that sort of understanding, people may perhaps be a little surprised when hearing there is such a thing as “research in logic” - for weren’t the fundamental principles known since, like Aristotle? Or, they might think, that’s all really about solving logic problems - peharps some more complicated and tricky ones than the ones they’re used to.

That isn’t quite the case though. Although some logic puzzles of that kind can prove quite important, it just happens that there are also plenty of fundamental problems in logic that aren’t really unsolved.

When one talks about “fundamental aspects of X” - for some appropriate meaning of “appropriate” - that can be usually termed “philosophy of X”. In that sense, these problems just mentioned are problems on the philosophy of logic. And this post is a (very) brief review on a selection of those problems - and I do intend to come back to some problems more in depth later, so have my apologies for not going too deep about any of them.

What is logic, or rather, what is logic supposed to be?

This is a contentious question, but in most generality, logic can regarded as being about truth preservationg. You’d like to perform some operations on statements¹ you have or make such that their truth is preserved ~ that is, if you started with some true statements, you want to be have some confidence that the outcome of those operations will be also true in the end, given the semantics of the operations.

The said operations usually take the form of connectives like “and”, “or”, “not”, and a few others. So, for instance, if statements a is true and b is true hold, you’d like for the statement a and b are true to hold as well.

That might seem a bit too vague to work with though. Philosophers and logicians usually like to say, that logic is about validity of statements, irrespective of their content. So, for example, if a is false and b is any proposition, a and b is false independent on whether a is about something you told your gramma to make her feel well or about some deep aspect of the universe, which everyone has reasons to believe in, but which is false nevertheless - one could say logic doesn’t care what the contents are - it doesn’t care about your gramma, and much less about the universe.

And then the bussiness of logic, so to speak, would be to come with appropriate rules and semantics for such “operations” (calling the such as “and” and “or” operations might not be entirely adequate, but never mind that).

And why would one be interested in that? Well, it sure looks like a truth preservation apparatus would be quite a useful one.

One would usually prefer to act according to some truth than to some falsity, so, in the least, logic in some way could provide us with some kind of norm for distinguishing some adequate actions from some inadequate ones, right? Let’s try to look into that a bit more closely.

What is the normative status of logic, if any?

There is this idea that in order to be deemed rational beings we think, or ought to think, according to laws of logic. Or, at the very least, logic is supposed to somehow guide our thoughts - “otherwise, of what use would it be?”, you and many people, including logicians and philosophers could say. Some would go so far as saying (I’m looking about you, Frege) that “for a thought to be a thought at all, it must be constrained by the laws of logic”. That is to say that logic is normative for rational thought, meaning that is somehow norms how reason is to take place.

But, is it so? If you think it is, then you’ve got a some of problems in hand. To name a few:

1. In logic, it’s usually held that if in every case that “a is true” “ b is true”, one can infer “a entails b” (written \(a \vDash b\)), or “a implies b”²
   • But one would hardly accept an argument such as “in every case that ‘Socrates drank the hemlock’ is true, ‘\(2 + 2 = 4\)’ is true as well, therefore it holds that ‘Socrates drank the hemlock implies that \(2 + 2 = 4\)”;
2. In logic, it’s usually held that from an inconsistent set of propositions, one can infer anything (this is called the “law of explosion”, ex falso quodlibet, “from contradiction, anything follows”)
   • But, for someone (that’s not too skilled in maths) to get (by whichever methods) to the conclusions that \(2 + 2 = 5\) and \(2 + 2 = 4\) and \(4 \neq 5\), can hardly seem to provide reason for inferring that “Bolsonaro is a good politician”;
3. In logic, it’s usually understood that, if “a entails c”, then “a and b entails c” (\(a,b \vDash c\)) where a, b and c are propositions - meaning that by coming with new propositions, one can perhaps infer more propositions, but not less³;
   • But in life we could have, for instance, a = “I have two eyes”, c = “I may be able to see”, and have \(a \vDash c\). But then, if I add b = “my eyes are blind”, it’s not the case that \(a, b \vDash c\) - that is, reasoning isn’t, and isn’t supposed to be, monotonic;
4. If you, in good faith, write a book on (say) Ornithology, then you (supposedly) have good reasons to believe that any given sentence in the book is true; however, you could also, quite reasonably, add to the preface “in a work of this magnitude, it’s hardly possible not to incur in error at some point, for whathever reason; in case you find some error, feel free to let us know”
   • That is, you have enough reason to believe any given sentence found in the book is correct, but not enough reason to believe all sentences found therein are correct - on the contrary, you have good reason to believe at least some of them are false (be it by mispelling or wathever). This is known as the preface paradox.

There are many other such problems of course.

To be sure, for any of those problems, you can probably find a good enough solution within the realms of logic. For instance, there are plenty of logics that don’t have explosion, so maybe you could pick one of those. Likewise, there are such non-monotonic logics, which could also be of use, and “relevant logics”, where “b is always true” doesn’t necessarily entitle one to derive “a implies b” for any a.

But then, you can hardly deal with all problems that arise that way in a uniform manner, and any solution that you find to one of them can create other problems further down.

But well, is logic supposed to deal with that kind of thing anyway? If not, why we’d care about it in the first place?

One possible answer is “logic hasn’t really anything much to do with reasoning - or, at least, not more than any other discipline”, as argued most notably by Gilbert Harman (see his Change in View).

Alternatively, one could say that logic isn’t quite supposed to deal with reasoning directly, but we can still come up with so-called “bridge principles”, principles that purport to, well, bridge logic and reasoning. That is something that John MacFarlane tries to set up in his paper In What Sense (If Any) Is Logic Normative forThought?.

But this is still an open problem. Most people, it seems, would be inclined to say that “yes, logic is normative”. But how so? And which logic? Would you go with classical logic, or some non-classical logic, such as FDE (as perhaps some buddhist would be inclined to accept)?

Either way, there seems to be plenty of problems to sort out there - and I will probably come back to this in a future post.

Is logic exceptional? How so?

The previous discussion hints at another problem, this one possibly a bit more philosophical. If you subscribe to logic as a foundation for how we think rationally (although you might be starting to doubt that), or, even if not entirely that, at least for (say) the methods of sciences, then you might be inclined to see logic as exceptional in some way.

That is, if it’s on the foundation, it better be a strong one. But what is securing the foundations? “Well,” you might say, “it’s supposed to secure itself”. Which might be sensible enough first answer.

But, if you think about sciences, such as biology, physics, chemistry and whatnot, you can see each of them has their methods which change and evolve with time, so as to make use of better understanding of “how things work”.

Underlying that is supposedly a set of principles that is, again, supposedly, unchanged, namely, the principles of logic - our secure foundations. For (say) a physicist to decide on holding to or dropping a theory, she should do so logically, in some sense - otherwise, how is science supposed to improve on itself?

In spite of that, we know the conception of logic varied greatly over time - at least from Aristotle onwards in the westerner part, or at least from Nagarjuna in the Indian subcontinent. The nowadays mainstream classical logic is in many ways different from aristotelian logic, and took its current shape only recently, after much toil.

But then it would seem to be just like another science: one has theories, carries experiments (for some appropriate sense of “experiments”), and then hold on to, or drop said theories according to the outcomes of the experiments. In principle, it wouldn’t seem unfair to lay the development of logic (whathever it is) in that manner (though it does raises problems, to which I will come back later). But then it looks just like the development of any other science.

One possible reply would be to say

“You see, the understanding of logic might have changed, but the fundamental underlying principles have been, and still are, the same, and unchanged. That is because logic is supposed to be universal, and apply to all domains equally - and it wouldn’t be sensible to say that kind of universal prinicple would be subject to change. With that in mind, logic could be said to be exceptional to other disciplines, in that we can hopefully find said fundamental underlying principles without the need to do material experiments - and those make up what we could rightly call ‘the one true logic’ or, not to incur in circularity - although that hardly can be escaped -, the ‘correct logic’.”

That is to say that there is a “ultralogic” that after, by whichever considerations, one finds it, one need look no further: those are the foundations. And there seemingly are good reasons to follow that line.

But another view is that there actually is no such “ultralogic”, and that what one actually needs are various logics, one for each use: so, you could prefer using something like classical logic for, say, mathematics, but some paraconsistent logic for when one need to deal with real contradictions (about contraditions, see for instance, Priest’s Can contradictions be true?)

This view is known as pluralist view about logic: there are possibly more than “one true logic”, and it lends easily to anti-exceptionalism.

But you could be anti-exceptionalist about logics whilst holding there is only “one true logic” (though some suggest that, well, not quite), all you need is to say “logic is like any other science”.

That is an open question, philosophically speaking. Hopefully I was able to impart some of its significance though.

By the way, apart from anti-exceptionalism/exceptionalism/pluralism/non-pluralism about logics, there is also the view (not held by too many, as far as I know) of nihilism about logic, with the idea roughly as

“It makes no sense to talk of pluralism about logic, for logic is supposed to be universal, and hence it should apply in every situation as a “truth preserving thing” - if it doesn’t, it isn’t logic, but something else. There, however, doesn’t seem to be a logic that applies in every situation. Hence, there doesn’t seem to exist anything worth the name of logic.

I won’t go much into that last option, but if you’re interested, you might want to check this, or this nice lecture by Gillian Russel.

Why study logic(s) philosophy

It turns out the study of logic (or of logics, if you’re a pluralist) can reveal many subtleties that would hardly be apparent otherwise. As Graham Priest said in a lecture, “logic acts like a magnifying glass, allowing us to see in details structures we didn’t know existed”.

And being more aware of what sort of foundations we find ourselves in be lead to a better understanding of wathever structure (or lack thereof) we have in our lives, and in our world. It does seem to me a worthy topic of consideration.