Why is first-order logic interesting to philosophers?

This site had a question: Is First Order Logic (FOL) the only fundamental logic?

Let me ask the opposite: Why is FOL still interesting or useful to philosophers? For example, the "ancestor" relationship cannot even be defined in it. This alone shows that it's extremely limited and different from human thought. Why study it or teach it to students then?

It's been argued that certain properties of FOL offset its limitations, but I have my reservations:

• Completeness. This only applies to Tarskian semantics. FOL with Herbrand semantics is not complete, yet there is no loss of inferential power. I would argue that being able to prove more things is useful. Completeness sounds like that ("we can prove everything") but isn't.
• Semidecidability. Searching for proofs in FOL is still computationally intractable. Semidecidability is a curious theoretical property, but I don't think it gives us anything in practice.
• Because it is computationally tractable, unlike general HOL, and still sufficient for science and mathematics. Ancestor relation, truth predicate, etc., can be easily handled by using meta-language, and other tractable parts of HOL are reproduced by FOL based set theory. Second order logic is "set theory in sheep's clothing", as Quine put it. So when taken with the usual blows and whistles FOL isn't limited at all, and it is unclear that HOL can add much to it. Related How did first-order logic come to be the dominant formal logic? – Conifold Nov 5 '20 at 1:39
• @Conifold `So when taken with the usual blows and whistles FOL isn't limited at all` Can FOL + ZFC define `ancestor`? – bobcat Nov 5 '20 at 1:49
• When relations are interpreted in ZFC the usual way, as sets of pairs, trivially. – Conifold Nov 5 '20 at 1:58
• @Conifold Blows and whistles? – user4894 Nov 5 '20 at 6:33
• Proof checking is algorithmic, completeness is what allows for that, and tractable in practical cases (satisfiability of something like huge Boolean tautologies is rarely of interest). HOL is not even recursively axiomatizable, there is no compactness, and it gets even worse with non-logical axioms on top of it. Proof search can not be algorithmic even in FOL, but is at least semi-algorithmic, and there are well worked out techniques like semantic tableaux. There is nothing like that for HOL. – Conifold Nov 6 '20 at 21:28

Firstly, the fact that the ancestor relation cannot be defined in FOL is not itself a philosophical difficulty. It relates mainly to the issue of consistency and completeness and their omega counterparts over infinite domains. It hardly means that FOL is extremely limited.

Your question could reasonably be split up into separate components.

1. Why are philosophers interested in logic at all?
2. Why predicate logic, as opposed to type theory, lambda calculus, category theory, or some other formulation?
3. Why first order logic as opposed to second order?
4. Why classical logic as opposed to non-classical logics?

#1. Philosophers have long been interested in logic: since Aristotle at least. Logic helps to sharpen up the formulation of an argument, so that we can see clearly exactly what is being expressed. It aids in distinguishing valid arguments from invalid ones. It helps to break down a complex proof into individual steps that are more obvious. It helps to identify assumptions and hidden premises. Modal extensions to logic and the possible world semantics that go with them have proved very fruitful in various philosophical theories.

#2. Many systems of logic perform these tasks more or less well, but first order predicate logic hits a kind of sweet spot in the history of logic. Logics prior to the invention of predicate logic and model theory were just too weak. On the other hand, the logics that were developed later in the 20th century are more complex and are perhaps difficult for undergraduate philosophy students to understand.

#3. Some philosophers who use logic do indeed use SOL, but again, it is more difficult to understand, and it brings additional issues. SOL has no general axiom system for its semantics and no general proof theory. FOL has all kinds of nice properties that are listed in the question you linked. Also, much of what is typically expressed using SOL can be handled using plural quantification.

#4. Philosophers are typically more open to the use of non-classical logics than mathematicians in my experience. There have been philosophers who advocate particular logics, such as Michael Dummett with intuitionistic logic, Stephen Read with relevance logic and Graham Priest with dialethic logic. There are also philosophers who take a pluralistic approach to using logics. The application of different logics has interesting consequences in the philosophy of language and in metaphysics.

FOL is a simple model of human reasoning, and much like simple models in general, it is a pedagogical aid in introducing students to the formal aspects of logic without being unwieldy and overcomplicated. One, after all, could make the argument, why teach many formal logics since they are clearly a limited aspect of human reason itself which is largely defeasible and uses natural language.

Your question goes to multiple aspects of philosophy, including the pedagogical, historical, and technical aspects of logic. Let's start with a simple question:

Why teach a child to count, when clearly engineering requires a healthy use of higher math?

In this vein, it is obvious why FOL is taught given its inherent limitations in describing human reason. For starters, how can one teach SOL if FOL isn't understood? So, in your original language, it's not a question of interest, but of utility. Any formal system when first encountered might seem interesting, and then become uninteresting once mastered (and taught to undergrads repeatedly). But it is much, in the same way, most of us math-inclined folk find little joy in counting and arithmetic, it is absolutely a vital theoretial and practical building block for assessing the cardinality of infinite sets, determining loci of intersection in topology, and assessing the monotonicity of infinite series.

There's a point of view that topics of inquiry move out of the realm of philosophy and into the realm of science when they become codified, standardized, well-understood and reliable. In contrast, live philosophical topics are speculative, open-ended, dimly understood, and controversial, almost by definition. In other words, philosophers invent sciences, they don't, by-in-large, practice them.

Given that modern formal logic is arguably the youngest major science to have been birthed directly out of philosophy, we might say that a well-understood logical system like FOL is of diminishing philosophical interest for precisely the same reasons it has proven so valuable in fields such as mathematics and computer science.

Logic is still often considered as a philosophical topic, because it was part of the philosophical realm for such a long time, and has been a science for such a comparatively short time. But most of the actual philosophical work is now being done in the less standardized logics.

• Well put. Is there a reading that inspired this observation, or would this just be considered a generally-acceptable claim by reasonable persons in logic? – J D Nov 5 '20 at 19:49
• I know, I really should provide some cites. Well, the idea that philosophy is the mother of sciences is a commonplace --I can't really cite a singular source for it. The rest more or less follows from that. – Chris Sunami supports Monica Nov 5 '20 at 19:57
• I upvoted because a variety of answers are useful for explaining. I'm also interested in your views on some threads I've started in meta. As a fellow midwesterner, I think this site may have become a bit too exclusive, and as one of the few members of this forum with extended experience and a reputation over 20k, I would appreciate any insights you have about reinvigoration the culture here to increase traffic and participation. – J D Nov 7 '20 at 2:05

First of all, there's an implicit assumption in your question that philosophical interest comes from strength. This is unjustified, especially given the general tradeoff between strength and tameness. Weaker logics correspond to simpler types of argument, and that might be a very interesting sort of thing in a given context.

Second, first-order logic isn't really as weak as it looks; rather, it's context-sensitive. For example, it is true that if S is a structure and R is a binary relation on S which is first-order definable in S, the transitive closure R* of R need not be first-order definable in S. However, if instead of limiting ourselves to S itself we look at what we can say with first-order logic in the entire set-theoretic universe V in which S lives, defining transitive closures is trivially easy. The point is that first-order logic, by not a priori having too powerful tools, lets us keep track of what information we're using when defining an object: the contrast above demonstrates in some sense that defining transitive closures requires nontrivial information beyond what the structure need provide us in general, and this is an interesting thing to note.

A couple quick remarks on this point:

• Compare Quine's criticism (whether you buy it or not) of second-order logic as "set theory in sheep's clothing" - the point being that second-order logic arguably reaches outside the given structure under consideration to an unacceptable degree.

• This is related to the role of ZFC as a foundational system; I said a bit about this in an answer to a question of yours at math.stackexchange. The idea that first-order logic + ZFC functions as a foundation for mathematics is a sort of simultaneous having and eating of cake: we benefit from the tameness of FOL while the ZFC axioms guarantee us enough expressive power for what we actually want to do.

So to summarize: strength isn't the end-all-be-all, and the weakness of first-order logic is just one facet of a more complicated story. Beyond this, first-order logic has additional interest for its more technical properties (compactness, completeness, Lowenheim-Skolem, incompleteness, interpolation, ...). It also has some interesting meta-properties provided by Lindstrom's theorem and its variants. And whether these are desirable or unfortunate, they're all certainly interesting.

Finally, the history of first-order logic will I think further motivate it as a topic; there's a lot written on this, but the SEP article is a good starting point. This paper of Ferreiros is also a great source, notwithstanding that its overall goal is to motivate logics other than first-order logic.

There are certain limitations to FOL, particularly the Lowenheim-Skolem theorem which is why we have to use HOL for models which are uncountably infinite because using an countably infinite number of sentences we can always construct a countable model. For very elementary definitions in Mathematics such as the least upper bound property for real numbers (or Dedekind cuts) we have to use second order logic. First order logic suffices for most of arithmetic, but mathematical induction is second order (omega incompleteness comes to mind), which we frequently use in arithmetic, which in turn is equivalent to the axiom of choice and the well ordering principle (which intuitionist's reject).

Having said that, first we have to address the question why any of us should be interested in any Symbolic Logic at all. Many professional mathematicians don't find symbolic logic interesting or useful either. Most of the time we use a metalanguage to just how that a proof exists in the object language by useful metalogical theorems and subsidiary deduction rules (Defined in Kleene, Stephen (1980). Introduction to meta-mathematics. North Holland. pp. 102–106. ISBN 9780720421033).

The primary reason we developed symbolic logic at all was to just concentrate on the syntax and not consider the semantics at all, do mechanical symbol shunting and yet be able to reason correctly, viz. soundness. One could argue that the motivation of developing symbolic logic was enabling a Turing machine to reason for us. David Hilbert had already shown that in Plane Geometry (Euclid) you need not understand what a point or a line means, but yet be able to prove correct theorems just by syntactic manipulation.

First order logic is philosophically interesting when it comes to understanding the limits of Turing machines against human cognition, because it exhibits both soundness and completeness. There has been much speculation in this problem, even by Kurt Godel himself, who gave the disjunction that either the mind is a machine or there exists infinitely many diophantine equations which cannot be solved, as a corollary of omega incompleteness of FOL. It's also handy when you are arguing or checking arguments. The short answer is, despite it's limitations, FOL is useful. We are perfectly aware of its limitations, and we are also aware that if we are to circumvent it's limitations, soundness and completeness have to be sacrificed. Whenever a certain argument is effable in either FOL or propositional logic, one should go with that, because it is much more reliable. I personally think, as Poincare opined that logic is good for checking things, but it's not useful for creating new things. There may be differences of opinions, but we already know that 3-SAT is NP-complete, so we have to wish ourselves luck in deriving semantically true statements using a computer. As far as the "ancestor" relationship goes in defining FOL, I do not see that as a problem. What I can say is simply using FOL and the compactness theorem that ∃ x ∀ n ∈ N x < 1/n, which was I believe what Leibniz argued both in his calculus and monadology, but was unable to prove. One of the consequences of this result is now the philosopher and the theoretical physicist has to consider infinitesimals in their science, metaphysics, and pataphysics.

In conclusion, philosophers are interested in FOL because there has been positive results in studying it by philosophers, model theorists, proof theorists and so on. There are some truth's, given we have defined our semantics, we can conclusively show which remains dubious in any metalanguage. It is alive and there are yet things to understand about it, and interpret about it.

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