# Can inductive arguments be made in first order logic and, if not, why not?

After reading a question by rus9384 Why is faulty generalization called an informal fallacy? I wondered whether induction can be part of any argument in first order logic (FOL).

rus9384 symbolized an example of an inductive argument: ∃x: F(x) ∴ ∀x: F(x)

Given the rules for existential elimination and universal introduction, I don't think this argument can even get started, but I may be wrong.

Clearly, we can go in the opposite direction. Here is a proof of going from a universally quantified sentence to an existentially quantified one: ∀x: F(x) ∴ ∃x: F(x)

I am not concerned with whether induction is a fallacy, formal or informal, but to what extent induction arguments can be symbolized or made at all in FOL.

QUESTION: Can inductive arguments be made in FOL and, if not, why not? If they can, an example of their use would be helpful.

Reference

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

• Initial thought is No because FOL is a deductive system, but I am open to being proved wrong. – virmaior Jul 28 '18 at 23:14
• But my point was that faulty generalization contradicts deductive reasoning. Inductive reasoning focuses on premises and deductive on the derivation. Exactly we assume here the first part of the "argument" to be true (for some x ...), just like in deductive reasoning. – rus9384 Jul 28 '18 at 23:41
• @rus9384 This is a different question from your question. I am trying understand how to answer yours, but that is separate. Your question seems to depend on the definition one has for formal fallacy as distinct from an informal one. However, could one call faulty generalization a formal fallacy with the formal referring to FOL when FOL cannot even address induction? But then I wondered what is it about existential elimination and universal introduction that makes induction not work in FOL. That became my new question. – Frank Hubeny Jul 29 '18 at 0:31
• Also, I'm wondering if "Fa" was required statement in the proof, since ∀x: F(x) ∴ ∃x: F(x) is valid from the definition. Or... Hm, maybe in order to prove this one we must use logic of sets or so. – rus9384 Jul 29 '18 at 1:06
• "Induction" in FOL is Fa,Fb,... Fz ∴ ∀xFx, where of course the number of premises is finite. IF the "individuals" : a,b,...z are not all the objects of the universe, then of course the argument is not valid. – Mauro ALLEGRANZA Jul 29 '18 at 13:22

You can formalize inductive logic, but it is usually though to require the introduction of an ambient Bayesian probability theory. The reason first-order logic alone doesn't work is because first-order logic examines whether an argument is valid or invalid. An argument is valid if and only if the truth of the premises guarantees the truth of the conclusion. By contrast, an inductive argument provides probable support for the conclusion. To evaluate inductive arguments you need a way to evaluate the degree of support a set of premises provide for the conclusion. First-order logic makes no such provision. Hence to formalize inductive logic probability theory is used. Intuitively, what you want to know is how probable a state of affairs P is given the premises.

As a corollary, the argument form you cite (∃x: F(x) ∴ ∀x: F(x)) does not really capture the form of inductive reasoning. First, the existential quantification requires only one entity that is F to be satisfied. Almost no inductive arguments have that form, and usually involve a large number of observations. In fact, in the sciences, inductive arguments typically make testable hypotheses relative to background hypotheses from other scientific domains (This, by the way, is another reason first-order logic cannot really "formalize" inductive reasoning adequately). Hence the form you list here is at best a highly idealized way of thinking about inductive reasoning that is so divorced from inductive reasoning in practice as to be useless. Secondly, inductive arguments most often have the form of predictions about what our future observations will likely be, and don't require a universal quantification over a domain. For example, if you argue, "I've tasted thousands of lemons and every single one has been sour. Therefore, the very next one I taste will likely be sour." This is a perfectly cogent inductive argument that does not require a universal quantification over a domain.

Check out the following page if you want to know more: https://plato.stanford.edu/entries/logic-inductive/

• I'll take a look at the SEP link. – Frank Hubeny Jul 29 '18 at 18:47
• I am going to accept this because of the reference you provided. It allows me a way to explore this further. – Frank Hubeny Jul 31 '18 at 17:08
• Probability alone is not enough. You want statistics and credibity analysis. But even tgis is not enough to distinguish causes and effects, which is a part of human logic as well. – rus9384 Jul 31 '18 at 22:39

You can represent inductive arguments using the various notations used to symbolize first-order logic. However, there is no other sense in which you can “make” them in first-order logic.

Some examples of inductive arguments represented in a common notation for first-order logic, in which they are all invalid, include:

1. Pa, Pb, Pc, Pd, Pe |= Pf
2. ∃xPx |= ∀xPx
3. A→B, A→C, A→D, A→E |= A→F

There are infinitely many examples.

• I can see how the first three are invalid because I could come up with interpretations in which they are not valid. I can also see how the fourth would be valid because of the complete specification of the domain. I wonder if there is something about the existential elimination rule and universal introduction rule that guarantees such inductive arguments are not possible in FOL? – Frank Hubeny Jul 29 '18 at 1:02
• The argument Frank showed also is valid in FOL. But is it then deductive? – rus9384 Jul 29 '18 at 1:03
• @rus9384 Yes, “valid in FOL” simply means “deductively valid in FOL.” Inductively persuasive arguments are generally described as having inductive “strength,” rather than validity. – ChristopherE Jul 29 '18 at 1:52
• @FrankHubeny Yes. There are lots of different ways to construct legitimate sets of deduction rules for predicate logic that work. (They work if they are sound and complete.) But any legitimate set of deduction rules for FOL that includes a universal introduction move must allow it only in very specific circumstances. One way, for example, the move from Pa to ∀xPx could be legitimate is if “a” is merely a placeholder variable, not a specific member of the domain. So, some systems will allow a move like this: 1. ∀xPx 2. Pa (from one, universal elimination), 3. ∀xPx (from two, univ intro) – ChristopherE Jul 29 '18 at 1:59
• Do you have a reference where I could get more information? I am not expecting induction to be possible in FOL, but I am trying to get a better feel for why one should not expect it to work. This would help me better understand why we need a second order logic for, say, mathematical induction. I am hoping those "very specific circumstances" for universal introduction might help explain why. Regarding "a" as a placeholder rather than a specific member of the domain, isn't that effectively what is being done by not allowing UI with "a" in an undischarged assumption? – Frank Hubeny Jul 29 '18 at 13:08