# Proof that using only logical form is valid?

I'm studying logic. One of the fundamental things that I find everywhere is the claim and I'm quoting wikipedia:

"The concept of logical form is central to logic, it being held that the validity of an argument is determined by its logical form, not by its content."

And a logical form is defined (on Wikipedia) as follows:

"The logical form of a sentence (or proposition or statement or truthbearer) or set of sentences is the form obtained by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks on a form."

And they go on to say that:

"The importance of the concept of form to logic was already recognized in ancient times. Aristotle, in the Prior Analytics, was probably the first to employ variable letters to represent valid inferences."

But why does it work? Why is only the form important and not the subject matter? I mean, is it a fact? Is there a proof that shows conclusively that this is the case? Or is it just something that we found to work?

• For an interesting study in addition to the answers, take a look at First Order Logic vs. Second Order Logic. Even within logic, there are nuances which make an argument logical in one context, and not in another. Commented Aug 14, 2015 at 4:50
• Are you aware that your first quote is a definition NOT a judgement? Commented Jul 21, 2023 at 10:34
• There are many different ways of characterising logical validity and it is not necessary to refer to 'form' though many accounts do. See for example my answer to this question: philosophy.stackexchange.com/questions/93546/… Commented Jul 21, 2023 at 13:05

Logic is rooted in the claim that consistently useful generalizations can be made about an argument based solely on the form, independent of content. It's a process of abstraction, very similar to the way numbers are an abstraction of our experience of quantity. It's possible to study logic as a purely abstract, mathematical exercise, but if you want to view it as the science of argument, you can't escape that claim.

Thinkers such as Tarski, and many others, have done their best to establish firm foundations for logic at a fundamental level, but you still essentially have to accept as a given that formal concepts such as logical truth and validity do reliably map back to the real world (much the same is true in mathematics).

The main issue is that logic is our chief system for constructing proofs --we cannot use it to justify itself, or its own axioms.

• Thank you! So just to reaffirm, Logic is a claim and it's very likely that it can't be proved, at least by itself (which makes sense). Commented Aug 13, 2015 at 21:15
• The problem is that Kant, Frege, Brouwer, Carnap, Tarski, Quine, Kripke, etc. had very different ideas about what "logic" is, what constitutes its "form", and how it is to be separated from "subject matter". At best there is a vague ideal, not a claim. Tarski's version is one among many, and it is far from "firm" even in mathematics, where it presupposes platonism philosophy.stackexchange.com/questions/24885/… Commented Aug 14, 2015 at 1:25
• @Conifold I've edited in line with your comment. Commented Aug 14, 2015 at 2:09
• @shobhu I may have overstated the case a little. I've edited to be a little more nuanced. Commented Aug 14, 2015 at 2:21

There is no "proof" because this is supposed to be a convention. The intuition is simple enough, logic is supposed to encode "basic laws of thought", some arguments depend on empirical evidence while others can be done "by logic alone", those are the formal ones. Unfortunately, philosophers could never agree which "laws of thought" are basic enough to be "logic". Aristotle and Kant only included syllogisms, Frege added to that the calculus of quantifiers, all of arithmetic and much of set theory, while Brouwer thought that the law of excluded middle was too much already. Quine went all the way and argued that all "laws of thought" derive from experience, vastly generalized and abstracted to be sure, but ultimately they are adopted because they work. See Is logic empirical?

What one takes to be "logic" determines which arguments are valid, 7+5=12 follows from empty premise "by logic alone" for Frege, but not for Kant. It also determines how one separates "logical from" from "subject matter". There are two main approaches, deductive and semantic. In the deductive one we simply list all "recognized" formal rules (such as A and A->B infer B), and then construct arguments where each step conforms to a rule. The problem is that different logical systems have different rules (and axioms) and adopting one or another is clearly context dependent and open to discussion.

In the semantic approach developed by Tarski one says that the inference is logically valid if it holds when all non-logical symbols in it are interpreted in any "possible world". There are as many ideas about what "possible worlds" might be and how they might be accessed, as there are about logic, so at best this provides some vague guidance as to what formal rules to adopt. Many philosophers believe that such analytic/synthetic distinction simply can not be drawn in a principled way, "logical form" is as impossible to separate from "subject matter" as physical form from physical matter.

On other hand, the tower of knowledge clearly has many floors, some more empirical than others. While all of it is ultimately exposed to the winds of experience we may wish to designate the relatively stable lower floors as "logical forms". See Friedman on relativized a priori.

What it says, in my perspective is, is analogous to the following: You don't specifically need software application A to show the logical behavior of transistors inside a CPU. You could also do that with software application B. So here software application A and B are the subject matter, or the content terms. Where as the behavior of the CPU is the abstract level you are proving the logical behavior of.

Here is another example. The concept of addition. Here I proof the logical validity of addition: "When I take 3 apples, and I take 2 apples more, and put these inside a bag, I have 5 apples in my bag." The logical concept proven here doesn't require an example with apples. It could have been any example to explain the concept op addition. So the subject matter of apples inside a bag is not important to proof the logical form of addition. If apples would be required, its not an abstract logical concept. As logic doesn't apply to specific matter. Math for example doesn't only apply to the stock market, but applies on the whole universe.

• This is fascinating ! :-) but this sentence: "So the subject matter of apples inside a bag is not important to proof the logical form of addition" .. how could you prove the logical form ? Isn't it just demonstrated by examples (apples/oranges etc) ? From a curious noob Commented Aug 13, 2015 at 11:38
• @Mike What about... When I take 3 ice cubes, and I take 2 ice cubes more, and put inside a bag. I've a no ice cubes left in a bag after some time! Then doesn't it depend on whether you are putting apples or ice cubes (i.e the subject) in the bag? Commented Aug 13, 2015 at 12:56
• @shobhu You are adding parameters, the time it takes to melt ice cubes. I could reply with, what if the bag was sub zero celcius? They wouldn't melt. In order to explain anything, you keep it as simple as possible. And I have the feeling your question is drifting off topic. Commented Aug 14, 2015 at 6:37
• @user2808054 I guess you are asking "How could you prove the existence of potentiality?" I mean, a formula exists in potency, It can only be actualized, brought into reality, when applied so something. So yes you need something to apply it to, to prove its existence. The effects of a black hole (bending light) on its environment proof its logical existence. I see similarity in your question with the following: "How can they proof the existence of the Higgs boson with measurement?" Commented Aug 14, 2015 at 6:41
• I think I see the nub of my question: you can only demonstrate a theory (formula) to be repeatably true in various examples, but can't ever prove it. Eg it's assumed that if 3 apples +2 apples = 5 apples then the same is true for all things, but that's just an assumption. I see what you mean though. Commented Aug 14, 2015 at 12:44

Your question is a compound one, and some of the bits have different answers. Some of the bits are debatable.

But why does it work?

This is because the formal system is such that building on a small set of operators, we can prove either through truth tables, truth trees, or inference rules that any time an argument is formally valid that were the premises to be true, the conclusion would also have to be true.

In a sense, this is more certain than "If I see a light bulb producing light, then there must be electricity flowing through it" because it is an abstraction such that by rule it will work.

Why is only the form important and not the subject matter?

Because the rules are such that they are truth-preserving, and the individual rules are not complicated. Or to put it another way,

if we stipulate

1. Every claim must be true or false.
2. Every term must be used consistently.
3. Contradictory outcomes are not considered acceptable outcomes.

and then we make some operators that have deterministic results (A & B is true only when A and B are both true; A v B is true if either A or B is true; A -> B is true except when A is true and B is false), then we can guarantee the results.

I mean, is it a fact?

This is a complicated question, because the definition of "fact" is complicated. It works in the same way that the rules of chess work.

Is there a proof that shows conclusively that this is the case?

No, and to a good extent there cannot be, because it's pretty much the minimal set of things we need to do proofs.

Or is it just something that we found to work?

In general, this is not held to be the case. Where this is some disagreement is whether our logic has anything to do with the world. Most philosophers think so. Some, like Richard Rorty, think that this is nothing more than a game we're playing, and we could be playing a different one, i.e. it's a self-contained novel of sorts. This is a big feature in some post-Modernisms.

• I find your analogy of chess very interesting. And I definitely want to know more about if logic has anything to do with the world. But if there is a disagreement about it, then can you please give me an example of something that doesn't follow the rules of logic in this world. Commented Aug 13, 2015 at 12:58
• well, one difficulty is that's not a view I ascribe to (or at least not one I would commit myself to), and I'm a bit lost as to how the world works if it doesn't follow the simplest laws of logic (we're not talking some esoteric deontic + modal logic). But at the simplest level, one gap between the world and 1st order logics is that it can be partly cloudy but logic at its simplest requires us to give everything a truth value or else it breaks. Commented Aug 13, 2015 at 14:03
• Do you mean we can't uniquely or precisely define "partly"? Because we can give a truth value to the statement that "Is the sky partly cloudy?" Correct? Commented Aug 13, 2015 at 21:03
• Given the statement: "The sky is cloudy." In the simplest versions of logic, this must be either true or false. Sure, we can add "The sky is partly cloudy" and give that truth values of either true or false. But there's no gradation vis-a-vis any claim, and that's a necessary attribute to make it work. Commented Aug 13, 2015 at 23:08

the validity of an argument is determined by its logical form. The logical form of a sentence (or proposition or statement or truthbearer) or set of sentences is the form obtained by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks on a form.

This definition does not had anything to the standard notion of form. We might just as well look up the word "form" in an English dictionary.

The idea is that logic concerns the relations between terms or their referents, rather than the terms or their referent themselves. Take for example:

(C) If a = 2, then if x = a, then x > 0

This clearly doesn't say anything about a or x because it doesn't say whether a = 2, x = a and x > 0 are true or false. Yet, it does say something, and something rather interesting, and what it says is easily understood by anyone who is proficient in, in this case, English. Broadly, what conditional (C) above says only concerns the relation between the possible truth values of a = 2, x = a and x > 0. These possible truth values depend on the semantic of the language used, and therefore, crucially, on the semantic of the symbols =, >, 2 and 0. If we don't equivocate, then (C) is true, and it is true irrespective of whether a = 2 and x = a would be actually true or not. Given that we don't assume anything as to a = 2 and x = a prior to considering conditional (C), we actually don't assume that a = 2 and x = a are true. Yet, conditional (C) is undoubtedly true.

But why does it work? Why is only the form important and not the subject matter? I mean, is it a fact? Is there a proof that shows conclusively that this is the case? Or is it just something that we found to work?

It is definitely something we find works. Conditional (C) certainly works, given the semantic we use, and I don't think anyone could reasonably disagree.

The broader explanation is that logic is a cognitive capacity, somewhat like vision or memory. And these work because they are the result of natural selection and of the evolution of species. If human logic didn't work, our ancestors presumably wouldn't have bene able to survive, prosper and reproduce, and then we would not be here to talk about it. So, they did survive, which suggests human logic works fine. That is, logic is a selective advantage, it is an adaptation to our environment, and we inherited it from our ancestors through our DNA, and our species probably inherited it from its ancestor species. This is the only scientific explanation as to why humans are logical.

Something else is the fact that we cannot seem to be able (reasonably) to fault logical truths. This is only the case because human logic is the only logic native to our own brain. We are unable to imagine any colour which would be outside the spectrum of colours we normally experience, and this even though other animals seem to experience other colours (ultraviolet). Some people also do, but they are in a tiny minority, and they themselves cannot imagine any colour outside the ones already familiar to them. The colours we perceive seem to be genetically determined. And so seems our logic.

This does not mean that there is no alternative logic, but there is certainly no alternative logic that we know of and validated by natural selection over the entire biosphere over a period of possibly 525 million years since the first proto-neuron cells, and possibly longer, broadly 2 or 3 billion years, if we go back to the first unicellular organism with something resembling a capacity to learn from experience.