Proof that using only logical form is valid?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.