How does one actually settle an argument with logic?

Question

If we had the following situation:

Two students are arguing (Alice and Bob). The actual argument they are having is about whether or not a specific word is offensive. It's not a new argument, it's an argument that's been had many times in many settings, and the two students aren't getting very far.
Alice says: "A large portion of people find it offensive"
Bob replies: "How do you know that?"
Alice says: "Because this argument has been had over and over again it is proof that there are enough people that find it offensive"
Bob replies: "That's ridiculous, if we start a conversation about the world being flat, and the fact that other people have conversations about the world being flat, doesn't mean the world is flat"
Alice agrees: "Of course it doesn't, but it does mean that there are many people who believe the world is flat and that's the point I'm making"
Bob replies: "Your logic is flawed, there's no point in having this conversation"

How do the students settle the argument with logic? Is there a set of basic rules that once agreed on they can use to determine who is actually being logical?

EDIT: To clarify, I don't think either side can ever prove whether the word is offensive or not, the statement that Alice wants to defend is:
"Because this conversation has been had many times, many people find it offensive"
if Alice were using this as proof that it's offensive, I would agree that it would be a flaw, but Bob thinks the statement itself isn't true. Bob claims that even if the conversation has been had over and over by different people, it's not proof that there are people who find it offensive.

Answer 1

There's many logical issues with the argument. As should be expected. It is very rare that logic actually wins an argument, and it's usually because there's a referee ensuring the argument stays logical.

The particular issue you will likely want to have attention drawn to is the difference between "a specific word is offensive" and Alice's statements such as "A large portion of people find it offensive" and "Because this conversation has been had many times, many people find it offensive". These statements are being made in English, rather than a language with explicit logical constructs, so its expected that people will take leaps.

The first leap is from "a word is offensive" to "a large portion of people find it offensive." This statement is not justified by any further logic. If anything it's being presented as a definition for a word having the property of being offensive. In a truly logical argument, both Alice and Bob would have agreed upon a list of these definitions and axioms ahead of time. In a real argument, like this, Alice may introduce a new axiom. For this to be "logical," Bob would have to accept the statement as true, or Alice would need to justify the statement using logic and relying on statements that Bob has already accepted as true.

The second piece is "Because this conversation has been had many times, many people find it offensive." This is another implication that doesn't have a logical justification included. Bob could easily argue that 2 devil's advocates could have this conversation without finding it offensive. There's also a trend to saying less and less. We started with "the word is offensive" then changed to "a large portion of people find it offensive," and now we're down to "many people find it offensive." Define "many." 100 people may be "many people," but it's a small portion of a city of 1,000,000. Alice is assuming that these statements are justifying her more sweeping claims, but in fact they are not logically justifying them. She is relying on the human side of Bob to look at a not-quite-logical-argument and accept it as justification in the larger argument.

Bob, on the whole, is much more logical. His argument is basically taking the approach of proving Alice's argument to be non-logical at every step without adding any non-logical statements of his own. However, he did make one similar mistake. When he talked about the flat earth arguments, he made the assumption that there was a connection between flat earth arguments and offensive word arguments. While it's natural for one to argue "Alice, if you invalidate a similar argument, you invalidate your own," that statement is not logically justified unless the agreed axioms for the debate all treated offensive word arguments and flat earth arguments equally.

Because we don't have a preamble where Alice and Bob lay down the axioms and define the domain of discourse, we can only guess as to whether Bob's argument is logical or not. If I were to pick up the argument from here, I'd invoke Godwin's Law as paradoxically as I can, but pointing out that in many debates where the logical axioms are not stated, arguments about Hitler are often placed in a separate category such that one cannot logically infer anything about a Hitler argument from a non-Hitler argument and vice versa. I'd say this is typically justified because it's hard to specify logical axioms which work well in corner cases with individuals who are considered to be that evil.

The real moral of the story is "Debate using logic, not English." Just kidding! The real moral is that real debates are never 100% logical, because the devil is always in the details. Even if the debate starts as 100% logical, someone will eventually attack one of the axioms, and demand it be justified instead of simply assuming it was true. This process will continue until the debate ceases to be 100% logical, or until Baron Münchhausen pulls himself up out of the mire by his own hair.

Answer 2

Wittgenstein has noted that all of logic is a set of tautologies. If you want something other than tautologies, you need an inductive base.

From a basic logic point of view, statistics never support an argument. So all observations of frequency are not conclusive. But we generally live in a world that also acknowledges science, at least at some basic level shared by both Aristotelian and modern scientific standards.

You can actually count observed instances and say 'there are N observed instances'. If you have enough resources, you can find that 'm +/- s% of the population with a certainty of d%' does whatever. To declare arguments like that illogical holds one far too close to a place where absolutely nothing that is not tautological can be proved.

And their informal equivalents are not illogical, they are just informal. Whether N is 'enough', or whether it is 'large' are subjective, but declaring talk about them 'illogical' as the base of an argument because they lie outside the strict bounds of philosophical logic is simply evasive.

One can agree to deal with likelihoods, or one can discard science as a whole. Since modern life depends upon science to a ridiculous degree -- agree to deal with likelihoods. At that point this argument has an obvious logical means of solution. Use the word on people to whom it applies, and see if insult results in a large enough part of the population to establish its effect with an acceptable p-value.

Answer 3

Alice and Bob are making fundamentally incomparable propositions, however at least one of them believes they are making fundamentally comparable propositions (hence their arguing). In order to "settle" the (non)argument, perhaps we would need to formulate the propositions of either side such that both Alice and Bob could recognise their propositions are fundamentally incomparable with each other

For example, we could say that Alice is making at least three claims: (1) use of term x is objectively offensive to some people, (2) many people have had similar conversations to the one we are having, and (3) the fact that many people have had similar conversations to the one we are having is sufficient grounds for the claim that use of term x is objectively offensive to some people.

We could say that Bob appears to be making just one claim: even if we grant that many people have had similar conversations to the one we are having, this is insufficient grounds for the claim that term x is objectively offensive to some people, which claim relates on an epistemic level only to claim 3 of Alice's overall argument

So, we could "settle" the argument through showing that the terms of the argument are different for either side, where the emphasis for Alice is on claim 1 of her overall argument and the emphasis for Bob relates on an epistemic level only to claim 3 of Alice's argument

Answer 4

"A large portion of people find it offensive"

This is a logical fallacy. It's called an argumentum ad populum.

All of Alice's statements are in fact a mere repetition of the same logical fallacy.

For Bob and Alice to settle the argument with logic, Alice will first have to acknowledge that her argument is fallacious.

Then, she'll have to listen to the arguments Bob has for his position and look for fallacies or other errors in those arguments.

Then, Alice should provide her counter-arguments against Bob's arguments. Bob should listen to those arguments and look for errors in those.

Then, Bob should counter those counter-arguments with counter-counter-arguments. etc.

Alice and Bob should continue this process of each coming up with counter-arguments against the arguments of the other until one of them no longer is capable of providing any counter-arguments.

The person who is incapable of finding any flaws in the arguments of the other and therefore of providing valid counter-arguments is the person who loses the argument.

Alternatively, you may come to a point where you feel that this process is too tedious or where you realize that both of you simply don't have sufficient data to back up your claims. In that case, you agree to disagree.

Answer 5

Calvin: The statement itself.

The Taj Mahal is white.

Most people would say "T". An artist may say "F" because a little black in certain places brings out the white, so it isn't all white. "F"

What will it be?

We have to have something to plug into the truth table, then our logic can begin. The table doesn't care, just give it a T or F!

I think there is a logic of 1, 0 and 1.5 too, the name escapes me. Denotological logic? Lukasiewicz?

So is the problem on this level? Perhaps we could be on the level of the proposition itself? Edit:The parties must come to a T or F on Bob's statement before we can begin to speak of logic. I am not a logician (which is probably obvious).