The Problem of Deduction?

[Fifty]

So the problem of induction gets a lot of airtime, but there's a problem of deduction too! I'm curious about what people think about this, especially people who have lots of math education. Before I begin, you should be aware that Lewis Carroll put it much more eloquently and amusingly than I am going to.

How do you convince someone of the rationality of the following argument:

1) A
2) A --> B
Therefore B

Suppose someone said "I accept 1) and 2), but then why should I accept the conclusion?" Then you could say well, this is also an obvious rule:

3) (A & (A --> B)) --> B)

So the person goes "ok, I accept 3), that seems clear enough. But then why should I accept the conclusion from 1), 2), and 3)? Then you could just point out that the following is also a clear rule:

4) (A & (A --> B) & ((A & (A --> B)) --> B)) --> B

You can see how this would keep going on and on and on. So what's going on here? I mean, to people who aren't thick headed its just obvious that 1) and 2) imply the conclusion, without the addition of 3) or 4), but its also not clear that the person is doing anything wrong by asking the questions, unless you theorize that we are just supposed to intuitively get inferences like that. But logic people surely don't want their discipline to rest on something like intuition do they? I mean logic people (at least ones who aren't very sophisticated about logic) hate words like intuition! It's all about rationality and proving and reason and premises and assertions and rules and yada yada yada. So what's going on?

 

[Ayatollah So]

I take the following moral from Carroll's story (and never mind what Carroll himself intended): a norm != a rule. Modus ponens is a norm that licenses, or perhaps better, moves us to infer B. Whereas, the rule
(A & (A --> B)) --> B)
is a statement that might help explain what we are doing when we follow the norm, but you have to have some degree of a clue (i.e. at least some tentative grasp on the norm) to understand the explanation.

Inductive/abductive logic is like that too, only the "move" is less decisive.

 

[warpus]

How do you convince someone of the rationality of the following argument:

Simple. You draw a truth table for the 'implies' function.

A | B | A --> B
----------------
F | F | T
F | T | T
T | F | F
T | T | T

This is all so far by definition of --> (implies)

So now we pull out the rows that are of interest to us... namely, where A is true and A -> B is true
Ahh.. and the only possible row is one in which B is true.

Therefore B is true, whenever A and A --> B are also true.

 

[Masquerouge]

Is it something that can be explained in normal terms?

 

[Fifty]

I take the following moral from Carroll's story (and never mind what Carroll himself intended): a norm != a rule. Modus ponens is a norm that licenses, or perhaps better, moves us to infer B. Whereas, the rule
(A & (A --> B)) --> B)
is a statement that might help explain what we are doing when we follow the norm, but you have to have some degree of a clue (i.e. at least some tentative grasp on the norm) to understand the explanation.

Inductive/abductive logic is like that too, only the "move" is less decisive.

So when you say "have some degree of a clue", is that the sort of "logical intuition" I talked about? Of course, I personally don't think the logic intuition is a bad thing, its just people who know a little but not a lot about logic seem to think that its this wholly intuition-free. I think that the tortoise is mistaken, but only just as much as you're supposed to just see that modus ponens is true, or as you said, you have to just have a clue!

I'm mainly just curious about how math people with no substantive philosophy education (i.e. not already-clued-in people like you ) feel about this issue. I don't really think its a problem in the "OMG MODUS PONENS IS IRRATIONAL! " sense.

Simple. You draw a truth table for the 'implies' function.

No, that doesn't work at all. It can just be reapplied to propositions about the contents of truth-tables, and so on. If the solution were that simple, you can bet your arse a smart dude like Carroll would have thought of it.

EDIT:

Is it something that can be explained in normal terms?

Try reading the Carroll dialogue, it is much more well written than my OP.

 

[brennan]

This is a problem?

 

[Fifty]

This is a problem?

Well, like I said, I don't think its a problem in the sense that it poses any serious dilemma with logic or whatever, but it does pose a problem for some people's foolish and naive view of logic!

PS: I'm supposed to be on ignore!

 

[warpus]

No, that doesn't work at all. It can just be reapplied to propositions about the contents of truth-tables, and so on.

When you ask the following question:

Given:
1. A is true
2. A --> B

We assume that --> takes on the truth table I provided. Without this assumption, the problem would be unsolvable.

The 100% proper way to state the problem would be to include this assumption.. but it is never done, as truth tables can be easily looked up.

So either:
A. We do not assume what --> means when the problem is stated, rendering it unsolvable
B. You solve the problem by (implicitly or explicitly) referencing the truth table for -->

It seems as though you are going for A. here.. Are you really asking: "How do we know what --> really means" ?

 

[Fifty]

We assume that --> takes on the truth table I provided. Without this assumption, the problem would be unsolvable.

That isn't really true (not only because truth tables are a recent invention!), but I think I get what you mean, so moving on...

A. We do not assume what --> means when the problem is stated, rendering it unsolvable
B. You solve the problem by (implicitly or explicitly) referencing the truth table for -->

You still aren't getting the heart of the issue. Suppose I gave the tortoises question to you, and you showed me a truth table. I ask, "why suppose that a truth table at that configuration yields the conclusion?" and you say, "well, look at that freaking box on the truth table!" Ok, so you seem to be asserting the proposition "If such-and-such configuration on a truth table yields a 'T', then that conclusion follows from the conjunction of the expressions in the top of the columns that you are using". So that's another hypothetical proposition to add, and so on and so on. The problem remains. Truth tables are just shortcuts for looking at all the possible truth conditions of the premises and conclusion, they aren't like some grand foundational thing. Frege, for instance, did pretty uch all of what we would call propositional logic without truth tables.

PS: School time, I'll get back to this later folks!

 

[warpus]

You still aren't getting the heart of the issue. Suppose I gave the tortoises question to you, and you showed me a truth table. I ask, "why suppose that a truth table at that configuration yields the conclusion?" and you say, "well, look at that freaking box on the truth table!" Ok, so you seem to be asserting the proposition "If such-and-such configuration on a truth table yields a 'T', then that conclusion follows from the conjunction of the expressions in the top of the columns that you are using". So that's another hypothetical proposition to add, and so on and so on.

That's not a hypothetical proposition though. Whatever logic symbol we use (be it -->, or AND, or OR, or XOR, or NAND, etc.) has a corresponding truth table that goes with it.

In this case --> is well defined, which is why I jumped on this right away with the truth table..

The problem remains. Truth tables are just shortcuts for looking at all the possible truth conditions of the premises and conclusion, they aren't like some grand foundational thing. Frege, for instance, did pretty uch all of what we would call propositional logic without truth tables.

Truth tables are just tools we can use to help us solve problems like these.
Are you asking where the values in these tables actually come from?

I don't see this as a problem at all.. It just seems to me like you could use this reasoning to recursively ask "well how do you know that?" until you end up asking: "Well, how do you know that the world really exists?" and "how do you know you can trust your senses?"

At some point you're going to have to make some assumptions, and by asking a logic question involving --> you assume that it comes with its truth table, one that is widely used and accepted. With this assumption, there is no 'problem'. Without it, the problem is not solvable.

If this is a problem, then asking any sort of question about anything is a problem.

 

[brennan]

Well, like I said, I don't think its a problem in the sense that it poses any serious dilemma with logic or whatever, but it does pose a problem for some people's foolish and naive view of logic!

Have you met a real-world example of this?

PS: I'm supposed to be on ignore!

You are, it's called 'view post'

I'd ask your awkward non-logician to explain how they can accept 1 and 2 but reject the conclusion. To do so rather requires you to not accept 2.

If A is true then B must be true.
A is true.
To reject 'B is true' then requires you to not accept 'then B must be true' which was part of you previous proposition.

(In the absence of any other variables I would consider that first proposition to be equivalent to (A->B)

 

[warpus]

Brennan, you could accept 1. and 2. while ignoring the meaning of -->

In that case, you might reject the conclusion that B is true.

 

[brennan]

Then you're not really accepting 2 are you?

 

[warpus]

Then you're not really accepting 2 are you?

In a sense you are. You can mistakenly assume the every day meaning of the word "implies", which is vague, and thus open to interpretation.

Which is likely what the issue here really is..

 

[Fugitive Sisyphus]

If someone doesn't comprehend simple logic it really isn't worth arguing with them.

 

[brennan]

If someone doesn't comprehend simple logic it really isn't worth arguing with them.

...the irony of that comment here in OffTopic is immense.

 

[GoodGame]

This is fundamental logical reasoning. The only reason it should fail is:

1. people are incapable of understanding logic (e.g. Venn diagrams, etc..).

2. people doubt the veracity of one or more of the premises.
e.g. They don't believe that A is true,
or they don't believe that A causes B.

Some people prefer emotional and intuitive reasoning to logical reasoning.
Not using logic doesn't guarantee that one will reach false conclusions, just that they will look like idiots to people well versed in logic!

So the problem of induction gets a lot of airtime, but there's a problem of deduction too! I'm curious about what people think about this, especially people who have lots of math education. Before I begin, you should be aware that Lewis Carroll put it much more eloquently and amusingly than I am going to.

How do you convince someone of the rationality of the following argument:

1) A
2) A --> B
Therefore B

<snip>

I mean logic people (at least ones who aren't very sophisticated about logic) hate words like intuition! It's all about rationality and proving and reason and premises and assertions and rules and yada yada yada. So what's going on?

 

[warpus]

That's why philosophy and formal logic don't mix

 

[Fifty]

Are you asking where the values in these tables actually come from?

No, I'm saying that a similar proposition can be made when the argument is put in truth-table form just as well as it can when it is put in premise-conclusion form.

Have you met a real-world example of this?

Oh yeah, all the time! It seems to be zooming right past warpus' head right now, for instance. More generally, though, it seems like you often encounter people who are opposed to answering a question by saying "well, it just seems obvious that it's right! I can't get more than that and I don't think I need to!" Such people often dismiss those answers as "illogical", yet I'm trying to show that logic rests on similar grounds.

You are, it's called 'view post'

Ahh cool, I never ignored anybody so I didn't know what it was like.

I'd ask your awkward non-logician to explain how they can accept 1 and 2 but reject the conclusion. To do so rather requires you to not accept 2.

If A is true then B must be true.
A is true.
To reject 'B is true' then requires you to not accept 'then B must be true' which was part of you previous proposition.

(In the absence of any other variables I would consider that first proposition to be equivalent to (A->B)

No it doesn't. There are three things operative in the argument: the premises, the rules of inference, and the conclusion. What that person is doing is accepting the premises, then wondering how those premises lead to that conclusion. It just seems like they are rejecting 2, because to us normal people it seems so fantastically thick-headed to accept 1 and 2 but reject 3.

If someone doesn't comprehend simple logic it really isn't worth arguing with them.

Yeah but that's not really the question.

...the irony of that comment here in OffTopic is immense.

I dunno, I think this sort of logic is extremely uncommon in OT, and so it doesn't really apply much. To be sure, you have this silly contingent of kids who've taken a few math classes and so run around parroting terms like "proof" and "assertion" and "logic" at everything that moves, but that doesn't mean that there's really some giant deficit of propositional logic understanding in OT.

1. people are incapable of understanding logic (e.g. Venn diagrams, etc..).

But the question is what does it mean to understand logic? Does it mean that you have to just sort of look at it and see that its obviously true? That is, do you have to just intuitively grasp that its true? If not, then what is going on?

2. people doubt the veracity of one or more of the premises.

Not in this example. The tortoise accepts the premises every time, what he rejects are the rules governing the leap from premises to conclusion.

That's why philosophy and formal logic don't mix

That's an ironic comment seeing as how thus far in the thread, the only person who seems to grasp what is going on is the dude with the philosophy education (Ayatollah). Philosophers have driven a good bulk of the advancements in logic.

 

[GoodGame]

Well my understanding of logical deduction is that it's on same level as geometric reasoning. It's one part arithmetic (summing up parts), and one part set theory (Venn diagrams). Logic is often separated from truth, since it's possible to have a completely logical chain of reasoning based on false premises, and thereby reach a false, yet logical conclusion.

Intuition doesn't figure into logic, except perhaps as the explanation for using a poorly defined, or unstated, assumption. Possibly some use of the word 'intuition' is perhaps as a codeword for, "I'm not going to explain my assumptions".

Psychology posits that there are intuitive and emotional forms of reasoning. My guess is that people who 'do not get logic' are actually using semi-conscious reasoning based on experience and a poorly characterized estimation of probabilities.

Logic is about trying to be cognizant about one's reasoning, even if it doesn't guarantee reaching true conclusions. So people who 'don't get it' aren't really trying to analyze their reasoning, they're just trying to reach a true conclusion.

But the question is what does it mean to understand logic? Does it mean that you have to just sort of look at it and see that its obviously true? That is, do you have to just intuitively grasp that its true? If not, then what is going on?