Blind Reasoning

[aleph_0]

Paul Boghossian has a paper on our method of using Modus Ponens titled Blind Reasoning (http://philosophy.fas.nyu.edu/docs/IO/1153...ndreasoning.pdf) and I'm wondering if anybody else has any ideas on it. It examines the paradox (?) that might have been introduced by Lewis Carroll, but which is all the same exemplified in his story What the Tortoise Said to Achilles. The point is this: Suppose you have two known facts, in the language of first-order logic, P and P --> Q. Now suppose that you had convinced someone that these two statements are true, but that person denies Q. Tragedy. You want him to accept Q, so you explain "Whenever you have any two statements of the form P and P --> Q, you can always derive Q!" And the person agrees--so he says, "Yes, I grant that (P & (P --> Q)) --> Q. I just deny Q." So you say, "But you've granted P and P --> Q. So you have the antecedent and so you must get the consequent." And again the person agrees, "Well yes, We have the antecedent P, and P --> Q, and that whenever you get these two you get the consequent. I agree with all that. So we can derive ((P & (P --> Q)) --> Q) --> Q. We're still lacking Q, silly man." In general, our list of proven statements beings:

1. P
   
2. P --> Q
   

And then,

1. P
   
2. P --> Q
   
3. (P & (P --> Q)) --> Q
   

And then,

1. P
   
2. P --> Q
   
3. (P & (P --> Q)) --> Q ~(~P v ~(~P v Q)) v Q
   
4. ((P & (P --> Q)) --> Q) --> Q
   

Obviously this can carry on through to infinity and we never actually get to Q. So a person could consistently refuse to believe in Q while accepting P and P --> Q. In fact, even if you stopped him and said, "Now wait a minute. You're denying Q. So we can also write ~Q. And whenever you write a sentence of the form P --> Q and another ~Q, you get ~P. So you're saying ~P and yet you started off agreeing to P. There's your contradiction!" And the person will deny that he ever assented to ~P. He will simply say, "Yes, whenever you have ~Q and P --> Q you get ~P. But I never gave you ~P. I simply agree that (~Q & (P --> Q)) --> ~P. ~(Q v (~P v Q)) v ~P And moreover, ((~Q & (P --> Q)) --> ~P) --> ~P, and so on." So the list of the derived is now

1. P
   
2. P --> Q
   
3. (P & (P --> Q)) --> Q ~(~P v ~(~P v Q)) v Q
   
4. ((P & (P --> Q)) --> Q) --> Q
   
5. ~Q
   
6. (~Q & (P --> Q)) --> ~P
   

You might notice that (P & (P --> Q)) --> Q and (~Q & (P --> Q)) --> ~P are contraries, but how are you ever going to prove it? You'd first have to get this problematic man to see that the former is equivalent to P and that the later is equivalent to ~P, and then prove that if two statements are equivalent and you can prove the first one, then you can write the second one.

This is different from denying an axiom. We take as an axiom in first-order logic P v ~P, no problem. This is a problem with induction rules. Axioms are just the building-blocks, induction rules are the glue. Induction rules are what allow you to get from one or more lines of valid statements to the next. Modus ponens is different from the axiom (P & (P --> Q)) --> Q.

Now Boghossian goes into this long treatise on Reliablism and Internalism, but there are a couple points to be made which may obviate the whole discussion. First, the whole conversation relies on modus ponens to begin with. If we ever get an answer to the problem, it will be of the form "If such-and-such is true then (such-and-such)_2 is true, and such-and-such is true, thus (such-and-such)_2 must be true." So we're assuming the ability to derive the antecedent all along. But this may not get us where we want. We're perfectly convinced of Q--what we have to show is that there is a rigorous way of using this troublesome man's very own standard of writing down derivations that will get him to write down Q.

I bet I can pull a contradiction out of this troublesome man already. But before I do, would that even count as victory? Let's assume I prove a contradiction. I say, "If you have a contradiction then you must be wrong," he will assent, "Yes, so (R & ~R) --> W. And you've proved me (R & ~R), I'm just not yet convinced of the W..."

So there must be one and only one thing that counts as victory. Get him to write Q.

So the first question has to be, "What is this standard of 'So we can write...'?" At some point, he's willing to write stuff that logically follows--every time he's in some specific situation, he's willing to write down an inference. If you cannot apply this rule--if you say, "Well, he'll just come back and say, 'Yes, we're in situation S and I will write T, so S --> T and S, but I'm just not going to give you T...' " then you've defined the problem into impossibility. If there is no particular circumstance in which he is going to write any new derivation, we must give up all hope for him. He's lost his mind. The challenge has become to predict the unpredictable.

I have more to write about this later, but I have to run. I'll post a follow-up later, but I'd be interested to hear responses to this so far.

[aleph_0]

I should probably, by request, explain the logical symbols '-->', 'v', '~', and '&'. '-->' simply means 'if... then...', so where you have the statements P and Q defined as {P: Socrates is a man; Q: Socrates is mortal}, you can write "If Socrates is a man then Socrates is mortal," as P --> Q. '~' means 'not', so "Socrates is not a man" is represented as '~P'. 'v' means 'or' so "Socrates is a man or Socrates is mortal," is represented by 'P v Q'. '&' means 'and', so "Socrates is a man and Socrates is mortal," is represented by 'P & Q'.

As a very interesting note, here is a quote from Crispin Wright:

I think there is one line which is worth some attention. This would have it that a belief is default-justified just in case any attempt to make out that it was unjustified would have to presuppose it. Consider, for example, my belief that I am capable of rational thought. Anything that I might do, for whatever reason, to try to make a case that I have no justification for this belief would have to involve some process of ratiocination. And whatever that process was, I oughtn't to attach any credibility to it unless I take it that I am capable of rational thought. So, in the sense proposed, I am default-justified in holding that I am capable of rational thought. I cannot consistently suppose that any doubt I might entertain about that is rationally grounded.

One might well hold out some hope for a case that modus ponens too can be default-justified in this kind of way--that any case against it would somehow have to presuppose it. Pay attention, though, to the species of 'justification; which is delivered in this kind of case: it is not that I get a justification for thinking, for instance, that, as a matter of real fact, I am capable of rational thought. It is merely that I cannot, self-credibly as it were, take myself to have contrived a doubt about it. In parallel, this kind of default-justification of modus ponens would merely bring out that scepticism about it was self-undermining. Why, though, should the fact that a rule is so deeply entrenched in our procedures of argument that anything we'd recognize as a case against it would have implicitly to rely on it,--why should this fact be supposed to have any tendency to show its objective validity? Default-justification of this kind doesn't seem to be what a defender of the objectivity of logic should be looking for.

My question is--what more could be asked for? If it is impossible for us to think outside of these terms by definition, then it is nonsense to be disappointed that we cannot think outside of these terms. For us to even talk about something being objective we must suppose that it is susceptible to discussion. By the objectivity of logic, we can only hope to show that logic applies to what is within our realm of understanding. To hope for more is as silly as to be disappointed that one must exert effort to do work, to think in order to know, etc.

Edited November 18, 2006 by aleph_0

[aleph_0]

... Damnit, they deleted the wrong post. I wanted this post deleted and wrote a new one in its place... Hmm...

[Cogito]

If we have P ^ P --> Q, then Q will be true no matter how many fools claim it isn't. If someone attempts to deny it, you cannot reason with him. You can, however, show him reality. (You are made of matter. All things made of matter are affected by gravity. You can deny that you are affected by gravity, yet you still cannot jump to the moon).

[DavidOdden]

I have two questions. First, why do you want this "->"? Second, do you find this to be a useful way of looking at reasoning, and if so, why?

[aleph_0]

Yes, but how do you prove, rigorously, that P and P --> Q forces you to implicitly accept Q?

[aleph_0]

I have two questions. First, why do you want this "->"? Second, do you find this to be a useful way of looking at reasoning, and if so, why?

I'm not sure what you mean by both questions. Do you mean, why symbolize logic? If so, it is for the same reason that you symbolize mathematics. You could simply write "one plus one is equal to two, the square-root of four is both two and negative two..." but this gets difficult to read as it gets more complicated. So philosophers and mathematicians have decided to symbolize these points for rigorous clarity by adopting the linguistic convention that true statements can be written on a list and derivation rules take one or more of these lines on the list to produce new entries on the list, e.g.,

P

P --> Q

(thus) Q

or for a more complex example,

P --> (~(Q & (~R v T)))

Q & ~R

(thus) ~P

But now that we have this symbolized form of representing arguments, one may wonder how we get from some lines to one particular new line rather than another. More pointedly, since the rule of derivation is expressed in English as, "Whenever you have P on one line and P --> Q on another, you can write Q," the person might say, "Well sure. But that doesn't mean we can yet write Q. It just means (P & (P --> Q)) --> Q." It's a difference in how we transcribe the English sentence into a symbolized sentence.

But more to the issue, how do we justify our derivation rule MPP? For if we know it in the way that we know the content of our sight, then we must wait for something to cause us to see it. You cannot simply choose to see something, but you can simply choose to "see" (or rather, consider) MPP. And so it must be more internal. Moreover, it must be non-inferential, for if it depends on an inference then it is not the most basic derivation rule and we should instead train our sights on the most basic rule. So the goal becomes a non-inferential yet "automatic" justification of MPP.

Edited November 19, 2006 by aleph_0

[DavidOdden]

I'm not sure what you mean by both questions. Do you mean, why symbolize logic? If so, it is for the same reason that you symbolize mathematics.

I don't know how that would apply to the "->" question. Can you show me a reason why I need this "->" thing?

As for the question of symbolizing logic, okay, I have no problem with it, but now that I've gotten logic symbolized, what is the problem that you're trying to solve? For example, are you really asking "Do we need rules of inference?". The answer is "No", as Kleene shows in his textbook. Or, as you asking how you might persuade an irrational person to accept logic? That is a question about persuasion, not about logic.

I think that the "you can write" approach is not valid, i.e. the approach that says you can introduce any arbitrary statement you want. Properly done, you can't get him to accept "P->Q" without first accepting "Q". If he's gonna accept Q one moment and deny Q another, then I see no hope of using reason with this person.

[aleph_0]

I don't know how that would apply to the "->" question. Can you show me a reason why I need this "->" thing?

Like I said, I don't really know what you mean by the question. And like I said, you could do away with it and use the English version of "if... then..." though that tends to be less formally clear and less easily wielded in logical proofs. If it is the notation that has you worried/confused, think of the material implication of the "horseshoe", or reduce it to the Normal Form of a negated branch of an appropriate disjunction.

For example, are you really asking "Do we need rules of inference?". The answer is "No", as Kleene shows in his textbook.

Interesting, how does he show this? Need for what? As far as I know, axiom schemata cannot generate sufficient proofs as MPP can.

Or, as you asking how you might persuade an irrational person to accept logic? That is a question about persuasion, not about logic.

Not [necessarily] an irrational person--a person wholly willing to accept the very letter of what you say, but who does not come to your conclusion by admittedly valid methods of argument (for, indeed, we would agree that (P & (P --> Q)) --> ((P & (P --> Q)) --> Q, which is the new line written by our hypothesized recalcitrant friend). How do you show that the derivation rule requires Q?

I think that the "you can write" approach is not valid, i.e. the approach that says you can introduce any arbitrary statement you want. Properly done, you can't get him to accept "P->Q" without first accepting "Q".

The point is to suppose that he already accepts P and P --> Q and work from there.

But really, this is all side-line talk. I had wanted to delete this post and start a new one with a better focus, but the mod deleted the wrong post (the new and shinier one with a better focus) and left this old one. I've asked for the old one back, but I fear it may be gone for good and I'm not sure I want to write it out in whole, so I may just start a new one with relatively little content that I can just build off of in discussion.

[DavidOdden]

Like I said, I don't really know what you mean by the question.

Well, for example, is it your position that "->" is a logical primitive? Sticking just with symbolic logic, is there a reason why I have to have it?

or reduce it to the Normal Form of a negated branch of an appropriate disjunction.

For example. Don't you think that makes the problem go away?

Interesting, how does he show this?

Well, my copy is a few thousand miles away; I don't remember the axioms, or even whether he shows that the method is equivalent to the "lines of proof" deductive method.

The point is to suppose that he already accepts P and P --> Q and work from there.

That's the central point: he should not just accept "P->Q". Anybody who will accept "P->Q" for no reason is not being logical. So when I asked whether you think this (the arbitrary declarations and proof-lines way of "reasoning") is useful, I was referring to this kind of problem. I don't think that Kleene's axioms correspond to anything real in reasoning, and yet he does cleverly come up with condensed axiomatic system. As a mental exercise, a form of method-gymnastics, it's not immoral or anything; so I was just curious whether you thought this has some relationship to reasoning.

[KendallJ]

How do you show that the derivation rule requires Q?

Uh, this is called defining the rules of deduction and induction. The "->" thingy, means "results in Q being true". Well someone has to go out and develop the operators by which this works.

In the case of 2+2=4, this is true as a mathmatical statement once addition has been defined and shown to always lead to a correct result.

In the case of deduction, this is the laws of logic. Aristotle put a very nice start on this. Distinguish the function of the logic (i.e. it's validity) from the truth of the statements (truth). If statements are true, and the logic is valid, then the conclusion MUST be true. This is the purpose of the rules of logic.

All men are mortal [A]

Socrates is a man

therefore,

Socrates is mortal [C]

1. A is true

2. B is true

3. the logic (i.e. the attribution of general characteristics of a group to a particular member of that group) is valid

4. so C MUST be true.

If you "accept" 1,2, and 3, then denying 4 is irractional, as is asking why it is so that it must be true.

In the case of deduction, well that is a little trickier because no one has articulated the rules of induction as well as deduction. The scientific method is one subset of induction.

Edited November 19, 2006 by KendallJ

[aleph_0]

Okay, I think I understand your question now, and yes we take any complete theory of propositional logic--for instance, the axiom schemata:

(taken from Wikipedia's article on axioms)

With this, then assume some P and some P --> Q and modus ponens which gives you that, of any form P and P --> Q then you can derive Q. Well, strictly speaking, for such a basic inference you don't even need the axioms--the assumptions and MPP get you where you want to go anyway. I don't see how reducing this to Normal Form would make this go away, though, since it would just change your symbols and not the content of the argument. It would become P and ~P v Q thus Q. But the recalcitrant friend would say, "No, no. MPP just states that, whenever you have P and ~P v Q then you can write ~(~P v ~(~P v Q))) v Q. But we never get Q."

As for how we get P --> Q, you might as well ask how we get P. We just assume they are some justified beliefs. Boghossian uses the example of a man who is trying to learn whether the opera is tonight. He knows that the date is January 10th (say he looked at a newspaper he bought that day and trusts newspapers, provided no extreme circumstances), and he knows that if it is January 10th then the opera is performed that night because he read the brochure. So we take his premises to be justified and justified independent of any conclusion he might reach with them. So we have a P and an P --> Q. Now we want to know whether this man should reason that Q.

Well, my copy is a few thousand miles away; I don't remember the axioms, or even whether he shows that the method is equivalent to the "lines of proof" deductive method.

Since my copy is at least equally inaccessible--namely, since I don't have a copy --we'll have to ignore this point for now.

s a mental exercise, a form of method-gymnastics, it's not immoral or anything; so I was just curious whether you thought this has some relationship to reasoning.

I do think it correlates to reasoning in the above way. It is a formalized representation of proper inferences, perfectly equivalent to and more rigorously clear than English expressions of reason.

But I am now in the process of re-writing a better topic.

[Capitalism Forever]

The meaning of P --> Q is a proposition with the following truth table:

P   Q   P --> Q

---------------

T   T	  T

T   F	  F

F   T	  T

F   F	  T[/code]

There is no line in this truth table that has T for P, F for Q, and T for P --> Q. This proves that it is contradictory to say that P and P --> Q are true while Q is false.

[y_feldblum]

I think DavidOdden's question is about what is the underlying basis of logic, and what concepts of logic does this basis necessitate?

Where there is a supernatural or mystical or higher plane of existence which emanates truth, and where abstract knowledge comes from the direct intellectual perception of floating abstractions, logic is concerned with deducing from these floating abstractions yet further floating abstractions. This art of integration requires floating-abstraction rules of logic, which have no basis themselves but are taken to be the basis. Rules such as P & (P --> Q) --> Q are taken as the basis for logic and are not to be questioned, or understood by means of looking at reality. Logic is reduced to mental gymnastics with no basis in or application to concrete facts.

Where reality is reality, where entities have identity and act causally, and where the mind must observe and integrate the facts of reality to form abstract knowledge, logic is concerned with integration based on identity and causality, and every concept and rule of logic must be understood with reference to entities and to the laws of identity and causality. P and Q must be understood as describing certain properties of an entity, and P --> Q must be understood as describing a causal relation between these properties. Then upholding P and also P --> Q, while denying Q, becomes upholding a contradiction in reality, and entails violating A is A.

However, it is very important to realize that logic is a far, far broader art than just deducing abstractions from abstractions, which is what formal symbolized logic describes. Logic is concerned with integrating concretes into abstractions and applying these abstractions to concretes, most of which formal symbolized logic does not touch. Taking formal symbolized logic as a basic foundation for all of logic is a mistake: it is a derivative, and its validity rests on it being put in its proper hierarchical place with the rest of logic, and not being taken to be a floating primary.

To the OP, the first thing you want to do is identify the entities or the properties which P and Q mean, and identify the causal relation which P --> Q means. And then, when the law of identity is both asserted and denied at the same time and in the same respect (by upholding P and P --> Q, but denying Q), walk away from the conversation.

[DavidOdden]

I think DavidOdden's question is about what is the underlying basis of logic, and what concepts of logic does this basis necessitate?

Entirely correct, and I fully endorse your reply. I want to highlight the statement "P --> Q must be understood as describing a causal relation between these properties". This is the only sense in which "->" is meaningful. Furthermore, causal relations are not perceptual axioms, they must be induced. The question that should be asked is, how does one validly induce a causal relationship?

[Capitalism Forever]

To the OP, the first thing you want to do is identify the entities or the properties which P and Q mean, and identify the causal relation which P --> Q means.

Entirely correct, and I fully endorse your reply. I want to highlight the statement "P --> Q must be understood as describing a causal relation between these properties". This is the only sense in which "->" is meaningful.

Sorry folks, but this is totally wrong. Implication does not imply causation, just like correlation does not. It is true that "If there are drops of water on the window, it's raining," but it does not mean that "Because there are drops of water on the window, it is raining." (In fact, it's the other way around in this case!)

[DavidOdden]

Implication does not imply causation, just like correlation does not.

The part that you didn't seem to get is that a statement of the form "P->Q" is invalid unless it states a causal relationship. Please assume that I have enough knowledge of logic and language that I would not simply replace the arrow with the word "causes".

I'm curious whether you really hold that it is true that "If there are drops of water on the window, it's raining". I would simply have point to that as an invalid relationship that can be expressed by ramming sentences together.

[Capitalism Forever]

The part that you didn't seem to get is that a statement of the form "P->Q" is invalid unless it states a causal relationship.

I got that, and that's what you got wrong. Let me give you another example: "If people are using their umbrellas, there are drops of water on the window." Do you think that's an invalid statement? If yes, why? If not, does it mean that the use of umbrellas causes the drops of water on the window, or that the drops of water on the window cause the use of umbrellas?

One way to look at implication is as a 100% correlation. And as any good statistics textbook will tell you, correlation does not necessarily mean causation. If A is correlated with B, it could mean that A causes B, or that B causes A, or that some third factor C causes both A and B, or that it's a mere coincidence.

I'm curious whether you really hold that it is true that "If there are drops of water on the window, it's raining".

It's not exactly a statement of scientific precision, but I was trying to be concise rather than precise.

[y_feldblum]

Then what does P --> Q, "wet umbrellas --> wet windows", mean? What is the fact of reality to which it refers, or what is the sum of the facts which it unifies? How, and why, is this abstraction initially formed?

Edited November 19, 2006 by y_feldblum

[aleph_0]

The meaning of P --> Q is a proposition with the following truth table:

P   Q   P --> Q

---------------

T   T	  T

T   F	  F

F   T	  T

F   F	  T[/code]

There is no line in this truth table that has T for P, F for Q, and T for P --> Q. This proves that it is contradictory to say that P and P --> Q are true while Q is false.


Easily enough, our recalcitrant friend could deny this truth table.  He could say that it's possible for P --> Q to be true while P is true and Q false.  I agree, this runs into contradiction (I can actually prove it by a more complicated but direct way not using truth tables at all), but can you prove that this person must accept Q?  That is to say, if he accepts P and P --> Q, and MPP, then he [i]has to[/i] accept Q?  Let's assume he accepts a contradiction on top of all of this.  We still don't get Q even though he has accepted everything he's supposed to need in order to get Q, and then some (namely, a contradiction).


And if you tell him that, whenever you get a contradiction you must deny your antecedent, he will say, "Sure, sure.  And you've proved a contradiction.  So we get ~Q --> (P & ~P) (really, this should be in Greek letters to indicate axiom schemata and derivation rules, but the point need not be pressed too hard), and we know that if ever you derive a contradiction then you must deny the antecedent, so (~P & P) --> Q.  But we still never get Q."


I think DavidOdden's question is about what is the underlying basis of logic, and what concepts of logic does this basis necessitate?



Where there is a supernatural or mystical or higher plane of existence which emanates truth, and where abstract knowledge comes from the direct intellectual perception of floating abstractions, logic is concerned with deducing from these floating abstractions yet further floating abstractions. This art of integration requires floating-abstraction rules of logic, which have no basis themselves but are taken to be the basis. Rules such as P & (P --> Q) --> Q are taken as the basis for logic and are not to be questioned, or understood by means of looking at reality. Logic is reduced to mental gymnastics with no basis in or application to concrete facts.


So if I put in content for P and Q, there might be some situation for which you would assert P and deny Q?  The same "mental gymnastics" is performed in math, where we abstract from any particular number and represent it by an x.  Do you deny the basis of mathematics?


... entails violating A is A.
How are you not performing mental gymnastics by using this floating abstraction, "A is A"?


The part that you didn't seem to get is that a statement of the form "P->Q" is invalid [i]unless[/i] it states a causal relationship. Please assume that I have enough knowledge of logic and language that I would not simply replace the arrow with the word "causes".



I'm curious whether you really hold that it is true that "If there are drops of water on the window, it's raining". I would simply have point to that as an invalid relationship that can be expressed by ramming sentences together.


How about the true collection of statements, "Socrates is a human, and if Socrates is a human then Socrates is a mammal"?

[DavidOdden]

Let me give you another example: "If people are using their umbrellas, there are drops of water on the window." Do you think that's an invalid statement? If yes, why?

It is invalid, since it does not state a truth. It is simply not that case that "Umbrellas -> windows" corresponds to reality, even statistically. Very often, people may be using umbrellas, perhaps because there is a very light rain which doesn't get on the windows, or perhaps there is no rain at all and they are using the umbrella to keep the fierce sun off of their heads. Happens a lot on the golfing circuit.

We have been discussing the technical concept "logical implication", symbolized as horshoe or "->" which has the truth table that you gave. I would not deny that there are different but related concepts of "implication" that exist, but what the formal thing "->" expresses is causation (or domain restriction but let's put that aside for a moment because I think that can subsumed under causation). I think what you are focusing on is the meaning of "if", which includes a lot of things that aren't "->". I certainly don't hold that "if" means "causes".

[DavidOdden]

How about the true collection of statements, "Socrates is a human, and if Socrates is a human then Socrates is a mammal"?

The regular version of that goes "All men are mortals; Socrates is a man; therefore Socrates is a mortal". The answer, in that case, is that this is one convention that allows quantifiers to be integrated into formal logic. However, the generalized quantifier theory notion of domain restriction does as well, without using the horseshoe connective. That's basically the notion of "context" formalized. The reason why "Ax(Mortal(x))" doesn't work is that it says "everthing is mortal", not just men, but everything. Thus the horseshoe was introduced to allow quantifiers into formal logic. But with domain restriction, i.e. saying "In the context of 'man'", the horsehoe-free statement "Ax(Mortal(x))" does work. Now I don't know why you deviated from the standard example, but hopefully that will give you enough of an answer that you can clarify your question so that I can see exactly what you are focusing on.

[aleph_0]

Sorry, I seem to have skipped this post.

In the case of 2+2=4, this is true as a mathmatical statement once addition has been defined and shown to always lead to a correct result.

What would constitute being "shown to always lead to a correct result"? Take one object and move another up to it so that you have both in your visual field, and say, "Ah, that's two from one and one." And do this same process for 2 + 1, 1+ 2, 2 + 2, 2 + 3, 3 + 2, 1 + 3, 3 + 1, ...? And then, I suppose, you'd have to do the same thing with multiplication, exponentiation, factorialization, logarithms, etc.? Or am I reading you wrongly?

If you "accept" 1,2, and 3, then denying 4 is irractional, as is asking why it is so that it must be true.

Again, the point is to show, rigorously, that it is indeed irrational. What requires (4)? By (3) we could instead write the following in the place of (4): (1)--and if (1) then (2)--thus (3). But we don't actually give three. Just that whenever you have (1) and (1) --> (2) then (3).

The regular version of that goes "All men are mortals; Socrates is a man; therefore Socrates is a mortal".

I know, I just figured I'd use an example that was even more undeniable.

The answer, in that case, is that this is one convention that allows quantifiers to be integrated into formal logic.

I know, I just assumed keep it simple for people who hadn't studied predicate calculus. This collection of statements can just as well be represented in propositional logic, it's only by getting into "for all" and "there is" cases that it falls apart, so I've just avoided those cases.

The reason why "Ax(Mortal(x))" doesn't work is that it says "everthing is mortal", not just men, but everything.

Usually, when one has no access to special symbols, "for all" is represented by a simple '(x)' and "there is" is represented '~(x)~'.

Now I don't know why you deviated from the standard example, but hopefully that will give you enough of an answer that you can clarify your question so that I can see exactly what you are focusing on.

Nope.

[Capitalism Forever]

Then what does P --> Q, "wet umbrellas --> wet windows", mean?

It means that wet umbrellas imply wet windows; in other words, that the umbrellas cannot be wet without the windows also being wet.

What is the fact of reality to which it refers, or what is the sum of the facts which it unifies?

It refers to a correlation between the wetness of the umbrellas and the wetness of the windows. Correlation does not equal causation.

How, and why, is this abstraction initially formed?

The same way and the same reason concepts like "and" and "or" are formed. In fact, "if" is nothing more than just another conjunction: "If P, then Q" is a synonym of "Not P, or Q."

[Capitalism Forever]

Easily enough, our recalcitrant friend could deny this truth table. He could say that it's possible for P --> Q to be true while P is true and Q false.

Then what does he think P --> Q means? If he accepts P --> Q as a valid sentence, he must attach some meaning to it; some fact of reality it refers to. (Meaningless sequences of symbols are not valid propositions.) And the correct meaning of P --> Q is the truth table I provided. There is no way around that!

Let's assume he accepts a contradiction on top of all of this.

Then he is convinced! Although I know that he is not convinced, since A is not A, he is also convinced.

And if you tell him that, whenever you get a contradiction you must deny your antecedent, he will say, "Sure, sure. And you've proved a contradiction. So we get ~Q --> (P & ~P) (really, this should be in Greek letters to indicate axiom schemata and derivation rules, but the point need not be pressed too hard), and we know that if ever you derive a contradiction then you must deny the antecedent, so (~P & P) --> Q. But we still never get Q."

Well, (~P & P) also implies ~Q (as a false statement implies any statement), so that is not really the way to go about it. If I say "...you must deny your antecedent" and he accepts that, he has just accepted that he must deny Q.

May I ask why you are interested in this issue? Are you yourself unsure of the validity of the use of implication in reasoning? Or do you hope to derive some benefit to your life from this? I just don't see why anyone would want to spend any of his time on such a question.