The essence of logic is to find out what argumentative structures

[quantumdude]

This will be the new version of my "Logic" thread in PF v2.0. I'll get my logic notes pasted into this forum ASAP, along with some of the more useful discussion from the old thread.

 

[Kerrie]

i did not read through every single post of your last thread on logic tom, however, it would seem "right" to me that logic is subjective, at least when i am referring to the reasoning and rationalizing form of logic...

 

[ahrkron]

I disagree. In a sense, the essence of logic is to find out what argumentative structures can definitely be "trusted" regardless of content, so that arguments can be analysed independently of how you "feel" about the conclusion they seem to produce.

 

[Kerrie]

ahrkron...

after plugging in "is logic subjective", i came up with this link...

http://www.angelfire.com/md2/timewarp/logic.html

any comments? i am more interested in understanding logic completely...

 

[quantumcarl]

Originally posted by ahrkron
I disagree. In a sense, the essence of logic is to find out what argumentative structures can definitely be "trusted" regardless of content, so that arguments can be analysed independently of how you "feel" about the conclusion they seem to produce.

We have learned logic from the way the universe works. We are mimicing what we see in the sequence of events with which the universe unfolds.

We have called logical sequence "logical sequence" in an attempt to harness the incredible logic witnessed in the structure and efficency of the universe.

Thats what I think about logic.

 

[damgo]

"I know what you're thinking about," said Tweedledum: "but it isn't so, nohow."
"Contrariwise," continued Tweedledee, "if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic."

 

[quantumdude]

Hey folks, glad to see you all talking. My notes are coming up, slowly but surely, here:

https://www.physicsforums.com/showthread.php?s=&threadid=111

It is taking me a little while to translate all the color and smiley brackets to the new forum.

The more meaningful discussion in the PF v2.0 version of this thread centered on...

1. The Principle of Charity
This is an admonishment to make another person's argument as good as possible. This means, first and foremost, try to make the argument deductively valid whenever possible. If not possible, then try to make the argument a strong inductive argument. A strong inductive argument is preferable to a valid deductive argument with questionable premises.

Why do all this? Because the whole point of debate is to learn, not to be agreed with. If you make your opponent's argument as good as possible and, in the process, discover that he is correct, then you have learned something. If, on the other hand, you discover that his best argument is fallacious, then again, you have learned something.

2. The Difference Between Deductive and Inductive Arguments.
This was done via exercises that I posted. Audacity Dan posted his solutions, but unfortunately I did not copy them.


I'll put the exercises back up shortly, along with the rest of my notes.

 

[Les Sleeth]

Originally posted by quantumcarl
We have learned logic from the way the universe works. We are mimicing what we see in the sequence of events with which the universe unfolds.

We have called logical sequence "logical sequence" in an attempt to harness the incredible logic witnessed in the structure and efficency of the universe.

Thats what I think about logic.

Excellent. I agree logic "works" because it reflects the order/symetry of the universe. It is the same quality that allows math. It works with everything . . . except that which isn't within the boundaries of order and symetry.

By the way, what turned you quantum? Photon bombardment? Blackbody abuse? H??

 

[wuliheron]

In a sense, the essence of logic is to find out what argumentative structures can definitely be "trusted" regardless of content, so that arguments can be analysed independently of how you "feel" about the conclusion they seem to produce.

Like anything else, logic depends upon context. For example, the liars paradox:

"Everything I say is a lie."

Makes perfect sense if everyone who hears it knows you happen to be a chronic liar. Strictly logically speaking, however, it makes no sense whatsoever. Thus, even whether or not we should use of logic depends upon the context. This also applies to how you "feel" about the conclusions of logic. The context can over-rides the content of logic and, in fact, whether something is consider a content or context just depends on how you want to look at it.

Therefore the essense of logic is not so much to find out what argumentative structures can be trusted, but more how it fits into the various contexts life presents us. That is of course not to downgrade the incredible usefulness of logic for making sure your check book is balanced or whatever. Just to put it in perspective.

 

[quantumcarl]

Originally posted by Tom
Hey folks, glad to see you all talking. My notes are coming up, slowly but surely, here:

https://www.physicsforums.com/showthread.php?s=&threadid=111

It is taking me a little while to translate all the color and smiley brackets to the new forum.

The more meaningful discussion in the PF v2.0 version of this thread centered on...

1. The Principle of Charity
This is an admonishment to make another person's argument as good as possible. This means, first and foremost, try to make the argument deductively valid whenever possible. If not possible, then try to make the argument a strong inductive argument. A strong inductive argument is preferable to a valid deductive argument with questionable premises.

Why do all this? Because the whole point of debate is to learn, not to be agreed with. If you make your opponent's argument as good as possible and, in the process, discover that he is correct, then you have learned something. If, on the other hand, you discover that his best argument is fallacious, then again, you have learned something.

2. The Difference Between Deductive and Inductive Arguments.
This was done via exercises that I posted. Audacity Dan posted his solutions, but unfortunately I did not copy them.


I'll put the exercises back up shortly, along with the rest of my notes.

Tom, that is a good reading! I really like the slow going through definitions. I think its so important to stop and define every word we use before continuing with a discussion so that there is a common ground for the participants to work. Its (yes) very logical.

I am still convinced that if a logic does not reflect, in every detail, a law or a state that exists in nature, then it is faulty logic and will not stand up to the rigors of time or scrutiny.

Here we are, members of an intricate existence, witnessing the precision and the efficiency of nature supporting our being. How can we not have learned logic from the lessons in nature that we study.

The more we study nature, the more complex and in depth our reasoning becomes and the more stable our logic.

 

[CJames]

Nice to see the most important thread for the philosophy forum reinstated. I love the new look.

 

[quantumdude]

I've finished posting my Chapter 0 notes. I'll repost the exercises in this thread, in case anyone wants to take a crack at them. My notes for Chapter 1 are also finished, but I'll give everyone a little time to catch up first. By staying a chapter ahead of you, I hope the progression won't be as jerky this time out.

https://www.physicsforums.com/showthread.php?s=&threadid=111

 

[quantumdude]

In PF v2.0, I posted some exercises to reinforce the material covered in the notes. Audacity Dan (now Dissident Dan) posted his solutions to the first set. If you would like to see them, then cough up the $20 for the archive CD.

Here are some exercises that cover Chapter 0 pretty comprehensively.

Disproof by Counterexample
Each of the following deductive arguments is invalid. Provide a counterexample for each.

Argument 1:
All paramecia are single-celled organisms.
No sea urchins are paramecia.
Therefore, no sea urchins are single-celled organisms.

Argument 2:
Some Englishmen are Protestants.
Winston Churchill was a Protestant.
Therefore, Winston Churchill was an Englishman.

Argument 3:
If an animal is a mammal, then it bears its young live.
A gorilla bears its young live.
Therefore, a gorilla is a mammal.

Argument 4:
Some dogs are good pets.
Some dogs are terriers.
Therefore, some terriers are good pets.

The above exercise set sparked some interesting conversation in PF v2.0. The noteworthy thing about these arguments is that, although they are invalid, every statement in them is true![/color] This highlights the necessity of logical rigor.

Deductive or Inductive?
Examine each argument below. Is the argument deductive or inductive? Explain.

Argument 1:
All human choices are determined, since all events in the universe are determined and all human choices are events in the universe.

Argument 2:
All birds can fly. I’ve never seen one that can’t.

Argument 3:
Today is Wednesday. You came 4 days ago, so that means you came on Saturday.

Argument 4:
I sent her the letter 3 weeks ago and have still received no answer; therefore, my letter must have been lost in the mail.

Argument 5:
A=B and B=C, therefore A=C.

Argument Analysis and Charity
Assuming ordinary context, examine each of the following arguments. Identify the conclusion and the premises, and supply a missing premise that would make the argument deductively valid.

Argument 1:
Bats are not birds, because birds have feathers.

Argument 2:
The baseball game was dull, since both teams played poorly.

Argument 3:
This liquid is not acid, for the litmus paper we placed in it did not turn red.
Argument 4:
He passed the examination; therefore, he must have lied.

My first set of notes for Chapter 1: Categorical Statements is now up.

 

[CJames]

Argument 1:
All paramecia are single-celled organisms.
No sea urchins are paramecia.
Therefore, no sea urchins are single-celled organisms.

All cattle are animals.
No cats are cattle.
Therefore, no cats are animals.

Argument 2:
Some Englishmen are Protestants.
Winston Churchill was a Protestant.
Therefore, Winston Churchill was an Englishman.

Some people use logic.
Lifegazer is a person.
Therefore, Lifegazer uses logic...I couldn't help it. I'm so rude. That was uncalled for.

Argument 3:
If an animal is a mammal, then it bears its young live.
A gorilla bears its young live.
Therefore, a gorilla is a mammal.

If a lifeform is a plant, then it reproduces.
A human reproduces.
Therefore, a human is a plant.

Argument 4:
Some dogs are good pets.
Some dogs are terriers.
Therefore, some terriers are good pets.

Some oranges are rotten oranges.
Some oranges taste good.
Therefore, some rotten oranges taste good. (Granted, this is a tad bit subjective but...)

Take care. --Carter

 

[quantumdude]

Originally posted by CJames
Argument 2:
Some Englishmen are Protestants.
Winston Churchill was a Protestant.
Therefore, Winston Churchill was an Englishman.

Some people use logic.
Lifegazer is a person.
Therefore, Lifegazer uses logic...I couldn't help it. I'm so rude. That was uncalled for.

LOL

Alas, in your zeal, you got this one wrong. The predicate term of the first two premises ("people who use logic") should be the same, as it was in the original argument, whose schema is:

Some p are q.
r is a q.
Therefore, r is a p.

Otherwise, good job.

Tom

 

[Lifegazer]

That guy was voted Britain's greatest hero. So thanks CJ.

 

[quantumdude]

Kerrie, could you split off the "Quantum Mechanics vs. Logic" posts into a separate thread? (It starts with heusdens' first post).

Thanks,

 

[quantumdude]

Back to the topic...

I will post more of the Chapter 1 notes tomorrow. I was holding off on it because it is about Venn diagrams. Since I cannot make those in this forum, I thought I had to skip it and go directly to the part on immediate inferences.

Then I found this website:
http://www.venndiagram.com

Neato.

 

[kyleb]

Originally posted by Kerrie
i did not read through every single post of your last thread on logic tom, however, it would seem "right" to me that logic is subjective, at least when i am referring to the reasoning and rationalizing form of logic...

that is not an easy one to explain but i will give it a try, hopefully most people will agree with this explanation:

ideally, logic is objective; however, our perceptions are bound to be subjective due to our nature as individuals. so in practice, logic is objectivity used as a tool to lessen the effects of our inherent subjectivity.


does that seem logical to everyone?

 

[quantumcarl]

oops

I seem to have left this topic in the duff.

Sorry about that.

When a person feels bad about leaving a topic in the duff they are sorry.

When a treeplanter leaves a seedling behind they leave it in the duff.

There for, treeplanters are sorry by up to 600 times a day.

Tom, this is good learning...
must find time...
must learn...
I learnink...
must spend more time learning...
must read logic topic more often...

 

[wuliheron]

Logic is the science of the absurd.

 

[quantumdude]

Last week (I think) I posted the first two sections of Chapter 1: Categorical Statements. The reading is a bit dry, but it really is necessary to learn what the building blocks of syllogisms are before looking at their structure.

Before I move on to the next part on immediate inferences, let me give some exercises on what has been presented so far.

Indicate whether each of the following sentences expresses an A, E, I, or O statement. When necessary translate the sentence into standard form. Indicate whether any meaning is lost in the translation. Write an abbreviation for each sentence, indicating which term each letter represents. Also give the schema for each statement.

1. Lassie is not a cocker spaniel.
2. Most records cost less than five dollars.
3. Sixty percent of all college students work part-time to pay for their education.
4. Almost all professional basketball players are over six feet four inches tall.
5. All politicians are not dishonest.
6. War is not healthy for children and other living things.
7. Only those who bought tickets in advance were able to get seats.

I'll get the next set of notes up tomorrow.

 

[quantumdude]

The next set of notes is up. It's not much, but that's because I am still writing the proofs of the propositions that appear in the next section (Section 4: Immediate Inferences on the Aristotelian Interpretation).

In the mean time, you might try diagramming a few categorical statements from the previous exercise set.

 

[quantumdude]

The next set of notes is up, and it is the first installment of the notes on immediate inferrences. As explained in detail in the notes, immediate inferrences are conclusions drawn from exactly one premise. As one would expect, there are definite laws prescribing which ones are valid and which are not. It is to those laws that we now turn.

As practice, I advise all interested parties to complete the following exercise:


1. Draw the Venn diagrams for each of the 4 types of categorical statement.
2. Use the diagrams to prove Propositions 1.7, 1.8, 1.10, and 1.12 in the notes.


I will post the proofs after a few days. It will be a real help to your learning to try them in the meantime (they aren't very hard).

 

[Royce]

Tom thanks again for the link.
I'm posting a reply to see if I can move this thread up to the current page. Right now the only way I have found to get here is through the link you gave me in you PM to me.

 

[Mentat]

Originally posted by Tom
In PF v2.0, I posted some exercises to reinforce the material covered in the notes. Audacity Dan (now Dissident Dan) posted his solutions to the first set. If you would like to see them, then cough up the $20 for the archive CD.

Here are some exercises that cover Chapter 0 pretty comprehensively.

Disproof by Counterexample
Each of the following deductive arguments is invalid. Provide a counterexample for each.

Argument 1:
All paramecia are single-celled organisms.
No sea urchins are paramecia.
Therefore, no sea urchins are single-celled organisms.

All humans are multi-cellular.
No dogs are humans.
Therefore, no dogs are multi-cellular.

Argument 2:
Some Englishmen are Protestants.
Winston Churchill was a Protestant.
Therefore, Winston Churchill was an Englishman.

Some PF members are atheists.
Futurist was a PF member.
Therefore, Futurist was an atheist (this one is for those of you who have been here for a while).

Argument 3:
If an animal is a mammal, then it bears its young live.
A gorilla bears its young live.
Therefore, a gorilla is a mammal.

If an animal is a bird, then it lays eggs.
An Iguana lays eggs.
Therefore an Iguana is a bird.

Argument 4:
Some dogs are good pets.
Some dogs are terriers.
Therefore, some terriers are good pets.

Some fish are fearsome predators.
Some fish are goldfish.
Therefore, some goldfish are fearsome predators.

Deductive or Inductive?
Examine each argument below. Is the argument deductive or inductive? Explain.

Argument 1:
All human choices are determined, since all events in the universe are determined and all human choices are events in the universe.

Deductive. You are taking two propositions (that all events in the Universe are determined, and that all human choices are events in the Universe) and finding the proposition that logically follows.

Argument 2:
All birds can fly. I’ve never seen one that can’t.

Inductive. You are reasoning only on an observed pattern.

Argument 3:
Today is Wednesday. You came 4 days ago, so that means you came on Saturday.

Deductive. You are again taking two propositions (that I came 4 days ago, and that today is Wednesday), and finding the proposition that logically follows.

Argument 4:
I sent her the letter 3 weeks ago and have still received no answer; therefore, my letter must have been lost in the mail.

Hmm. Inductive, I guess - though there this is not really a logical conclusion, but just one of the possibilities.

Argument 5:
A=B and B=C, therefore A=C.

Deductive. This is a mathematical approach. Plus, you have taken two premises, that must always be true, and found the proposition that logically follows.

Argument Analysis and Charity
Assuming ordinary context, examine each of the following arguments. Identify the conclusion and the premises, and supply a missing premise that would make the argument deductively valid.

Argument 1:
Bats are not birds, because birds have feathers.

Propositions:
1) Birds have feathers.

3) Bats are not birds.


Missing proposition:
2) Bats don't have feathers.

Argument 2:
The baseball game was dull, since both teams played poorly.

Propositions:
1) The game was dull.

3) Both teams played poorly.

Missing proposition:
2) Games, in which both teams play poorly, are dull.

Argument 3:
This liquid is not acid, for the litmus paper we placed in it did not turn red.

Propositions:
1) This liquid is not acid.

3) The litmus paper didn't turn read.

Missing proposition:
2) When litmus paper is introduced to an acid, it turns red.

Argument 4:
He passed the examination; therefore, he must have lied.

1) He passed the exam.

3) He must have lied.

Missing Proposition:
2) He cannot pass without lying.


These are probably all wrong, but it was fun trying.

 

[quantumdude]

Ah, thanks for the memories Royce. I now remember why this thread dropped so far down the list: I was waiting for someone to try the exercises to make sure that I wasn't just talking to myself. LOL

Thanks Mentat for "keeping me company".

Originally posted by Mentat
These are probably all wrong, but it was fun trying. [/B]

No, they are all correct.

 

[Mentat]

I'll try the next exam later, but I have a question for Tom. Does the book ever touch on the fact that Inductive Logic shows Deductive Logic to be paradoxical, and that Deductive Logic does the same to Inductive Logic? I apologize if this was already covered in your link, as I have not read the whole thing yet.

 

[Mentat]

Originally posted by Tom
Thanks Mentat for "keeping me company".

No problem. I hadn't realized before, that this thread contained exams, as I hadn't actually had a chance to even visit the thread before.

No, they are all correct.

REALLY?! Cool!

What is the name of the book that you were studying (it seems extremely informative).

 

[quantumdude]

Originally posted by Mentat
Does the book ever touch on the fact that Inductive Logic shows Deductive Logic to be paradoxical, and that Deductive Logic does the same to Inductive Logic?

No, and I've never heard of that. Can you explain?

What is the name of the book that you were studying (it seems extremely informative).

It's called Logic, by David Baum. Just a first textbook on the subject.

So far, we slowly crawled through the introduction, and we are about half way through the first of two chapters on syllogistic logic. I'll post some more by the end of the week.

I really want to get to symbolic logic, because that's where the power of deductive reasoning gets a huge boost.

 

[Mentat]

Originally posted by Tom
No, and I've never heard of that. Can you explain?

Sure, but you'll probably recognize it, before I'm through.

The paradox of Deductive Logic
Let's take, for example, Euclid's rule: "If two sides of a triangle are equal to the same, they are equal to each other" (I think that's how it goes).

For the purpose of this example, let's say that there is a triangle, where it can be shown that the two sides are equal to the same, but I refuse to believe that they are equal to each other.

Now, you would wish to use deductive logic to show me that it must be so, but...

Proposition 1 is "The two sides are equal to the same"
Proposition 2 is "If two sides are equal to the same, they are equal to each other"

Now, I'll accept those two, but it is another proposition altogether (Proposition [oo]) to say that "Therefore, the two sides are equal to each other". I refuse to accept Proposition [oo], and have no reason to yet. So, you say, "if you accept Propositions 1 and 2, then you must accept Proposition [oo]", which we'll call Proposition 3.

Well, now I'll agree to Propositions 1, 2, and 3, but I still disagree with Proposition [oo], and I don't have to agree with it, because you have yet to say that "if you accept 1, 2, and 3, you must accept Proposition [oo]".

And so it goes on. This is an Inductive approach, in that I am telling you that, no matter how many new propositions you produce, you will still never resolve this paradox.


The paradox of Inductive Logic
This is much more simple. Basically, deductive logic tells us...

1) Inductive Logic is based on learning from observed patterns.
2) What we think is a "pattern" is not necessarily a pattern (it could be a coincidence every time) unless you have tried it as many times as possible (which is infinite, obviously).
3) Therefore, Inductive Logic is based on trying something an infinite amount of times, and is thus not "proof" of anything.

It's called Logic, by David Baum. Just a first textbook on the subject.

So far, we slowly crawled through the introduction, and we are about half way through the first of two chapters on syllogistic logic. I'll post some more by the end of the week.

Thanks. It's very interesting to me.

I really want to get to symbolic logic, because that's where the power of deductive reasoning gets a huge boost.

Yeah, I've had some dealings with symbolic logic before (mostly in Raymond Smullyan's books, which I highly recommend, btw), and I think it's probably one of the most interesting things I've studied.

 

[quantumdude]

Originally posted by Mentat
And so it goes on. This is an Inductive approach, in that I am telling you that, no matter how many new propositions you produce, you will still never resolve this paradox.

It seems that all this does is highlight incompleteness: that the formal system has axioms that cannot be proved within the system. That's not a paradox.

I don't want to get into Goedel here at all in this topic. Indeed, I have another dormant topic in the Math forum that is moving towards that (I think it's time to revive that one, too). This thread is about using logic, whereas discussions such as this are more along the lines of proving things about logic, which is not what I'm after here. Basically, I started this thread for people like Lifegazer and Alexander so I could stop repeating the same explanations of their fallacies, and instead cut and paste sections from the Logic Notes.

Both of them are gone now, but I think it will still be a good idea to teach anyone here who wants to learn.

3) Therefore, Inductive Logic is based on trying something an infinite amount of times, and is thus not "proof" of anything.

Now this is mentioned in the Introduction, and I covered it in the Logic Notes. It is openly admitted that induction is not proof, and the book stresses the fact that terms such as "sound" are reserved strictly for deductive arguments, and that such absolute terms would be misplaced on an argument that gives only partial support for its conclusion.

 

[Mentat]

Originally posted by Tom
It seems that all this does is highlight incompleteness: that the formal system has axioms that cannot be proved within the system. That's not a paradox.

True. However, it does show that Inductive Reasoning is required to show the flaw in Deductive Logic, and vice versa.

I don't want to get into Goedel here at all in this topic. Indeed, I have another dormant topic in the Math forum that is moving towards that (I think it's time to revive that one, too). This thread is about using logic, whereas discussions such as this are more along the lines of proving things about logic, which is not what I'm after here. Basically, I started this thread for people like Lifegazer and Alexander so I could stop repeating the same explanations of their fallacies, and instead cut and paste sections from the Logic Notes.

Alright, I just wondered if the book mentioned it.

Now this is mentioned in the Introduction, and I covered it in the Logic Notes. It is openly admitted that induction is not proof, and the book stresses the fact that terms such as "sound" are reserved strictly for deductive arguments, and that such absolute terms would be misplaced on an argument that gives only partial support for its conclusion.

But you can only reach the conclusion of Inductive Logic's incompleteness through Deductive Logic, which is itself incomplete (by virtue of Inductive Logic). It's rather circular, and that's why I called it a paradox. However, we needn't discuss this at all, unless the book happens to bring it up.

Side Note: Science is based on the Inductive Method, so shouldn't it be viewed as giving only "partial support", as you put it? I covered this in "A Universe Without Logic" where I said that every case of "cause-and-effect" that we've ever observed could be coincidence. However, this is off-topic, so I'll leave it alone.

 

[quantumdude]

Originally posted by Mentat
True. However, it does show that Inductive Reasoning is required to show the flaw in Deductive Logic, and vice versa.

I haven't read the Goedel's entire proof yet, but I do know that he proved his theorem deductively, so it seems that induction is not required to show incompleteness.

Side Note: Science is based on the Inductive Method, so shouldn't it be viewed as giving only "partial support", as you put it?

Yes. We will get to the scientific method of falsificationism in Part III: Induction.

I'll try to post something more in the notes this weekend.

 

[Mentat]

And I'll try to solve some more of your exams, as soon as I get my glasses fixed.

 

[drnihili]

With regards to the paradox of deductive logic. The most famous presentation of this is from Lewis Carroll (aka C.L. Dodgson) in What the Tortoise said to Achilles. It's reprinted in one of Hofstadter's books, I forget which one though. THe problem has very little to do with incompleteness in the Goedelian sense. Instead it's primarily a problem with the philosophical notion of a rule.

Any logical system must use rules to allow the transition from one statement to another. But these rules can also be written explicitly as a statement of the logic, typically as an axiom. Systems using "natural deduction" have lots of rules and no axioms. Axiomatic systems have lots of axioms and very few rules. The problem comes when one insists that every rule be reduced to an axiom. It turns out that's just impossible to do and still have anything resembling a logical system.

One way of trying to resolve the problem is via metalogic. In metalogic you prove things about a logical system, for example that any inference made in it is valid. In a sense though this merely puts the problem off for a bit. Metalogic must of course make inferences, and those inferences are subject to the same sort of objection which can only be resolved by appeal to meta-metalogic. And the list goes on.

Interestingly, this issue receives very little attention and is generally considered to not really be a problem. It's just accepted that no reasonably powerful logic system can show it's own soundness in the relevant sense. Every starting point can be questioned, but we do have to start somewhere.

The problems with inductive logic are far more trenchant. Perhaps the best read on those is Goodman's The New Riddle of Induction

 

[Mentat]

Originally posted by drnihili
With regards to the paradox of deductive logic. The most famous presentation of this is from Lewis Carroll (aka C.L. Dodgson) in What the Tortoise said to Achilles. It's reprinted in one of Hofstadter's books, I forget which one though...

Godel, Escher, Bach: An Eternal Golden Braid. That's actually where I first read about it.

THe problem has very little to do with incompleteness in the Goedelian sense. Instead it's primarily a problem with the philosophical notion of a rule.

Any logical system must use rules to allow the transition from one statement to another. But these rules can also be written explicitly as a statement of the logic, typically as an axiom. Systems using "natural deduction" have lots of rules and no axioms. Axiomatic systems have lots of axioms and very few rules. The problem comes when one insists that every rule be reduced to an axiom. It turns out that's just impossible to do and still have anything resembling a logical system.

One way of trying to resolve the problem is via metalogic. In metalogic you prove things about a logical system, for example that any inference made in it is valid. In a sense though this merely puts the problem off for a bit. Metalogic must of course make inferences, and those inferences are subject to the same sort of objection which can only be resolved by appeal to meta-metalogic. And the list goes on.

Interestingly, this issue receives very little attention and is generally considered to not really be a problem. It's just accepted that no reasonably powerful logic system can show it's own soundness in the relevant sense. Every starting point can be questioned, but we do have to start somewhere.

This is a very good explanation of the problem of Deductive Logic, drnihili! If I might point out - though you probably already noticed - you did run into something similar to Godel's Incompleteness, when you said that "it's just accepted that no reasonably powerful logic system can show it's own soundness in the relevant sense". This is probably why Tom had associated the problem of Deductive Logic with Godel.

The problems with inductive logic are far more trenchant. Perhaps the best read on those is Goodman's The New Riddle of Induction

Really? I guess there's more to it than I had imagined (I had just assumed the obvious incompleteness of Inductive Logic: namely, you can't try anything an infinite number of times). I'll look that up that book as soon as I can. Thanks .

 

[Mentat]

Oh, and belated welcome to the PFs!

I guess I just hadn't seen you post before (or I would have extended this welcome earlier).

 

[quantumdude]

Now that Mentat's mom has let him come back, maybe we can pick this thread up again.

 

[Mentat]

Originally posted by Tom
Now that Mentat's mom has let him come back, maybe we can pick this thread up again.

I will start as soon as I can, but I have to get off-line in a couple of minutes, so it can't be today. I'll try for tomorrow.

 

[Dissident Dan]

Originally posted by Tom

Thanks Mentat for "keeping me company".

No, they are all correct.

Oh, contraire!

You caught this error when CJames made it, but you let it slip by this time:
---------------------------------------------------------------
quote:
-----------------
Argument 2:
Some Englishmen are Protestants.
Winston Churchill was a Protestant.
Therefore, Winston Churchill was an Englishman.
------------------

Some PF members are atheists.
Futurist was a PF member.
Therefore, Futurist was an atheist (this one is for those of you who have been here for a while).
--------------------------------------------------------------

The original example had an object-object link. The one that Mentat gave had a subject-object link. If the first sentence was "Some atheists are PF members.", then it would have been a correct counter-example.

 

[Mentat]

Originally posted by Dissident Dan
Oh, contraire!

You caught this error when CJames made it, but you let it slip by this time:
---------------------------------------------------------------
quote:
-----------------
Argument 2:
Some Englishmen are Protestants.
Winston Churchill was a Protestant.
Therefore, Winston Churchill was an Englishman.
------------------

Some PF members are atheists.
Futurist was a PF member.
Therefore, Futurist was an atheist (this one is for those of you who have been here for a while).
--------------------------------------------------------------

The original example had an object-object link. The one that Mentat gave had a subject-object link. If the first sentence was "Some atheists are PF members.", then it would have been a correct counter-example.

I know that I may be wrong, but I disagree with what I understand from your post. I don't see why "Some atheists are PF members" would have been better, since the first proposition of the book's example is "Some Englishmen are protestants". Is it just because the "atheist" part was supposed to correspond to the "Protestant" part?

In that case, I still disagree, since I didn't hold in the second proposition that Futurist was an atheist, but that he was a PF member. The fact that mine places the religion aspect at different parts of the issue didn't/doesn't seem relevant to me. IOW it didn't/doesn't seem relevant to me that what I took for granted was Futurists being a member of a certain group to deduce his being a member of a certain religion, while the book's example did it vice versa.

 

[Mentat]

Oh wait, your right, Dan. I was supposed to keep the scheme as:

some p are q
r is a q
therefore r is a p

Sorry about that.

Here's another possible counter:

Some PF members are very intelligent
Edward Witten is very intelligent
Therefore Edward Witten is a PF member.

Is that better?

 

[Dissident Dan]

The new example is correct.

 

[quantumdude]

Originally posted by Dissident Dan
Oh, contraire!

You caught this error when CJames made it, but you let it slip by
this time:

OK, good catch.

Argument 2:
Some Englishmen are Protestants.
Winston Churchill was a Protestant.
Therefore, Winston Churchill was an Englishman.
------------------

Some PF members are atheists.
Futurist was a PF member.
Therefore, Futurist was an atheist (this one is for those of you who have been here for a while).
--------------------------------------------------------------

The original example had an object-object link. The one that Mentat gave had a subject-object link. If the first sentence was "Some atheists are PF members.", then it would have been a correct counter-example.

I just want to note that that is not a counterexample of any argument because the conclusion is true.

Just to review, and to refresh my own memory, the easiest way to do a counterexample is start with a conclusion that is false, and construct true premises using their terms.

If I start with: Therefore, Greg Bernhardt is a woman.

Then I can build the argument around it using true premises.

Some women are PF members. True[/color]
Greg Bernhardt is a PF member. True[/color]
Therefore, Greg Bernhardt is a woman. False[/color]

In this argument, as well as in Argument 2 above and in Mentat's do-over, the middle term is in the predicate of both premises.

edit: fixed a typo

 

[Mentat]

Originally posted by Tom
Before I move on to the next part on immediate inferences, let me give some exercises on what has been presented so far.

Indicate whether each of the following sentences expresses an A, E, I, or O statement. When necessary translate the sentence into standard form. Indicate whether any meaning is lost in the translation. Write an abbreviation for each sentence, indicating which term each letter represents. Also give the schema for each statement.

1. Lassie is not a cocker spaniel.

I need a little help here, should this be represented as (having first indicated that "Lassie" = "L" and that "c" = "cocker spaniel"):

No L are c. That is an A statement. It's schema would just be: no p are q.

Is that the way we are supposed to do that, or did I miss something (or the whole point )?

If this is right, then I will attempt the rest of the statements.

 

[Mentat]

Uh...Tom? I need verification that I'm doing this right, or I can't continue.

 

[quantumdude]

Sorry Mentat, I've been busy but now I'm back.

Originally posted by Mentat
I need a little help here, should this be represented as (having first indicated that "Lassie" = "L" and that "c" = "cocker spaniel"):

No L are c. That is an A statement. It's schema would just be: no p are q.

Actually, that's not an A statement. "No p are q" is an E statement. An A statement is "All p are q". However, you got the correct schema.

Is that the way we are supposed to do that, or did I miss something (or the whole point )?

You got it, aside from the label (swap E for A and you have it). In this case, "Lassie" is singular subject, and it is to be understood as "All of the members of the class of which Lassie is the only member". That said, this is to be understood as a universal negative[/color] statement, aka an E statement.

Here it is completely worked out:

Original Sentence:
Lassie is not a cocker spaniel.

Standard Form:
No dogs that are Lassie are cocker spaniels.

Abbreviation:
No L are C.

Schema:
No S are P.
[/color]

edit: fixed bold font bracket

 

[Mentat]

Good to see you back, Tom.

I'll get started immediately.

Originally posted by Tom
Indicate whether each of the following sentences expresses an A, E, I, or O statement. When necessary translate the sentence into standard form. Indicate whether any meaning is lost in the translation. Write an abbreviation for each sentence, indicating which term each letter represents. Also give the schema for each statement.

Number 1 was covered so I will move on to...

2. Most records cost less than five dollars.

This is an I statement.

Standard Form:
Most records cost less than five dollars.

Abbreviated Form:
Some r are l.

Schema:
Some p are q.

3. Sixty percent of all college students work part-time to pay for their education.

This is an A statement. (I'm not really sure, but I think so )

Standard Form:
All students belonging to the class "sixty percent of college students", work part-time to pay for their education.

Abbreviated Form:
All S are W.

Schema:
All p are q.

4. Almost all professional basketball players are over six feet four inches tall.

This is definitely an I statement.

Standard Form:
Some pro. basketball players are over 6'4" tall.

Abbreviated Form:
Some p are o.

Schema:
Some p are q.

5. All politicians are not dishonest.

This is an O statement. Using the principle of charity here, since this could mean two different things.

Standard Form:
Some Politicians are not dishonest.

Abbreviated Form:
Some P are not d.

Schema:
Some p are not q.

6. War is not healthy for children and other living things.

This is an E statement.

Standard Form:
No War is healthy for children and other living things.

Abbreviated Form:
No W are h.

Schema:
No p are q.

7. Only those who bought tickets in advance were able to get seats.

This is an A statement.

Standard Form:
All buyers [trying not to use the same letter twice] who bought tickets in advance were able to get seats.

Abbreviated Form:
All b are a.

Schema:
All p are q.


That's it. I'm pretty sure I got some of those wrong (as I wasn't certain when answering some of them), but that was pretty fun.

 

[quantumdude]

It was tough, but I actually resisted the urge and did not log on once the entire long weekend.

2. Most records cost less than five dollars.

This is an I statement.

Right. This is an example of one of those cases in which meaning is lost in formalization ("some" does not mean the same as "most"). It highlights the tradeoff we make when going from informal language to standard form logic: We lose shades of meaning from the intended thought, but we gain the ability to unambiguously evaluate the logic of the translated sentence. It is unfortunate, but it would not be possible to test syllogisms for validity if we did not translate sentences, as we will see in Chapter 2.

3. Sixty percent of all college students work part-time to pay for their education.

This is an A statement. (I'm not really sure, but I think so )

It can't be an A statement, because that is the "Universal Affirmative" statement, and 60% of a whole is not universal. This is a tricky one, because it is a combination of two types, I and O. That is, it expresses the I statement...

Some college students work part-time to pay for their education.[/color]

...but not exactly. That is because the logical quantifier "some" includes the case of "all". But, we are told a very specific "some" (60 percent) that most definitely does not include "all". So, the above also expresses the O statement...

Some college students do not work part-time to pay for their education[/color]

This is called an "exceptive statement", and is discussed in the Logic Notes in the 12th post from the top (do a "control-F" for the word "exceptive" and you'll be taken right to it).

4. Almost all professional basketball players are over six feet four inches tall.

This is definitely an I statement.

Almost. It is another exceptive I + O statement. "Almost all" communicates the idea that it is "some but not all"

5. All politicians are not dishonest.

This is an O statement. Using the principle of charity here, since this could mean two different things.

Very good. You recognized that this could mean either "All politicians are honest" or "Some politicians are not dishonest". When there is ambiguity, it is always best to ascribe the weaker of the two positions to an opponent, and so avoid mistakenly increasing his burden of proof.

6. War is not healthy for children and other living things.

This is an E statement.

Right. I would even go so far as to say that it is two[/color] E statements:


War is not healthy for children.
War is not healthy for other living things.
[/color]

Since the predicate is compound, so is the statement. But, since we are not on Quantificational Logic yet the distinction is not yet important.

7. Only those who bought tickets in advance were able to get seats.

This is an A statement.

Right. This is an "exclusive statement" that actually expresses a "universal affirmative".

I'll get going on the next installment of the Notes ASAP.

Thanks for playing!

edit: fixed a bold font bracket