XPost - ? re: starting Algebra and Logic simultaneously

[Robin in Tx]

I am trying to decide on next year's workload, and would really appreciate the wise advice/opinions of the experienced moms on this board!

 

My post:

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I'm trying to decide on possibly slowing dd down in math a bit and introducing Logic first. Probably Cothran's course. I'm thinking this will make the math all the easier.

 

Or maybe I should put the emphasis on math and spend sometime in Algebra before introducing Cothran?

 

To me, it seems that introducing both at the same time might be a little overwhelming.

 

Dd is 13, will have completed Saxon 87, and first year latin (LITCT). My plans are to use Saxon for Algebra. The other possibility would be to introduce Cothran and do a "rigorous" prealgebra course... one that takes her more in depth on topics she already understands instead of introducing a lot of new material. Does such a course exist (for some reason I'm not thinking Saxon 1/2 would fit this description... any comments on that)? The BJU PreAlgebra would be available to me, but I'm not sure about it, either.

 

Another consideration (to me) is we'll be getting more in depth with music theory next year. Advanced theory, formal logic, algebra, and 2nd year Latin just seems like a LOT to bite off at one time. To me it does. Does it sound reasonable to you?

 

Thanks for any input!

Robin

[Myrtle]

The algebra program we explicitly taught logic before it began algebra. It then used the logic to explain the algebra, but the kind of logic that was taught was mathematical logic. Cothran teaches informal logic and specifically eschews mathematical logic or symbolic logic, Venn diagrams, logic tables, etc, which is exactly the kind of formal logic that rigorous math programs depend on when they teach the subject or use symbolic logic to express and teach math (quantification, "for every epsilon there is a delta..." all done up in symbols) And as a digression, since Cothran brought up Bertrand Russell, I think, and this is speculation, that he might really be reacting against logical positivism that used symbolic logic, because of the role that logical positivism played in various assaults on theology.

 

Here is how logic is applied when we teach algebra. For example, when we explained why it was that the square root of 9 was not -3 we did it using the language of logic, via a discussion of bidirectional implications and discussion about assuming the converse etc. (the book does this, so we do this as well) But you could just state the whole lesson on the square root function as a rule to be memorized without the in depth discussion in logic. It is in this way that studying logic helped him, it helped him to understand the rule and why it worked that way. It wasn't so much that learning logic made him more logical, but that it gave him a set of concepts and terms by which we could explain to him errors in thinking, otherwise we are stuck with, "That's wrong and I don't know how to explain to you why it's wrong, just keep practicing and you'll get it."

 

That being said, we started logic late last spring and worked on specifically those concepts that we knew we'd be touching on in algebra rather, which is more like studying "applied logic" rather than studying logic as a subject for its own sake. So this year, we specifically looked for a curriculum that teaches logic with the intent on applying it to math and we'll start that in the fall.

 

You are right that starting up two heavy duty subjects like concurrently this can be overwhelming. The other sweet thing about beginning the study of logic slightly before the study of algebra was that my son was already familiar with how to work with defintions, axioms, and derivations and only had to apply that to a new field. I think having issues with this is probably a common problem, and in my unsolicitied opinion, one that has more to do with familiarity with these kinds of systems rather than problems with abstraction, so that being familiar with logic and its formal rules made it easier to deal with algebra, but in retrospect had we done both at the same time it wouldn't have removed the hurdle; I think it might just give him the same hurdle in two different subjects, whereas if you did it one at a time, the second go around with the second subject is a bit easier.

 

Next year we'll pick up on logic again only this go around it will be an entire course of mathematical logic. Rather than teaching only the basic concepts, vocabulary, and symbols, the course will teach him how to account for assumptions being made. This translates into becoming more aware of assumptions that are being made while solving a complex problem or proving a theorem.

 

Historically, algebra was not taught with mathematical rigor (logic) but rather heuristically, I'm thinking 19th century algebra, and in fact you can see gaffes in the instruction and even errors in solutions in these books. Logic was taught prior to geometry, though, since it was geometry was taught rigorously, and so you see older geometry texts giving lessons in logic before the geometry is taught.

[Cedarmom]

What about doing The Fallacy Detective? It is an easy,enjoyable, introductory book on informal logic. The book teaches how to "detect" fallacies of logic. We did the book as a read aloud and then did the questions orally. My son was then excited about logic and better prepared when we did Cothran's course the following year.

 

Cedarmom

[Laura K (NC)]

I haven't yet reviewed different (formal) logic books for my son for next year. I'll keep this in mind as I look.

 

Laura

[Jane in NC]

Myrtle makes a good point:

 

The other sweet thing about beginning the study of logic slightly before the study of algebra was that my son was already familiar with how to work with defintions, axioms, and derivations and only had to apply that to a new field. I think having issues with this is probably a common problem, and in my unsolicitied opinion, one that has more to do with familiarity with these kinds of systems rather than problems with abstraction, so that being familiar with logic and its formal rules made it easier to deal with algebra, but in retrospect had we done both at the same time it wouldn't have removed the hurdle; I think it might just give him the same hurdle in two different subjects, whereas if you did it one at a time, the second go around with the second subject is a bit easier.

 

 

 

What I may have been blaming as a problem with abstraction may indeed be a problem with logic or lack thereof. For example, many students automatically assume that if something is true, then the converse is true. They see cases where this follows (Right Triangle implies the Pythagorean Theorem holds; if the Pythagorean Theorem holds, then the triangle must be a right triangle.) But there are many conditional statements for which the converse is not true: If a quadrilateral is a square, it is a parallelogram. This is true, but its converse--If a quadrilateral is a parallelogram, then it is a square--does not necessarily follow. What about its inverse: if a quadrilaterial is not a square, it is not a parallelogram. Then there is the contrapositive: if a quadrilaterial is not a parallelogram, then it is not a square.

 

A student's work in geometry will certainly be easier if they understand the difference between these sorts of statements. But logical equivalency will also lead a student to see which algebraic maneuvers are "legal", so to speak.

 

I have a logical mind so I expect everyone else to have a logical mind. Something is flawed with this statement!

 

Jane

[Jean in Wisc]

I've been told that when a child's brain is ready for algebra, he will also be ready for logic.

 

If you do Cothran's logic, you could plan to spread it over 2 years (and still call it one credit). I've never gotten a child through both books in 1 year so far.

 

I don't think there is a right or wrong answer with this, however, I would not want to drop math for a year. My kids might never get up and going again :scared:. Perhaps you could do something like the Keys to... series--I've not used Saxon, but a good pre-algebra might do what you want, too--having those basics down pat never hurt a child!

 

??

 

Good luck!

 

:-)

 

Jean

[Laura K (NC)]

I have a logical mind so I expect everyone else to have a logical mind. Something is flawed with this statement!

 

 

:D

[Robin in Tx]

What a helpful post! So much to think about...

 

The reason I'm using Cothran's book is because I want dd to study Aristotelian logic, and I want that because of our emphasis on the use of language ... my end goal is Corbett's Rhetoric book, and it's going to take a series like MP's logic and rhetoric to get us there, I think. I am not at all opposed to learning symbolic logic, I just have goals for the classical study of the syllogism that I don't think symbolic logic will help meet... unless I'm ill informed, that is. I can see wanting to understand both, but for different reasons. Does that make sense, or do you think they would be interchangeable for our purposes?

 

Either way, though, I can see how the exposure to any formal logic (and Cothran would argue that his is a formal logic in the classical sense... as opposed to informal logic) would make algebra easier. You make a very interesting point/case. You always do. :) You're like SWB to me. Everytime you write something, I have to read it 20 times before it all sinks in.

 

I would be interested in knowing about what courses teach the symbolic logic as it teaches the algebra. Is it Gelfand?

 

THanks so much for your reply. I'll let you know what I do, but you've about convinced me to go ahead and start the logic and let the algebra come later.

 

Robin

[Robin in Tx]

Thanks for reminding me about this. I actually own the book, and had intended to read it this summer wtih dd. We just moved, and I had forgotten about it... packed away somewhere! I'll find it this weekend and we can start looking at it... if it goes very well, and is easy for dd to grasp, I'll know that I don't need to put off TL for another year or so. Good idea... thanks again!

 

Robin

[Robin in Tx]

Thanks, Jean. Yes, that's what I've always heard, too (I think that Jim Nance got that idea started a while back). I probably didn't make myself very clear... I wouldn't actually not do any math at all - we are in Saxon 87 right now, but we are behind schedule due to our move, and I could easily slow things down and take a nice summer break, and spread the rest out over part of next year, supplementing with enrichments. Maybe start Algebra in the Spring of 2009?

 

Or, I could go ahead and plow ahead as originally planned (doing about 25% of the book or more over the summer) and be ready for Algebra in the fall. In which case I'd delay the start of logic for at least a semester, maybe a year.

 

Or, I could finish up as much as we comfortably can, take a summer break, not worry about finishing the book and start in the fall with a bonafide, good quality pre algebra program (but then I'd have to shop for another publisher, I'm afraid...).

 

Dd will only be in 7th grade next year, so I'm not concerned about what track she's on. She takes to math very easily, but I would be surprised if she pursued a math/science path/career.. This is more my trying to decide just how much time and energy next year I want to spend/focus on the sort of thinking and work required to tackle these logical "hurdles" (as Myrtle cleverly calls them! - she's done a good job of convincing me that trying to tackle the same hurdles in two different subjects at the same time might not be ideal). You are right about the basics being down pat... if we just did that next year in math, it would give us some time practicing those hurdles, and she'd still be in Algebra by 8th grade...

 

Like you said, "???" :)

 

Thanks again,

Robin

[forty-two]

The reason I'm using Cothran's book is because I want dd to study Aristotelian logic' date=' and I want that because of our emphasis on the use of language ... my end goal is Corbett's Rhetoric book, and it's going to take a series like MP's logic and rhetoric to get us there, I think. I am not at all opposed to learning symbolic logic, I just have goals for the classical study of the syllogism that I don't think symbolic logic will help meet... unless I'm ill informed, that is. I can see wanting to understand both, but for different reasons. Does that make sense, or do you think they would be interchangeable for our purposes?

[/quote']

I don't think symbolic logic and Aristotelian logic are interchangeable. I studied symbolic logic in college, and now that I am working through Socratic Logic, I don't really think there is all that much overlap. While I suppose they both help train the brain to think logically, they really aren't all that similar. Honestly, I will probably have my kids study both, so they will be able to utilize logic well in the language arts and in math. I think both are valuable.

[Myrtle]

I don't think symbolic logic and Aristotelian logic are interchangeable. I studied symbolic logic in college, and now that I am working through Socratic Logic, I don't really think there is all that much overlap. While I suppose they both help train the brain to think logically, they really aren't all that similar. Honestly, I will probably have my kids study both, so they will be able to utilize logic well in the language arts and in math. I think both are valuable.

 

 

Forty Two, By any chance do you remember what book you used in college?

[forty-two]

Myrtle,

 

Unfortunately it wasn't a specific logic book, but was a substantial section in my Artificial Intelligence text - we must have spent a good third or half the class on symbolic logic. I don't know the title offhand, but if you are interested, it is on a shelf at my parents', and I can ask them to check the title and author. I just remember that it had a maroon cover =). I'm pretty sure that wasn't the only class I took that involved symbolic logic, but I can't remember which ones.

 

Wait, I think I remember the other big one - one of my EE classes was all about doing things with the basic logic functions: AND, OR, NOR, XOR gates, etc. So there was a LOT of truth tables and logic in that course. I can't remember if there was a separate logic text for that course or not - I didn't use it much if there was - but it would be on the same shelf with the rest of my college texts, and it would be easy to check.

[Myrtle]

Either way, though, I can see how the exposure to any formal logic (and Cothran would argue that his is a formal logic in the classical sense... as opposed to informal logic) would make algebra easier. You make a very interesting point/case. You always do. :) You're like SWB to me. Everytime you write something, I have to read it 20 times before it all sinks in.

 

I would be interested in knowing about what courses teach the symbolic logic as it teaches the algebra. Is it Gelfand?

 

THanks so much for your reply. I'll let you know what I do, but you've about convinced me to go ahead and start the logic and let the algebra come later.

 

Robin

 

I will have to go back and look at how Cothran characterizes his logic.

 

There seems to be two meanings to the word "formal." We average chickens tend to mean "officially and directly teach a subject" when we say formal..as in "to formally study ballet" or some such thing. However, I think that formal and informal when used to describe logic mean something else. I'm hoping someone can jump in and help out...A "formal" system is a bunch of symbols along with some very strict definitions and rules about how those symbols can be manipulated. Mathematical logic would be formal logic. Informal logic is the study of argumentation using natural language. A high falutin' distinction is the following, my emphasis on the last sentence...

 

Wherever there is reasoning, there is a logic that seeks to articulate the norms for that type of reasoning. Informal logic differs from formal logic not only in its methodology but also by its focal point. That is, the social, communicative practice of argumentation can and should be distinguished from implication (or entailment)—a relationship between propositions—which is the proper subject of formal deductive logic; and from inference—a mental activity typically thought of as the drawing of a conclusion from premises. Informal logic may thus be said to be a logic of argument/ation, as distinguished from implication/inference.{/QUOTE]

 

As an aside, I'm going to have to go think about all of this some more. I've always thought of formal logic as "stuff you do with symbols" and informal logic as "stuff you do with words." I'm wondering if what Forty Two said might not be a good idea given our educational objectives, but I'm also wondering how I'm going to fit yet another year of logic into an already crowded curriculum.

 

Gelfand does not teach logic, just algebra. He does proofs which require the student to use implications and inferences but he doesn't come out and directly give what those rules are. The algebra program which we use does this but it is old, out of print, and hard to find.

[forty-two]

Re: formal and informal logic, I've always seen traditional logic, like Cothran's books, described as formal, and the study of fallacies as informal. I'm pretty sure Cothran describes it that way. I must admit, while I think I have a grasp on the reason for the distinction, I can't explain it well, and so obviously don't really get it yet. It seems like trad. logic is a cohesive study from the ground up, so it is formal, as symbolic logic is, and fallacies can be studied in a more ad hoc fashion, thus informal.

[Robin in Tx]

Here's a cut from their article on logic. It is more or less how I've always understood it - that formal logic is the study of the laws of inferences.

 

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Form is central to logic. It complicates exposition that 'formal' in "formal logic" is commonly used in an ambiguous manner. Symbolic language is just one kind of formal logic, and is distinguished from another kind of formal logic, traditional Aristotelian syllogistic logic, which deals solely with categorical propositions.

 

Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato [3] are a major example of informal logic.

 

Formal logic is the study of inference with purely formal content, where that content is made explicit. (An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The first rules of formal logic that have come down to us were written by Aristotle. [4] In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal language captures all of the nuance of natural language.)

 

Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.[2][5] Symbolic logic is often divided into two branches, propositional logic and predicate logic.

 

Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.

 

"Formal logic" is often used as a synonym for symbolic logic, where informal logic is then understood to mean any logical investigation that does not involve symbolic abstraction; it is this sense of 'formal' that is parallel to the received usages coming from "formal languages" or "formal theory". In the broader sense, however, formal logic is old, dating back more than two millennia, while symbolic logic is comparatively new, only about a century old.

******

[Robin in Tx]

And here's Cothran's explanation of formal logic (as opposed to material logic, but not informal logic). Cut frm one of Cothran' articles on logic at the MP website. His comment about content and truth makes me think that he is not against symbolic logic for the reason you suggested... what do you think?

 

********************

 

The two main branches of logic, one called formal or minor logic, the other material or major logic, are quite distinct and deal with different problems.

Material logic is concerned with the content of argumentation. It deals with the truth of the terms and the propositions in an argument.

Formal logic is interested in the form or structure of reasoning. The truth of an argument is of only secondary consideration in this branch of logic. Formal logic is concerned with the method of deriving one truth from another.

The distinction between these two branches of logic was nicely described by G. K. Chesterton:

 

Logic and truth ... have very little to do with each other. Logic is concerned merely with the fidelity and accuracy with which a certain process is performed, a process which can be performed with any materials, with any assumption. You can be as logical about griffins and basilisks as about sheep and pigs ... Logic, then, is not necessarily an instrument for finding out truth; on the contrary, truth is a necessary instrument for using logic--for using it, that is, for the discovery of further truth ... Briefly, you can only find truth with logic if you have already found truth without it.

This last remark of Chesterton's is important. It is not the purpose of formal logic to discover truth. That is the business of everyday observation and, in certain more formal circumstances, empirical science. Logic serves only to lead us from one truth to another.

That is why it is best to study formal logic first. In formal logic you study the form of an argument apart from or irrespective of its content, even though some content must be used in order to show the form.

[Myrtle]

It's as clear as mud to me now.

 

"It is not the purpose of formal logic to discover truth. That is the business of everyday observation and, in certain more formal circumstances, empirical science. Logic serves only to lead us from one truth to another. "

 

So is what is "true" only those things that correspond to physical entities? That's the "correspondence" theory...I think...goes back to Plato, and then there's a bunch of other theories. But there are philosophers who argue that something can be absolutely true without having a physical existence. And many philosophers would say that there is no certainty in truth from everyday observations and that empiricism is not trustworthy. So does Cothran not believe in a priori "truth"? If he doesn't, and he only believes in a posteriori truth or empiricism, that would support the view that he's hostile against mathematical logic as a method of acquiring truth, since mathematics is not an empirical science, but a priori.

[Robin in Tx]

Well, I might be completely off base here, but it is my understanding that in a formal logic syllogism, the propositions are accepted as true for the purpose of the exercise. So, in other words, when you evaluate it, you evaluate it for the process, not the truthfulness of the proposition.

 

He may very well be hositle to symbolic logic. Maybe I'm not clever enough to recognize it. I'm under the impression, though, that he is just trying to make sure that symbolic logic is not mistaken as the logic of the classical trivium.

 

Robin

[forty-two]

Myrtle,

 

I don't think he is hostile to the idea of a priori truth, but he is just saying that you don't come to a priori truth via logic, but instead logic is used to take a truth that came from somewhere to find out other true things, via logical rules. I think that mathematical logic works the same way: you have to know or assume *something* as true before you can use logic to find another true thing. Logic doesn't create truth by itself.

 

Cothran *is* hostile to symbolic logic, but I really doubt it is because he is hostile to a priori truth.

 

Edited to add: Here is a quote from Cothran's "Logic is not math" article: "As Peter Kreeft and Ronald Tacelli point out in their book, 'Many modern philosophers are suspicious and skeptical of the venerable and common sense notion of things having real essences or natures and of our ability to know them. Aristotelian logic (traditional logic) assumes the existence of essences and our ability to know them.'" I admit, I am dicey on what a priori means, precisely, but to me "the existence of essences and our ability to know them" sounds a lot like a priori truth, although I could be completely off here.

[Charon]

First of all, let me just give my $0.02 on the whole formal/informal thing. The formality is generally associated with the degree of precision involved in the subject. Specifically, when there is a great deal of philosophical vagueness in whatever you are talking about, you can no longer focus on just the form of your argument. (Because you are constantly at risk of equivocating.) So, what I think has kind of started happening is that the term "formal" is getting wrapped up in just that distinction. So, that is how we end up with "formal" largely just meaning that it uses a formal language -- because of the role this issue of philosophical vagueness has played recently philosophical matters.

 

At any rate, the standard philosophy explanation of all this is to generally characterize "knowledge" as "justified, true belief". (I always get heckled when I say that -- yes, I am aware of the Gettier Problem.) The point is that knowledge entails at least justification and logic is the study of what constitutes good justification. Epistemology is the study of what knowledge is, and so when looked at this way, it makes a clear distinction between semantics, the study of what truth is/means, and logic as two separate ares of epistemology. Another distinction that ought to be more clearly made, incidentally is between metaphysics, the study of what is real, and ontology, the specific subarea of metaphysics that is concerned with what exists. We are running headlong into both empiricism and foundationalism in some of these quotes here where we are wondering how to come to Truth (with a capital T), and I think that the lack of distinction between "what is real" and "what exists" as at least possibly disparate matters is part of that.

 

At any rate, consider the followiong proposition: "I think." Is it true? Can it ever be false? Do we know it empirically? And, yet, "I" and "think" don't seem to contain the truth of the statement simply in their definitions, like "bachelor" does for a statement like "Bachelors are unmarried males." This sort of thing is often used as an example of a possible synthetic a priori proposition. We are talking ourselves right out of the possibility of such a thing in this thread's general approach to the matter of logic. It is because we (very naturally) imagine some sort of a foundation from which we proceed using valid formal deductive logic to make further conclusions. We imagine this foundation of Truth (since it doesn't come from logic) to then have to be arrived at ("I guess") from our empirical observations of that which exists. We've failed to make a number of very slight subtle distinctions between what exists and what is real because we have trouble imagining how we could possibly talk meaningfully about something that does not exist. And, we have trouble imagining how we could "back in" to truth using reason or that reason might be bigger than logic. We just sort expediently equate all these things. But, consider, as I say, "I think." Is it analytic? More generally, do numbers "exist"? Isn't what we say about them true or false? Is the concept of number just a convention, perhaps?

 

(At any rate, you may have noticed that I am sympathetic to some particular conclusions about all this. I won't pretend I am giving an unbiased account of any of this....)

[Robin in Tx]

You know, the more I read of Cothran's articles, the more I am led to believe that he is hostile to the current "critical thinking" movement being held up as formal "logic." That seems to be what gets his goat. He clearly distinguishes between symbolic logic and socratic logic, as he should, and he clearly promotes the value of studying traditional socratic logic, but I haven't been able to discern that he is actually hostial to formal, symbolic logic... I do, however, pick up on a very negative tone when he suggests critical thinking programs (as published and marketed to educators these days) are misguiding when marketed as "logic" programs. That seems to be what he's "hostile" about (might be too strong of a word, though)... with regards to symbolic logic, his beef seems to be that he feels that it is no substititue for traditional logic in the classical trivium - with which no one seems to disagree.

[Beth in SW WA]

Thanks, Jean. Yes, that's what I've always heard, too (I think that Jim Nance got that idea started a while back).

 

Speaking of Nance, he is starting his first-ever online Intro To Logic course starting in the Fall. My son will do CD Alg 1 and this course simultaneously. We will most likely not do formal science so he can devote the necessary time needed for those 2 courses (as well as Omnibus 1 and Latin 1).

 

Last night we took a friend of my ds's to youth group and he just tested for a 9th grade Science/Engineering magnet program. He said it had tons of Logic questions on the test which he was completely unfamiliar w/. I thought that was interesting.

[Tina in Ouray]

Robin,

 

 

I can't really help you with your “workload†decisions because what is a reasonable load for one student might totally unravel another. What you have outlined sounds like a workable plan for a student who is capable, willing to work (with a little nudging), and not struggling or way behind in math or language skills. I've had several students work through a 7th grade schedule like this. The nice thing about mid-high is that if something doesn't pan out, or if you find mid-way through the year that you tackled too much, you can always slow down or delay something without it adversely affecting high school records.

 

 

Doubling up on algebra and TLI/II isn't going to be that demanding. If mathematics comes easy to your student, I'm guessing that TL won't be very difficult either. TLII gets a bit more challenging after chapters 5 and 6. If, at this point, you get bogged down in it at all, you could always slow down and bump work up into the summer or the following year.

 

 

It sounds to me like you have a good grasp on what you're doing in logic and why you're doing it. You're right: “formal logic†in the classical tradition is the study of the form or structure of the syllogism. Aristotelian logic is also sometimes called “categorical logic†because it traffics in categories of things rather than the relationships between propositions (as in modern logic). Modern symbolic logic is not a substitute for studying Aristotelian logic. But modern symbolic logic is VERY easy to grasp after studying traditional logic.

 

 

Someone mentioned Cothran's aversion for “critical thinking†programs. Here's an interesting article called “Critical Thinking in the Tower Ivory.†It might help to fill out some of the history on this movement.

 

 

You said, “[There] is no substitute for traditional logic in the classical trivium.†Amen.

 

 

Tina in Ouray, CO

[Robin in Tx]

Tina, thanks for the response (I was sorta hoping you'd join in - I knew you'd have something to add!). And thanks very much for the link. Interesting!

 

I have a daughter that is able, but not very willing. To be truthful, she's a bit on the lazy side. Math *can* come easily to her... often it does... but she has to be in a really good mood first :). Well, I guess this is a topic for another thread but that is definitely my biggest obstacle. Not a child who struggles or is behind, but a child who spends more mental energy on getting out of work than what the actual work would have demanded. I've never met a child so content to sit and do absolutely nothing!

 

Thanks again for the encouragement to move forward. Like you said, we can always step back if it's too much.

 

Robin

[Charon]

Robin,

 

 

I can't really help you with your “workload†decisions because what is a reasonable load for one student might totally unravel another. What you have outlined sounds like a workable plan for a student who is capable, willing to work (with a little nudging), and not struggling or way behind in math or language skills. I've had several students work through a 7th grade schedule like this. The nice thing about mid-high is that if something doesn't pan out, or if you find mid-way through the year that you tackled too much, you can always slow down or delay something without it adversely affecting high school records.

 

 

Doubling up on algebra and TLI/II isn't going to be that demanding. If mathematics comes easy to your student, I'm guessing that TL won't be very difficult either. TLII gets a bit more challenging after chapters 5 and 6. If, at this point, you get bogged down in it at all, you could always slow down and bump work up into the summer or the following year.

 

 

It sounds to me like you have a good grasp on what you're doing in logic and why you're doing it. You're right: “formal logic†in the classical tradition is the study of the form or structure of the syllogism. Aristotelian logic is also sometimes called “categorical logic†because it traffics in categories of things rather than the relationships between propositions (as in modern logic). Modern symbolic logic is not a substitute for studying Aristotelian logic. But modern symbolic logic is VERY easy to grasp after studying traditional logic.

 

 

Someone mentioned Cothran's aversion for “critical thinking†programs. Here's an interesting article called “Critical Thinking in the Tower Ivory.†It might help to fill out some of the history on this movement.

 

 

You said, “[There] is no substitute for traditional logic in the classical trivium.†Amen.

 

 

Tina in Ouray, CO

 

 

Modern symbolic logic is Aristotelian logic. That is to say, it is a subset of it. You really should do both "logic with words" and symbolic logic. And, truthfully, "formal logic" really should mean symbolic logic. Sure, when there was no such thing as symbolic logic, then when you are talking about the formal aspects of Aristotelian logic, you would be talking about his syllogisms and such (as opposed to, say, informal fallacies) all done in a natural language. But, symbolic logic is just the outcome of a bunch of work people have done in that specific area. It isn't like anyone rejected Aristotle's Laws of Excluded Middle or Contradiction. We should no more reject symbolic logic than we should start trying to use Aristotle's theory of the spheres again to do astronomy.

 

That said, certainly philosophical logic is more than just formal logic. If you just do what essentially looks like math all the time, then you will not necessarily walk away being very good at something like rhetoric. So, if you have an eye toward that sort of thing, then you would rightly be interested in "logic with words", too. More generally, as I say in my other post in this thread, when dealing with problems where philosophical vagueness plays a major role, you really need a lot of experience with "logic with words". You simply cannot "formalize" the objects you are dealing with and deal with them symbolically. Forcing the formality often tends to just end up begging the question. (That hasn't stopped people from trying to do such a thing with moral philosophy, though, for instance.) Nevertheless, though, you simply should not just reject out of hand some of the largest advances in logic in all of history (including Aristotle). It is true that a lot of modern empiricism is heavily associated with symbolic logic -- it's a real science-y thing to do and has been wielded in the past seemingly in an effort to do away with normal (informal) philosophy. I would agree that all of that sort of thing is wrong-headed. But, let's not throw the baby out with the bathwater, here. That doesn't mean that the logic, itself, is the bad thing, so much as, in this case, people's interpretation of it.

 

I think it is a big mistake to literally try to turn back time to 1800 and pretend that symbolic logic never happened by relegating it to some other subject as if "symbolic logic is really math and not logic at all, you see...." I mean, if you are going to do that, then do you just limit yourself only to the Organon, then? No one -- not even Ockham or any of the medievalists that came after Aristotle -- can add to it? Modern symbolic logic did not change Aristotelian logic. It's not like the Hegelian dialectic that actually rejects the Law of Contradiction or maybe popular ideas of "quantum logic" used to discuss events that both happen and don't happen all at the same time. Symbolic logic really just largely represents modern developments in Aristotelian logic. It's not analogous at all, for instance, to choosing to do modern languages over Latin or Greek.

[Robin in Tx]

Thank you for your responses in this thread!

 

You are right - I am definitely looking for the sort of logic study that fits between grammar and rhetoric. I am certainly not dismissing modern symbolic logic, though (and I don't think that the folks at MP are, either). I'm not "throwing it out" or trying to turn back time, and nothing I've read in this thread indicates that anyone is suggesting I do so... but perhaps something went right over my head (which wouldn't be the first time! LOL).

 

I picked up Cothran's text last night, and on the first page he briefly discusses traditional logic vs. modern logic. He does state that you will not find in his book truth tables or mathematical formulations, but then he states, "Despite a number of assumptions that traditional logicians find questionable, there is much in modern logic worthy of study." Then he says that to him, traditional logic has a closer relationship to ordinary human language and is a system unto itself which warrants separate study. Hence, he wrote a text. One with plenty of Venn Diagrams :).

 

I really don't know where the idea is coming from that he is hostile to modern logic. Perhaps his reference to other traditional logicians finding "assumptions questional" highlights a common rub in academia, and he was being charged of hostility by association :). I do think he would argue that there is no substititue for traditional logic (just as you would argue there is no substitute for modern logic), and he argues for the importance of traditional logic in the classical trivium... but that's not the same as dsmissing modern logic altogether. They obviously both have valuable place and function.

 

Oh well, I don't suppose it's my place to speak for Martin Cothran any further :)... I just want to make sure you understand that my original question was about when to begin the study of formal traditional logic (as opposed to informal traditional logic, not as opposed to modern logic). I apologize if it turned into something that put modern logic on the defense.

 

Robin

[Charon]

Thank you for your responses in this thread!

 

You are right - I am definitely looking for the sort of logic study that fits between grammar and rhetoric. I am certainly not dismissing modern symbolic logic, though (and I don't think that the folks at MP are, either). I'm not "throwing it out" or trying to turn back time, and nothing I've read in this thread indicates that anyone is suggesting I do so... but perhaps something went right over my head (which wouldn't be the first time! LOL).

 

I picked up Cothran's text last night, and on the first page he briefly discusses traditional logic vs. modern logic. He does state that you will not find in his book truth tables or mathematical formulations, but then he states, "Despite a number of assumptions that traditional logicians find questionable, there is much in modern logic worthy of study." Then he says that to him, traditional logic has a closer relationship to ordinary human language and is a system unto itself which warrants separate study. Hence, he wrote a text. One with plenty of Venn Diagrams :).

 

I really don't know where the idea is coming from that he is hostile to modern logic. Perhaps his reference to other traditional logicians finding "assumptions questional" highlights a common rub in academia, and he was being charged of hostility by association :). I do think he would argue that there is no substititue for traditional logic (just as you would argue there is no substitute for modern logic), and he argues for the importance of traditional logic in the classical trivium... but that's not the same as dsmissing modern logic altogether. They obviously both have valuable place and function.

 

Oh well, I don't suppose it's my place to speak for Martin Cothran any further :)... I just want to make sure you understand that my original question was about when to begin the study of formal traditional logic (as opposed to informal traditional logic, not as opposed to modern logic). I apologize if it turned into something that put modern logic on the defense.

 

Robin

 

 

 

No I don't think you are accounting for most of the issue here, Robin. For instance, read this. (I think this may have been posted already.) One major problem with this article is a false dichotomy. He makes it out like modern logic is just symbolic logic. In fact, "logic with words" is plenty much included in mainstream modern philosophical logic. Traditional logic is not actually supposed to just be modern philosophical logic, but rather more of a competing alternative to it with a specific disdain for symbolic logic. And, the way he talks about it, it really sounds like he really does mean it that way. On the other hand, Venn Diagrams are modern not traditional, so I'm not sure what that means.

 

There is a very good answer to his question of "why we would want to" (replace traditional logic with modern logic). Traditional logic is nothing more than modern philosophical logic sans some pretty major developments over the last couple centuries. You should no more exclude symbolic logic from a treatment of modern philosophical logic than you should exclude latin grammar from your latin program. Modern logic doesn't make additional assumptions. If anything, it is the other way around. Especially modern symbolic logic is just the essence of the syllogism without the additional baggage of things like the Categories, for instance, or the handicap of an imprecise natural langauge. You have to go through exactly the same logic to do it with imprecise natural language. Truthfully, there are a lot of good reasons to include modern symbollic logic and not doing so is up there with trying to use Aristotle's theory of the spheres for astronomy. I'm really not exaggerating that point at all, and, again, modern symbolic logic is not "the modern system" which includes informal logic and "logic with words" and so on.

 

And this:

 

"If logic is not math, then what is it? The answer, of course, is that logic is about finding truth with words, not symbols and with language, not math."

 

is just false. It isn't about finding the truth, as he, himself, said in some other quoted passage. And, even if it was, it wouldn't be about doing it with words and specifically not with symbols.

 

And this:

 

"The difference between the two systems of logic is quite dramatic, but most people can recognize the modern system because of its prolific use of symbols, in addition to common modern fixtures, such as truth tables and Venn diagrams."

 

is only superficially true. Modern symbolic logic is just Aristotelian Logic. So is Traditional Logic. The basic laws of identity, excluded middle, and contradiction are all the same. And, you say that he even has Venn Diagrams in his books, so what's the deal with that? What is really called for here is a standrad course of philosophical logic which includes "logic with words" and symbolic logic. Cothran's specific anti-math ranting and focus does him, I think, a great disservice. That was almost specifically why I bought Nance's book rather than his. At the time, I wasn't looking for mathematical logic, per se, but I certainly wasn't looking to exclude it either like Cothran seems clearly intent on doing if the above article is any indication. (And, incidentally, there is a whole lot more to his bringing up Bertrand Russell than just a name associated with modern symbolic logic. It is clear that Cothran's real beef is more with modern secular analytic philosophy. He has done just what I say he has -- thrown the baby out with the bath water.)