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Math Help - Is round a shape? (logic question)

  1. #1
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    Is round a shape? (logic question)

    Hey all,

    I'm trying to prove whether or not round is a shape.

    I have:

    StatementA: round is a shape
    ConverseA: A shape is round
    InverseA: If it is not round, then it is not a shape
    ContrapositiveA: If it is not a shape, then it is not round

    OR:

    StatementB: round is not a shape
    ConverseB: A Shape is not round
    InverseB: If it is not a shape, then it is not round
    ContrapositiveB: If it is a shape, then it is not round.


    I either need to show: A) round is a shape, can show ContrapositiveA is true, or ContrapositiveB is false

    B) round is not a shape, can show ContrapositiveB is true, ContrapositiveA is false.

    However, I am not sure if I've formed either contrapositive correctly, since they refer to objects as being round or shapes, and not to roundness itself as being a shape (do you see what I mean?). If that is incorrect, what would be the proper contrapositives? I could also use a proof by contradiction, so maybe my contrapostive approach isn't the best.

    Any thoughts?
    Thanks!
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  2. #2
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    Summary So Far:

    The original argument demonstarted the converse error:
    "Shape is an outline"
    "An outline can be round"
    -------------------------
    A Shape can be round
    This states that a shape can be round.
    We were trying to infer that:
    Round can be a shape
    Round cannot be a shape

    In otherwords, you cannot show something is true/false
    based on the true/false of it's converse
    -> Converse Error

    Define, "Shape",
    A shape is an appearance/outline.

    Note: appearnce/outline means appearnce or outline or configuration.
    (if it's an or clause, only one condidtion has to be correct).
    (Problem: If it's an OR clase, colors are also shapes).
    Possible Definition: appearance/outline -> (Appearnce AND outline) OR
    configuration

    Note: Point Raised: round is not a shape because
    round is a adjective that describes a shape.

    For this to be true, the argument follows the form:
    "Round is an adjective that describes a shape"
    "An adjective that describres a shape is not a shape"
    -----------------------
    "Round is not a shape"

    Is this a valid argument? Sure. Is it sound? No
    The reason it is not sound is because the second premise can be shown not to always be universally true:

    "An adjective that describres a shape is not a shape"
    Counter Example:
    Square is an adjective that describes a shape
    Square is a shape (it is an apparance/outline)
    Square is an adjective (a square corner.)
    Therefore, being an adjective that describes a shape does that
    mean that the adjective is not a shape.

    This is what I've come up with so far:
    Is an outline/apparance necessarily a shape?
    If all appearances/outlines are shapes
    round is a shape.
    Proof:
    "Round is an appearance/outline"
    "An appearance/outline is a shape"
    -----------------------------------
    Round is a shape.

    For this to be a sound argument, both premises have to be true.
    We have agreed (at this point) that #2 is true.
    Is number 1 true: "Round is an appearance/outline" ?
    round - Wiktionary
    round -> a circular object
    a circular object is an appearance of round
    ---------------------------------------------
    Round is an appearance

    round -> a circular object
    a circular object (abstration: circle) is an outline
    ----------------------------------------------
    Round is an outline

    If not all appearances/outlines are shapes:
    This does not force round is not a shape(at least some appearcenes/outlines are shapes).
    to do so would be converse error.

    Does round lack another condidtion that is necessary for being a shape?
    That is, if there is a property that shape must have, but round does not, then
    Round is not a shape.
    Can we find such a property?
    However, if we can agree that being an appearance/outline is a necessary and sufficient condidtion
    for being a shape, then round is a shape.
    (by Proof above)


    So, is my reason for round being an appearance/outline okay?
    does another condidtion exist that a shape must have to have that round does not?
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  3. #3
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    Quote Originally Posted by Richard Rahl View Post
    I'm trying to prove whether or not round is a shape.
    Any thoughts?
    Yes, I have lots of thoughts.
    First, what do you understand about proof?
    In formal systems, sentences are true or false. “Round is a shape.” is a sentence. Its truth or falsity depends on the system of axioms. What are the axioms of your formal system?

    Secondly, you have given several different forms of that sentence.
    However, each of those forms applies only to implications.
    Is your sentence an implication? Perhaps: “If something is round then it is a shape.”?

    Now again, sentences are true or false whereas arguments are valid or invalid.

    Are you saying that based on some axiom system, we can deduce the theorem “If something is round then it is a shape.” by some valid argument?
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  4. #4
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    Quote Originally Posted by Plato View Post
    First, what do you understand about proof?
    Everything I learned about proofs in Math 277 (2nd year university discrete math course) and Philosophy 100 (first year).

    I know about Proposistions (true/value), Sets of Propositions (consistent/inconsisten), arguments (vadid/sound), validitiy, soundness, proof by contradition (reductio ad absurdum), contrapostive, math induction, and direct proofs. I know all the logical connectors and theorms (demorgans theorm, distribution theorm), and well as boolean algebra, truth tables, etc. Also The converse and inverse errors, as well as modus ponus and modus tollens, that's what I can think of off the top of my head but i have my phil and math books and notes to refer back too.

    Quote Originally Posted by Plato View Post
    What are the axioms of your formal system?
    The definitions I gave for round and shape. To make things simple, they are basically the same as the ones in Wiktionary.

    Quote Originally Posted by Plato View Post
    Secondly, you have given several different forms of that sentence.
    However, each of those forms applies only to implications.
    Is your sentence an implication? Perhaps: “If something is round then it is a shape.”?
    I'm not trying to figure out if something which is round is also a shape, but rather if round, in and of itself, is (i.e. qualifies) as what we have defined to be a shape.

    Quote Originally Posted by Plato View Post
    Now again, sentences are true or false whereas arguments are valid or invalid.

    Are you saying that based on some axiom system, we can deduce the theorem “If something is round then it is a shape.” by some valid argument?
    Once again, not trying to argue if something that is round is a shape. I want to use the definitions and a logical proof technique to either assert or deny A) Round is a shape, or B) Round is not a shape

    Where those statements are the conclusion to a valid, and sound argument.
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  5. #5
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    In my view you are simply confused.
    You quote two different courses: a mathematics course and a philosophy course.
    I majored in philosophy as an undergraduate.
    That drove me to mathematics as a graduate student.
    Have you had a solid course on the foundations of mathematics?
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  6. #6
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    Quote Originally Posted by Plato View Post
    In my view you are simply confused.
    You quote two different courses: a mathematics course and a philosophy course.
    I majored in philosophy as an undergraduate.
    That drove me to mathematics as a graduate student.
    Have you had a solid course on the foundations of mathematics?

    Haha, I suppose I may be confused, but I'm not sure how. Isn't logic still logic regardless of if its in philosophy or mathematics? When I took my Discrete Math course in second year university, I found that much of it paralleled what I learned about logic in first year Philosophy, and likewise in my Digital Electronics/Digital Logic Course (and indeed, what I had intuitively reasoned for most of my life). There were some differences for sure (in discrete, we didn't really talk about soundness of an argument, just its validity (I asked my math prof about soundness, he said that in math it's considered "valid" if its both "valid" and "sound" as it would be defined in philosophy), and we didn't really do truth tables or boolean algebra in philosophy but the concepts still apply (which I discussed with my prof), but all the basics were still there, logic was still logic.

    I'm not sure what you mean by "solid course on the foundations of mathematics". I suppose the closest thing would be my Discrete Math Course which is defined in my school's Academic Calendar as:

    MATH 277 Discrete Structures
    An introduction to sets, binary relations and operations; induction and recursion; partially ordered sets; simple combinations; truth tables; Boolean algebras and elementary group theory, with applications to logic networks, trees and languages; binary coding theory and finite-state machines.

    The first chapter/few weeks or so in Discrete was solely on logic, and logical forms (everything I mentioned in the list is from discrete). My school does have a 3rd Math year course titled, "MATH 391 Mathematical Logic", and a 3rd year Philsophy Course titled "PHIL 340 Logic" but I have not taken them yet (still in second year). BTW: I've also taken Calculus 1 and 2, Matrix Algebra, and I'm currently in Linear Algebra and Statistics (second year).
    Last edited by Richard Rahl; March 8th 2007 at 05:57 PM.
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  7. #7
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    Quote Originally Posted by Plato View Post
    Have you had a solid course on the foundations of mathematics?
    It seems to me that the average person (not a math person) always views this as some philosophy of math.
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  8. #8
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    So am I completely wrong here or what?
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  9. #9
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    Well, to start off with, I would have as a postulate that a noun is not an adjective and vice versa. That immediately disqualifies round from being a shape, as round is an adjective and shape is a noun. Just because both words are concerned with the same idea does not make the one the other.

    Neither would I say that big is a size, for the same reason.

    Let's look at a couple of statements:

    The tabletop is round. (here round is an adjective, acting as a subject complement modifying the subject, tabletop.)

    Now lets force the word "shape" into the statment:

    The tabletop's shape is round. (i.e. round is still an adjective, acting as a subject complement modifying the subject, shape)

    Some adjectives can change to a different part of speech, becoming nouns while still retaining the same idea. For example, it is perfectly okay to say that red is a color, because red in its noun form still has the same idea as color does. However, if you check the dictionary, you will see that "round" loses all connotation to shape when used in its noun form.

    So, I would say that round is not a shape. Instead, it is a qualifier that describes a shape. Or to be more precise, round is a qualifier that describes the shape of something.
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  10. #10
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    Yea, that's the same argument I heard from everyone else. I still don't get it, perhaps I'll just have to concede that I (maybe) understand logic but obviously don't have a clue about grammer.

    Thanks,
    RR
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  11. #11
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    As far as logic goes, I'd say that you don't have to have an overall mastery of grammar, just the pattern of statements that especially use forms of the verb "be", because many arguments will contain a premise in the form of "A is B".

    For example, using round again:

    The table is round. (Here we are asserting something about the subject. Grammatically, table is the subject and we are predicating that its shape is round. Notice, however, that we are not renaming the subject, only qualifiying its shape.)

    On the other hand:

    Socrates is a man. (Here we are still asserting something about the subject, but now we are renaming it, calling Socrates a man. This is a true statement because Socrates has all the qualities that a man has. However, the statement is not reflexive, as we cannot say that a man is Socrates, as there are other men besides Socrates. A true statement of this form is transitive when going from the specific to the more general. For example, we can say that Socrates is a mammal, because a man is a mammal. We can say that Socrates is an animal, because Socrates is a man and a man is a mammal and a mammal is an animal.)

    The logic in use here is really just Aristotle's classification of things which exist, which is a hierachical structure that proceeds from general abstractions down through levels of narrowing abstractions, finally arriving at primary substances, the things that actually exist, from which no further declassifications can occur to the subject withoug making the subject cease to exist. i.e. In the hierarchy of animals, Socrates is a primary substance, because there is no further declassification that Socrates can undergo and still function as Socrates.

    In this scheme, only things that posess mass and dimension actually exist, and that is why words that are qualifiers or behaviors only have meaning when they are expressed within the context of some substance of matter.

    When dealing with words like shape and round, which are abstractions, looking at the parts of speech can resolve the identity crisis. If round in its noun form had a connotation that unequivocally expressed the same idea as shape, then you would have an argument.
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  12. #12
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    Yes, I understand your examples on Socrates, they are the similar to the ones we used to demonstrate the differences between True Premises & Conclusion, but Invalid Argument, False Premises & Conclusion, but Valid Argument, False Premises & True Conclusion, but Valid Argument, True Premises & Conclusion + Valid Argument = Sound Argument as well as going from the generic to the particular in class.

    Would you mind reworking the argument for me a little? What I mean is, I understand that such an argument is invalid:

    Shape is an outline and appearance (wikipedia)
    Round is an outline and appearance ( btw...I don't know if this is true)
    ------------------------------------------
    Round is a shape.

    But what of an argument of the following form:

    A necessary and sufficient condition for something to be a shape is that it is an outline and appearance (or configuration, according to the definition).
    Round is an outline and appearance.
    -------------------------------
    Round is a shape.


    Perhaps a looser condidtion on P1 is "
    A necessary and sufficient condition for something to be a shape is that it fits one of the definitions of shape".
    First off, is this argument valid? I think so, since whenever the premises are true, the conclusion must also be true (deductively). If it's not valid please feel free to show me why.

    Assuming it is, is it sound? In order to show it's sound, we must demonstrate both premises to be, in fact, true. So, can you show me that being an outline and appearance is not a necesseary or suffienct condidtion for something to be a shape (i.e. is there a property that something must have to be shape which round lacks). Second to that, can you demonstate that round is not an appearance and outline? (In my above posts, I demonstarted a possible reason that round is, based on definitions from wikitionary, but I am unsure if that is correct). Could you please rework the grammer argument to demonstrate that this argument is not sound (or please show me why its invalid)?. If you take the looser form of the P1, it would have to be demonstrated that round does not qualifiy for any possible definition of shape (argument by exhaustion). To show it does, we only have to find one definition that round qualifies for (which, atm, I'm thinking is the appearance and outline definition, which round may very well not be).

    tyvm,
    ~RR
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  13. #13
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    Some premises don't have to be proven. The premise that a shape is an outline and appearance is a definition, and so that premise you don't have to prove. For an argument I would get a little more specific with the premise:

    If a thing exists, then it is a shape.

    Here I mean shape to be a very general concept, and the concept from which matter derives. From the concept of matter, the levels would branch out to lateral concepts of inanimate and living, and then down through the branching levels, and finally to things that actually exist as individual specimens, which obey the chemistry definitions of possessing mass and taking up space. This is the only way a statement could be made that makes a thing be a shape. Shape must be in the hierarchy of matter and above it.

    Notice, in that premise, that the converse is also true, which is a property of all precise definitions:

    If a thing is a shape, then it exists.


    When the next premise is added however, with round as a shape, the argument stumbles before it can begin:

    If a thing exists, then it is a shape.
    Round exists.
    Therefore, round is a shape.

    But round does not exist. Round has no mass, takes up no space. You cannot point at anything that exists and say that that is a round. If you try to put round in the hiearchy just proposed, it would come under shape, no? But then would the classification of matter belong in that structure any longer? I think not. We would need to take matter and put it into a different hiearchy. The only way to put round in that hierarchy without changing the hiearchy that branches down from matter would be to insert it inside shape as a descriptor, to describe a type of shape. If you tried to take all the different types of shape and put them into the structure as independent classifications, then you would end up with an unworkable concept of existence, with types of shape driving the hierarchy as oppossed to types of matter.

    Now Plato would have no problem with saying that round exists, because of his idea of Forms existing in a transcendent realm that we cannot see. However, it was precisely because of the problems associated with that sort of thinking that Aristotle created his classification system. The transcendental things like round and big and so on were brought back to earth and inserted as qualities that existent things possess. So I would never make the argument that

    If a thing exists, it is a shape.

    I would instead say that if a thing exists, it has a shape, which is a big difference. Since round is one of the properties that shape has, then it is okay to say that some things have round shapes.
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  14. #14
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    The claim isn't that:

    if a thing exists, it is shape

    but rather:

    that any thing which is an outline and appearance is a shape.


    Of course, you are completely right that round is not a thing at all. So, in other words, in order for something to be a shape it must also physically exist, not just as a subjective concept?
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  15. #15
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    So, in other words, in order for something to be a shape it must also physically exist, not just as a subjective concept?
    That hierarchy that allowed the premise:

    If a thing exists, then it is a shape.

    was used only to support the premise. If I were to compose an argument, I would use a different hiearchy, and have shape as an abstract concept, from which other abstract shapes can descend from. For example, from shape would come a lower level of abstract shapes, i.e. round shape, square shape, etc., but not simply round or square. This is to stay consistent...nouns can only be nouns. You can't have a word that is a noun in meaning be equal to a word that is an adjective in meaning. There needs to be consistency there. And I would not try to insert this shape hiearchy into the matter hierarchy...because I think the result would be unworkable.

    Also, I would say that shapes can only be perceived as specific qualities within the context of actual physical things. In all cases, it is the thing that exists, not the qualities. Color for example. If I asked you to show me red, you might show me a red piece of paper. I will say no, you are showing me a piece of paper. You then might get a red ink pen and draw a line on a white piece of paper. I will say no, you are showing me ink. Forever and a day, you will never show me red. Such is the same with any abstract concept, say mass for example. Even the SI standard for mass is defined in terms of a specific piece of matter.

    Once you get this concept down and trusting its truth, it actually gets very easy to classify things in such a way that is both logical and common sensical. The two are not incommensurable.
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