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  1. #51
    Senior Member reason's Avatar
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    That is an awful lot of words, but you're still labouring under a misunderstanding of the original argument.

    Against my better judgement:

    Code:
    Ax[Fx -> Gx], Fa |= Ga
    
    1. Ax[Fx -> Gx]          Premise
    2. Fa                    Premise
    3. Fa -> Ga              Universal Elimination (1)
    4. Ga                    Modus Ponens (2, 3)
    
    Domain: organisms
    
    F : ... is a hamster
    G : ... is a member of Typology Central
    a : Provoker
    In short, Fa does not define the "Provoker" in the conclusion as a hamster.

    I don't know how tall Provoker is. I suppose he is from 5 to 7 feet tall, but my definition of Provoker doesn't include a specific height. He could be 5'1", 5'2", 5'3", 5'4", and so on up until about 7'0".

    If we write the statement "Provoker is 5'11"," are we defining "Provoker" as 5'11""? Not necessarily.

    If "Provoker" is defined as someone who is 5'11", then the statement "Provoker is 5'11"" is merely a tautology. (Like saying that an unmarried man is a man).

    However, if we can use the term "Provoker" to mean someone between 5'0" and 7'0", then the statement "Provoker is 5'11"" is not a tautology, i.e. it can be false without contradiction.

    The point is that in this latter case, should we use "Provoker" in another sentence like "Provoker has brown hair," the term "Provoker" will not carry with it the predicate "... 5'11"," but will continue to mean "... is from 5'0" to 7'0" tall."

    Back to the original argument, the term "Provoker" in the statement "Provoker is a member of TypologyCentral" does not inherit the property "... is a hamster" from the second premise.

    If "Provoker" was defined to be a hamster from the beginning, then the statement "Provoker is a hamster" would be a mere tautology, but that isn't what we do in quantificational logic, (because it would, besides other things, violate the definition of the semantic turnstile).

    A metalogic for QL under your interpretation could include additional rules of inference governing the situation where all the premises are false, but that would not be standard logic.
    A criticism that can be brought against everything ought not to be brought against anything.

  2. #52
    Tenured roisterer SolitaryWalker's Avatar
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    Quote Originally Posted by reason View Post
    If "Provoker" was defined to be a hamster from the beginning, then the statement "Provoker is a hamster" would be a mere tautology, but that isn't what we do in quantificational logic, (because it would, besides other things, violate the definition of the semantic turnstile)..
    In your universe of discourse, provoker is not defined as a hamster. In fact, no particular entity is defined as a hamster. We have however, the following original entities in our definition; a=provoker, Fx-is a hamster. So provoker isn't defined as a hamster from the very beginning. However, as the inquiry progresses, a synthesis of the two original entities defines provoker as a hamster. So Fx and a combined, create the expression of provoker is a hamster. Because the definition of provoker as a hamster required a process, or a synthesis of the two original definitions, it is necessarily the case that provoker wasn't defined as a hamster from the very beginning. There is a difference between having an entity defined a certain way in the universe of discourse and an entity defined a certain way in the premises. An entity that was defined a certain way in the universe of discourse is necessarily original (was defined this way from the beginning), an entity that was defined in the premises need not be original, as it may be a synthesis of the entities from the universe of discourse.

    In other words, there is a difference between having this figure;

    Universe of discourse.
    a=provoker
    F-is a hamster

    Premise 1. Fa-Provoker is a hamster

    and this figure,

    a=provoker is a hamster

    F-is a hamster

    Premise 1. Fa-provoker is a hamster is a hamster. (tautologous expression)

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

    In the first case, which is an excerpt from your argument, there is no tautology. What you seem to have in mind is the second case which is not part of your original argument. The tautology would occur if Fx-meant x is a hamster and a meant not just provoker, but provoker is a hamster. In other words, the expression Fa would be tautologous if and only if F and a are defined in the exact same way. They are not defined in the exact same way as F merely means something is a hamster, and a means 'provoker', nothing more and nothing less.

    The essence of a tautology is this, A=A, which simply means A is A. The only way you can have a tautology is if you use predicative relations to connect two entities that have the exact same definitions.

    In summary, the expression Fa is not tautologous because the two components of the expression (F and a), have different definitions and not the same. Fa is a synthesis of the two definitions (1. a=provoker, 2. F=is a hamster)

    Consider the following textbook example to back this point up.

    Source: Logic, Fourth edition. (Hurley)

    "P. 394, Exercise 8.1

    Translate the following statements into symbolic form. Avoid negation signs preceding quantifiers. The predicate letters are given in parantheses.

    1. Elaine is a Chemist ( C)"

    Lets turn to the very back of the book and check the answer this author provides for the given problem.

    P. 571.

    "8.1

    1. Ce"

    The rationale appears simple. C-something is a chemist. e-elaine. Hence Ce means that elaine is a chemist. Elaine isn't defined as a chemist from the beginning, she is defined as a chemist as the reasoning process progresses. She could have been defined as something else, possibly a doctor, but then a different symbolization would emerge, namely De (D-something is a doctor). Bottom line is, e, elaine is an unbounded variable, it can be synthesized with any expression, therefore no tautology emerges and subsequently no violation of reasoning takes place.


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


    The question of how provoker is defined is a question of epistemology, not formal logic. With respect to logic, however you define your terms, your definitions need to be listed in your premises or the universe of discourse. If you use a definition that is not listed under the same notation as the one that is listed, you're making an equivocation fallacy. It's exactly like saying, suppose I define John as a monster, and all of a sudden in my argument I say that John is a butterfly.

    So, if your definition says Fx-X is a hamster and then you interpret Fx as saying something other than X is a hamster, you're performing the action described above.

    Since your definition states only this "Fx-x is a hamster", you cannot interpret the statement as anything other that provoker is a hamster.

    However, if you had this in your definition.

    Fx-x is either a human or a hamster.

    or

    Fx- x is any creature, human or non-human.

    Then surely, you'd be free to interpret your conclusion as something other than that provoker is a hamster.

    Your proof could be fixed as follows.

    F-any creature.
    a-provoker
    M-member of the forum


    1. (x) Fx-Mx (any creature is a member.)
    2. Fa (provoker is any creature)
    3. Fa-Ma (1, universal elimination, because provoker is any creature he is a member of the forum)
    4. Ma (provoker is a member of the forum, this was entailed via modus ponens by virtue of Fa the supposition that provoker is any creature) The conclusion just states the provoker is a member, it does not state that provoker is a hamster, that is absolutely right. However, the step that led you to this conclusion does state that provoker is a hamster, namely Fa. You get the conclusion that provoker is a member under the condition that provoker is a hamster, as you could not have derived the conclusion that provoker is a member without Fa which states that provoker is a hamster.

    You are right that the conclusion alone does not show that provoker is a hamster, but the derivation process does. It is interpreted directly as provoker is a member (conclusion) because he is a hamster. (Fa-Ma. Fa. Ma This reads as if provoker is a hamster he is a member, but provoker is a hamster, therefore he is a member) He was defined as a hamster in the process of constructing the proof and only by virtue of this definition the conclusion has been reached that he is a member.


    In one sentence, provoker isn't defined as a hamster in the conclusion because (a) provoker wasn't defined as a hamster to begin with, the proof defined provoker as a hamster eventually (Fa), and without this definition the conclusion Ma is impossible.




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



    Quote Originally Posted by reason View Post
    A metalogic for QL under your interpretation could include additional rules of inference governing the situation where all the premises are false, but that would not be standard logic.
    That is something else. The question I am concerned with is whether or not it is possible to have a valid argument with a true conclusion that has at least one false premise, not all false premises.

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

    Additional points and recapitulations.

    Quote Originally Posted by reason View Post
    If "Provoker" was defined to be a hamster from the beginning, then the statement "Provoker is a hamster" would be a mere tautology, but that isn't what we do in quantificational logic, (because it would, besides other things, violate the definition of the semantic turnstile)..
    A statement of provoker is a hamster isn't a tautology, provoker is a hamster provoker is a hamster is.

    I do not see how a tautology violated the definition of semantic turnstile or any particular law of logic. A tautology is the opposite of a contradiction, it is by definition truth functionally true.

    Pull out a piece of paper and chart this argument in a truth table.

    A if and only if A. (It is symbolized as A triple bar A) Note, you will get all Ts in the middle which means that this expression is true in all cases. Now try, A and not A, you will get the opposite or all Fs in the middle which evinces that the expression in question is false in all cases.

    Although a tautology is generally not very informative, it is not in itself contradictory nor does it entail a contradiction. Although no tautologous expression is contained in your premises or original definitions, your argument has gone south for different reasons. Yet again, it needs to be explained how a tautologous expression violates laws of logic.
    Last edited by SolitaryWalker; 10-01-2009 at 03:49 PM.
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