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From: Vesa Monisto on 22 Jul 2010 13:20 "herbzet" <herbzet(a)gmail.com> wrote in message news:4C48457A.8F73A108(a)gmail.com... > ... > Or to put it differently, the relation of logical implication > holds between two statements X and Y when what X asserts > includes all of what Y asserts. No. That is called "equivalence". See case 3 and at the end: Formal fallacies of sentential (propositional) logic: 1) "Affirming the consequent": If p then q; q, therefore p. (We could have q even we didn't have p.) 2) "Denying the antecedent": If p then q; not-p, therefore not-q. (We could have q even we didn't have p.) 3) "Commutations of conditionals": If p then q; therefore if q then p. (Implication is not symmetrical, equivalence is.) 4) "Improper transposition": If p then q; therefore if not-p then not-q. (There could be other reasons than p to have q.) 5) "Affirming one disjunct": p or q; p, therefore not q. (p IOR q is not the same as p XOR q, i.e., 0111 isn't 0110) Formal fallacies of syllogistic logic: 6) "Undistributed middle term": --- 7) "Undistributed major term": --- 8) "Undistributed minor term": --- 9) "Two negative exclusive premises": --- 10) "Illicit negative/affirmative": --- 11) "Fallacy of existential import": --- Formal fallacies of predicate logic: 12) "Illicit quantifier shift": --- 13) "Unwarranted contrast (Some are / Some are not)": --- 14) "Illicit substitution of identicals": --- Excerpt from Quine's "Selected Logic Papers" (p.15): "... denial, alternation, and conjunction are expressed in the fashion not-p, p or q, p and q. Only the first two of these are taken as primitive, or undefined; the third is defined in terms of the other two as 'not-(not-p or not-q)'. Further defined compounds are 'p implies q' and 'p is equivalent to q'. These are defined respectively as 'not-p or q' and '(p implies q) and (q implies p)'." In terms of my "wigwam"-game (of syllogisms): /\ major (term implies middle term) / /\ middle (term implies minor term) / / /\ minor (term is implied by both major and middle terms), 'is true'. In set theoretic terms: 'intersection carries truth' ('is the same for all'). It is a metaphorical meme to talk about "having, containing, carrying". V.M.
From: Vesa Monisto on 22 Jul 2010 18:51 I add here the interpretations for the rest of formal fallacies: Formal fallacies of syllogistic logic: 6) "Undistributed middle term M": Some P are M, Some M are S, :. Some S are P. 7) "Undistributed major term P": All M are P, No S are M, :. Some S are not-P. 8) "Undistributed minor term S": All P are M, All M are S, :. All S are P. 9) "Two negative exclusive premises": No M are P, Some M are not-S, Some S are not-P. 10) "Illicit negative/affirmative": All M are P, Some M are not-S, :. Some S are P. 11) "Fallacy of existential import": All P are M, No S are M, :. Some S are not-P. Formal fallacies of predicate logic: 12) "Illicit quantifier shift": (Ax) (Ey) Fxy :. (Ey) (Ax) Fxy. 13) "Unwarranted contrast (Some are / Some are not)": (Ex) (Sx & Px) :. (Ex) (Sx & ~Px). 14) "Illicit substitution of identicals": Where f is an opaque(oblique) context and a and b are singular terms, to infer from fa; a=b :. fb. There are many other formal fallacies but those are most common ones. If there is a tiny possibility for to make a formal error, then the whole structure is said to be false. No gray / fuzzy cases are alloved in 'crisp' (binary) logic. 'Tertium non datur' (third is not a variate but a variable). As an example the case 7 analysed by Wigwam-game: /\P All M are P, / /\M No S are M, / / \/\S :. Some S are not-P. Here S is an 'outsider' of bot M and P. That's right by 'good will', which is not enough in binary logic, because /\P All M are P, / /\M No S are M, S/\/ \ :. Some S are not-P (is false because All S are P as 'insiders'). Well, I hope you see the reason why I said 'No' to your formulation: > "... the relation of logical implication > holds between two statements X and Y when what X asserts > includes all of what Y asserts." [and possibly more] That's true in a sense (with good will) but false without my addition []. There is no good will (gray areas) in black-and-white (binary) logic. Fallacies are possible in programming, too. -- Use for debugging skills! V.M.
From: Curt Welch on 23 Jul 2010 13:37 herbzet(a)cox.net wrote: > herbzet wrote: > > Wolf K wrote: > > > > The rules of inference in e.g. PQ (Boolean) logic are designed > > > to prevent transformation of a "true" sentence into a "false" one. > > > > Yes. (The question, for me, is how this magic is performed!) > > The broader question of this thread, of how truth or falsity is > assigned to sentences in the first place, is not my concern here. > > Given that a truth-value has somehow been assigned to a sentence, the > question then becomes, by what magic do these particular transformation > rules, the "logical" transforms, manage to pull off the trick of > preserving "truth" throughout all the allowed transformations of that > sentence, regardless of what truth-value was originally assigned? > > Let us recall that one and the same set of transforms allow: > > 1) the transformation of a false statement into another false statement > 2) the transformation of a false statement into a true statement > 3) the transformation of a true statement into another true statement > > but do *not* allow > > 4) the transformation of a true statement into a false statement. I don't get your point here. It's trivial to transform a true statement into a false statement. Just add not to it front of it. A -> not not A > How is this done? What is the common feature of these "logical > inference" rules by which they can magically distinguish and filter > out "truth" from "falsehood"? > > Let there be no mistake, this is a very non-trivial question -- > the nature of "logical implication" has been a matter of debate > for thousands of years, and is very much a live question still. > > My answer, arrived at after a great deal of consideration (trust me), > is as I posted the other day: > > "The only thing further I would want to say here is to broaden > somewhat the proposed definition of logical truth -- not only > to have the logical equivalence of statements X = Y, but also > to have the logical inclusion of one statement in another. This > is typically indicated with an arrow '->' and read as "implies"; > thus we have X -> Y (X implies Y) which IMO is the assertion > that what is meant by statement X includes what is meant by > statement Y. > > [We might say that Y is analytic with regard to X.] > > "For example, "John is a bachelor" includes as part of its > meaning "John is a man", so we can infer the latter from > the former. > > [I point out: this inference is valid /regardless/ of whether > 'John is a bachelor' is assigned the value "true" or "false".] > > "Again, it naturally falls out that if X is true, so too will > Y be true (though not necessarily the reverse). We can then > define the concept X = Y as meaning that X -> Y and Y -> X. > > "The "logical" transforms of a sentence X to a sentence Y then > are those transforms which preserve some (or all) of the meaning > of X in the meaning of Y, without adding anything extra." The word "meaning" doesn't fit there in my view. The total meaning in "John is a bachelor" is far more complex than any questions about the binary truth of the statement. Maybe, "logical truth value" is better than "meaning" in your sentence? > Or to put it differently, the relation of logical implication > holds between two statements X and Y when what X asserts > includes all of what Y asserts. In natural language, what a sentence asserts is it's meaning, and again, the meaning is far more complex than anything to do with trivially simple binary logic. Simple logic is not a good tool for understanding meaning. When we limit the discussion to simple binary logic, then all this becomes trivial. It's well know exactly what X -> Y means by it's truth table so there is nothing there that needs 1000's of years of debate to understand or explain. What people don't agree on, or fully understand, is meaning. But that has almost nothing to do with simple truth and logic. -- Curt Welch http://CurtWelch.Com/ curt(a)kcwc.com http://NewsReader.Com/
From: herbzet on 23 Jul 2010 16:19 Curt Welch wrote: > herbzet wrote: > > herbzet wrote: > > > Wolf K wrote: > > > > > > The rules of inference in e.g. PQ (Boolean) logic are designed > > > > to prevent transformation of a "true" sentence into a "false" one. > > > > > > Yes. (The question, for me, is how this magic is performed!) > > > > The broader question of this thread, of how truth or falsity is > > assigned to sentences in the first place, is not my concern here. > > > > Given that a truth-value has somehow been assigned to a sentence, the > > question then becomes, by what magic do these particular transformation > > rules, the "logical" transforms, manage to pull off the trick of > > preserving "truth" throughout all the allowed transformations of that > > sentence, regardless of what truth-value was originally assigned? > > > > Let us recall that one and the same set of transforms allow: > > > > 1) the transformation of a false statement into another false statement > > 2) the transformation of a false statement into a true statement > > 3) the transformation of a true statement into another true statement > > > > but do *not* allow > > > > 4) the transformation of a true statement into a false statement. > > I don't get your point here. It's trivial to transform a true statement > into a false statement. Just add not to it front of it. > > A -> not not A ??? 'A --> not A' is *not* standardly allowed as a logical transform. 'A --> not not A' *is* standardly allowed. > > How is this done? What is the common feature of these "logical > > inference" rules by which they can magically distinguish and filter > > out "truth" from "falsehood"? > > > > Let there be no mistake, this is a very non-trivial question -- > > the nature of "logical implication" has been a matter of debate > > for thousands of years, and is very much a live question still. > > > > My answer, arrived at after a great deal of consideration (trust me), > > is as I posted the other day: > > > > "The only thing further I would want to say here is to broaden > > somewhat the proposed definition of logical truth -- not only > > to have the logical equivalence of statements X = Y, but also > > to have the logical inclusion of one statement in another. This > > is typically indicated with an arrow '->' and read as "implies"; > > thus we have X -> Y (X implies Y) which IMO is the assertion > > that what is meant by statement X includes what is meant by > > statement Y. > > > > [We might say that Y is analytic with regard to X.] > > > > "For example, "John is a bachelor" includes as part of its > > meaning "John is a man", so we can infer the latter from > > the former. > > > > [I point out: this inference is valid /regardless/ of whether > > 'John is a bachelor' is assigned the value "true" or "false".] > > > > "Again, it naturally falls out that if X is true, so too will > > Y be true (though not necessarily the reverse). We can then > > define the concept X = Y as meaning that X -> Y and Y -> X. > > > > "The "logical" transforms of a sentence X to a sentence Y then > > are those transforms which preserve some (or all) of the meaning > > of X in the meaning of Y, without adding anything extra." > > The word "meaning" doesn't fit there in my view. You are not alone in that -- in modern logic we are used to abstracting away the meaning of propositions and dealing with their abstract forms -- that is, only the meanings of the "logical" elements of the propositions are retained for the purpose of analysis. Nevertheless, the net effect is as I have described above: some (or all) of the meaning of X is preserved in the meaning of Y, with nothing being added -- regardless of what interpretation (meaning) we may subsequently assign to the abstract logical forms of X and of Y. In fact, the word "effect" is misleading -- it is the preservation of meaning that determines what are the logical transforms, and a fortiori what are considered the logical elements of the language. > The total meaning in > "John is a bachelor" is far more complex than any questions about the > binary truth of the statement. Sure. > Maybe, "logical truth value" is better than "meaning" in your sentence? The logical transforms do not necessarily preserve truth-value -- in some instances they allow the derivation of true statements from false statements, e.g. from the false compound statement "Lassie is a man, and all men are mortal" we conclude correctly that "Lassie is mortal". > > Or to put it differently, the relation of logical implication > > holds between two statements X and Y when what X asserts > > includes all of what Y asserts. > > In natural language, what a sentence asserts is it's meaning Right. > and again, > the meaning is far more complex than anything to do with trivially simple > binary logic. Ok. > Simple logic is not a good tool for understanding meaning. But it is the reverse that is the case. Though we may have some doubt about what precisely it means to assert "John has a big ego", it is indubitable that this proposition includes, and hence implies, the proposition that "John has an ego" -- regardless of whether or not John has a big ego, or indeed whether he has an ego at all. > When we limit the discussion to simple binary logic, then all this becomes > trivial. It's well know exactly what X -> Y means by it's truth table so > there is nothing there that needs 1000's of years of debate to understand > or explain. Statements don't have truth tables -- formulas (abstract forms) with variable elements have truth tables. It's important not to confuse a statement X with the form of a statement X. Failure to maintain the distinction has contributed to the confusion that has prevailed in this matter. > What people don't agree on, or fully understand, is meaning. But that has > almost nothing to do with simple truth and logic. I agree that the relation of implication that holds (or fails to hold) between statements X and Y bears only a remote relationship to their truth or falsehood. -- hz
From: herbzet on 23 Jul 2010 16:20
Vesa Monisto wrote: > > "... the relation of logical implication > > holds between two statements X and Y when what X asserts > > includes all of what Y asserts." [and possibly more] ^^^^^^^^^^^^^^^^^ Right! |