[NT] Probability Relations and Induction

[SolitaryWalker]

No, SW, you don't understand. It's a disjunction.

Provoker has the property "... is either a hamster or a human."

Post your proof, in the mean-time I will compile what I think your proof would look like if you attempted to compile one.

Mx-Is a human
Fx-Is a hamster
Gx-member of typologycentral
a-Provoker

Premise 1: (x) Fx-Gx (all hamsters are members of typologycentral.
Premise 2: Ma V Fa (Provoker is either a human or a hamster)
Conclusion: Provoker is a member of typology central (Ga)
Step 3: Fa-Ga (1, Universal elimination)
Step 4: ?????? We've hit the dead end. In order to get the conclusion (Ga), we need Fa that provoker is a hamster. We don't have that. You can't say that Provoker is now a hamster (Fa), as he can be either a human or a hamster. The expression Ma V Fa merely states that at either one of the two (Ma, Fa) or both are true. You do not know which of the two is true and you do not know that both are true. Because of this you cannot assume that Fa is true.

I don't think that a disjunction aids the cause of your argument at all. On that note, we need a conjunction not a disjunction that provoker is both a human and a hamster. But in that case, we can only prove that provoker is a hamster because the conclusion that provoker is a member follows from the premise (Fa) that he is a hamster. This leads to a false conclusion. Your argument is missing the premise that all humans are members of typologycentral. (Mx-Gx) Only if you have that premise you could arrive at the conclusion of an argument with false premises and a true conclusion.

The bottom line is that your proof, featuring the disjunction symbol (V) does not give on the liberty to define Provoker as either a human or as a hamster. My proof (see post 38) does. The case is such because one of my premises states that provoker is a human and the other states that he is a hamster, I am free to use either one, or both.

 

[reason]

SW, the point of defining a domain in my previous proof was to not define "a" as either a human or hamster in particular, but merely as an organism. The 2nd premise states that orgnaism "a" is a hamster, but organism "a" is not thereby defined as a hamster. The conclusion then states that organism "a" is a a member of TypologyCentral.

So far as the conclusion is concerned, the issue of whether Provoker is actually a hamster or not is an open question, which is why it's impossible to deduce "Provoker is a hamster" from the conclusion alone.

 

[SolitaryWalker]

SW, the point of defining a domain in my previous proof was to not define "a" as either a human or hamster in particular, but merely as an organism. The 2nd premise states that orgnaism "a" is a hamster, but organism "a" is not thereby defined as a hamster. The conclusion then states that organism "a" is a a member of TypologyCentral.

So far as the conclusion is concerned, the issue of whether Provoker is actually a hamster or not is an open question, which is why it's impossible to deduce "Provoker is a hamster" from the conclusion alone.

Why don't you construct a proof to explain your reasoning process.

So far as the conclusion is concerned, the issue of whether Provoker is actually a hamster or not is an open question, which is why it's impossible to deduce "Provoker is a hamster" from the conclusion alone.

Lets see if that is true.

Fx-x is a hamster
Ma-x is a human.
Ox-x is an organism
Gx-x is a member of typologycentral.
a-Provoker

Premise 1- (x) Fx-Gx (all hamsters are members)
Premise 2- Oa (Provoker is an organism, any creature)
Premise 3-Cannot go anywhere from here. (Unless you use an unorthodox rule of derivation and substitute Oa with Fa, since an organism can be any creature. I do not think that this is an acceptable deduction of symbolic logic, but lets accept it for the sake of the argument.)
Premise 4-Fa-Ga (1, Universal elimination, if provoker is a hamster, he is a member)
Premise 5- Fa (rule of substitution, Oa substituted for Fa)
Premise 6-Ga

So far as the conclusion is concerned, the issue of whether Provoker is actually a hamster or not is an open question, which is why it's impossible to deduce "Provoker is a hamster" from the conclusion alone.

It is not an open question. According to the proof above, it has been deduced that provoker is a member because he is a hamster (See 4,5 modus ponens). This proof does not give one the option of deducing that provoker is a member because he is a human or because he is an organism (any creature), it only gives you the option of deducing that provoker is a member because he is a hamster. Conclusion alone does not show that provoker is a hamster, but the conclusion in synthesis with the premises does.

If your argument said this.

Fx-x is a hamster
Ma-x is a human.
Ox-x is an organism
Gx-x is a member of typologycentral.
a-Provoker

1. Ox-Gx (all organisms are members of typologycentral)
2. Oa (Provoker is an organism or any creature)
3. Oa-Ga ( 1, Universal Elimination. If provoker is an organism or any creature, then he is a member of typologycentral.)
4. Ga (2,3 modus ponens)

Now, in this argument we get the conclusion that provoker could be any creature, either a hamster or a human, or something else altogether because modus ponens (3,4) shows that provoker is a member because he is an organism. Since an organism is defined as any creature, then we are at liberty to say that provoker could be either a human or a hamster, but we do not know which he really is. This argument gets you the result that you were looking for in the last post, the one that you posted does not.

 

[reason]

SolitaryWalker,

I have provided a proof already and have nothing to change about it. The problem is not the proof, but your interpretation of it. You don't understand the logic and are deeply confused on the matter; with great pretension you have persisted in a fruitless line of criticism and questioning.

For the sake of argument, let us suppose that you are right about my argument. It would mean that SolitaryWalker, of TypologyCentral, has just rewritten the rulebook on deduction -- a valid argument of the form I presented is not only truth-preserving, but also falsity-preserving.

 

[SolitaryWalker]

SolitaryWalker,

I have provided a proof already and have nothing to change about it. The problem is not the proof, but your interpretation of it. You don't understand the logic and are deeply confused on the matter; with great pretension you have persisted in a fruitless line of criticism and questioning.

For the sake of argument, let us suppose that you are right about my argument. It would mean that SolitaryWalker, of TypologyCentral, has just rewritten the rulebook on deduction -- a valid argument of the form I presented is not only truth-preserving, but also falsity-preserving.

Figures, I've posted two or three diagrams outlining what appears to be your current argument and at least another three outlining how your argument could be substituted.

You obviously did not understand them. You persist in your typical mode, whenever you do not know how to solve a problem, you naively convince yourself that you do. Ignorance is a bliss afterall thankfully, as your inability to understand the views of others prevent you from being clearly aware of the fact that you don't know how to solve the problem in question. Do more proofs, perhaps then you will understand what this conversation is about.

As far as interpretation is concerned, the rules of inference I am using to interpret the argument come straight from this book. (The Logic Book by Merrie Bergmann (Used, New, Out-of-Print) - Alibris)

I am not altering the conventions of deductive reasoning, you are. You are the one maintaining that provoker can be defined as either a human or a hamster, yet you do not state such a definition in your universe of discourse, nor are you able to translate this expression into the language of symbolic logic, and most importantly, you are not able to construct a proof to support your conclusion.

Maybe, there is something wrong with the interpretation that I am using. In that case, cite a textbook where the argument interpretation that you are using is shown to be plausible. I will be pleasantly surprised if you find a single logic textbook that allows you to use terms in your argument that are not defined in the Universe discourse and do not directly correspond to any step of reasoning in the proof.

I will be even more surprised if you find the textbook that advocates your method of interpretation and also makes a plausible argument in favor of the plausibility of such a method.

But of course, that is a lot of hard-work isn't it? As usual, you can just pretend that your method 'somehow works' as in your mind it adds up somehow, and obviously whatever seems to be true to you is more likely to be true than the tedious methods of proof-solving described in authoritative textbooks.

You should teach me some of your method more, I think I would be much more content if I could assume that whatever I want to believe is true instead of wasting time on proving that it is true!

For humor's sake, lets try your proof again.

Fx: Is a hamster
Gx: Member of typologycentral.
a: Provoker

Premise 1: Ax (Fx-Gx)
Premise 2: Fa
3. Fa-Ga (universal elimination, if provoker is a hamster then he is a member)
4. Fa (2 reiteration, provoker is a hamster)
5.Ga(3,4 because provoker is a hamster, he is a member of typology central (modus ponens)

Hmm..something fishy about this. But hey, since I have reason on my team, it really doesn't matter. I am just going to say that provoker is defined as either a hamster or a human. It's okay that none of my definitions do not say that provoker is defined as a human or a hamster, or defined as anything other than a hamster. I will just assume that he is. It is all a matter of interpretation! Exactly like our friend Nietzsche said, "There are no facts! Only interpretations!". So, it doesn't matter what the proof really is, it only matters how we interpret it. (But wait a minute, if there are no facts, then there really is no such thing as a proof that is a certain way because only interpretations exist! Therefore the proof is whatever we imagine it as! In that case, there is no wonder that you are able to ignore the tedious proofs posted and assume that the truth is somehow on your side.)

There you go, so whatever it is that you don't understand or whatever it is that refutes your views, you can just assume you can 'interpret' it in a way that preserves your prejudices. So go ahead and assume that there is a definition of provoker as either a human or a hamster somewhere here, where we do not see, we need 'reason-vision' to see it. And only by virtue of this splendid piece of magic we will have the correct interpretation!

 

[reason]

Okay, I am not reading that. I only skim most of your posts, especially when you keep barking up a very wrong and verbose tree.

This proof, the one I provided earlier, is the only proof needed to make my point:

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

The problem is that you don't understand what this proof means. In short, the formula "Fa" does not define "a" as being "F", and so neither does the conclusion. To do so would be to violate so many laws of logic that it makes my head spin.

 

[SolitaryWalker]

Okay, I am not reading that. I only skim most of your posts, especially when you keep barking up a very wrong and verbose tree.

This proof, the one I provided earlier, is the only proof needed to make my point:

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

The problem is that you don't understand what this proof means. In short, the formula "Fa" does not define "a" as being "F", and so neither does the conclusion. To do so would be to violate so many laws of logic that it makes my head spin.

What laws of logic are you talking about? Cite your sources. You have no idea what you are talking about, do you?

In the expression Fa, a does not mean that it is F? In the case of Fa (F being hamster and a being provoker), this expression would be meaningless, as it would not say that provoker is a hamster. It would say something completely different (as after all, you just said that a does not mean that it is F, which shows that Fa does not mean that provoker is a hamster as you defined it in your universe of discourse.) If the case was such, then it would be impossible to symbolize the expression that provoker is a hamster, which is one of the premises in an argument. Apparently, merely symbolizing the premise that provoker is a hamster violates many laws of logic. Very interesting.


In any case my friend, come back in a year or so. For now, you're a hopeless case, not only do you need to do more proofs, but also enhance your reading comprehension, no wonder you say you haven't read most of those posts as they require an attention span far longer than yours. And as for all your remarks of the wrong proofs (they are not called trees, a truth tree is a technique that is used to test an argument for deductive validity, it is the same kind of a method as a truth table, it is not a derivation technique.), you have not pointed out a single error.

This thread was supposed to be for posters who demonstrate their knowledge by supporting their views with arguments, not those who cannot construct a single proof or write a paragraph furnishing the simplest of arguments supporting their views. In this case, I mean arguments, not just conclusions, and just conclusions is what you usually post.

In any case, there is hope for you, as you become more experienced by the time you come closer to graduating from your bachelor's program, your skills may improve. Perhaps in 2-3 years (when you are close to graduating) you will have the attention span that you need in order to read an essay of 500 words or to provide a basic explanation for how you read a proof. Hell, even that requires highly developed reasoning skills which you lack, all you have to do is cite an authoritative author who explains how proofs are to be interpreted.

So far, your views have been worthless as usual, as they were mere conclusions (your statement was that they violate laws of logic, yet no explanation was given to exlain why the case is such).

 

[reason]

SolitaryWalker, I have exerted as much effort as I care to explain to you why you are wrong.

 

[Provoker]

Provoker,

If John is a blacksmith and John is still John when he becomes a carpenter, then "is a blacksmith" was never a necessary property of being John. If the same is true of being Provoker and being a hamster, then these objections do not matter. In other words, the term "Provoker" just includes among its properties, "is either a human or a hamster."

Ok, this is an all new level of dubiousness. Here is what you are proposing: if Provoker is a hamster and Provoker is still Provoker when he becomes a human, then "is a hamster" was never a necessary property of being Provoker. (1) This is a fruitless analogy as a hamster cannot become a human the way a blacksmith can become a carpenter. In the former case, hamster and human are conditions of existence, while in the latter case, blacksmith and capenter are attributes that are applied to a particular condition of existence (2) You are treating "is a hamster" and "is a human" as if they are attributes of something, rather than essential conditions of existence that allow for attributes to apply to them. In other words, there is a person who occupies a place in spacetime who is called Provoker and posts at typologycentral. Provoker is not the condition of existence, (I can change my forum name and still be the same person who posts on the forum), Provoker is merely a name to refer to the existence of this human entity in spacetime that posts on typologycentral. It follows, therefore, that being human is a necessary condition of the existence of the Provoker we are referring to in the conclusion. This is not to be conflated with a hamster named Provoker that may very well existence, but whose existence depends on a different set of conditions than Provoker the forum poster. And this would have to be the case since it is a precondition of existence that something is itself: if it was not itself it would be something else and that something else is itself, and if something is not itself or something else it isn't. Given this basic law of logic, it should be impossible for Provoker the human forum poster and Provoker the hamster that may exist to be the same entity in spacetime. Your premise refers to one, while the truth of the conclusion depends on reference to the other. It follows, then, that nothing you have said demonstrates that your example is one in which false premises entail a true conclusion as it still makes the basic categorical error I outlined in my former post. Namely, it takes Provoker in the conclusion to apply to a different category than the Provoker in the premise. And this would have to be the case since the truthfulness of the conclusion requires that Provoker is necessarily a human and what is human cannot be hamster (as Provoker is defined in the premise). It is, therefore, an equivocation fallacy and not a case where false premises entail a true conclusion. That you have failed to recognize this and continue to introduce more names is beyond me. I do not care about names, what concerns me is the fallacy and basic categorical error you have thoughtlessly made and continue to parade.

 

[SolitaryWalker]

Ok, this is an all new level of dubiousness. Here is what you are proposing: if Provoker is a hamster and Provoker is still Provoker when he becomes a human, then "is a hamster" was never a necessary property of being Provoker. (1) This is a fruitless analogy as a hamster cannot become a human the way a blacksmith can become a carpenter. In the former case, hamster and human are conditions of existence, while in the latter case, blacksmith and capenter are attributes that are applied to a particular condition of existence (2) You are treating "is a hamster" and "is a human" as if they are attributes of something, rather than essential conditions of existence that allow for attributes to apply to them. In other words, there is a person who occupies a place in spacetime who is called Provoker and posts at typologycentral. Provoker is not the condition of existence, (I can change my forum name and still be the same person who posts on the forum), Provoker is merely a name to refer to the existence of this human entity in spacetime that posts on typologycentral. It follows, therefore, that being human is a necessary condition of the existence of the Provoker we are referring to in the conclusion. This is not to be conflated with a hamster named Provoker that may very well existence, but whose existence depends on a different set of conditions than Provoker the forum poster. And this would have to be the case since it is a precondition of existence that something is itself: if it was not itself it would be something else and that something else is itself, and if something is not itself or something else it isn't. Given this basic law of logic, it should be impossible for Provoker the human forum poster and Provoker the hamster that may exist to be the same entity in spacetime. Your premise refers to one, while the truth of the conclusion depends on reference to the other. It follows, then, that nothing you have said demonstrates that your example is one in which false premises entail a true conclusion as it still makes the basic categorical error I outlined in my former post. Namely, it takes Provoker in the conclusion to apply to a different category than the Provoker in the premise. And this would have to be the case since the truthfulness of the conclusion requires that Provoker is necessarily a human and what is human cannot be hamster (as Provoker is defined in the premise). It is, therefore, an equivocation fallacy and not a case where false premises entail a true conclusion. That you have failed to recognize this and continue to introduce more names is beyond me. I do not care about names, what concerns me is the fallacy and basic categorical error you have thoughtlessly made and continue to parade.

Good work, provoker. You have exposed an egregious epistemic error that reason has made. You have pointed out that an entity's identity would be changed if it was to become a different creature, reason overlooked this as he claimed that it is possible to say that provoker could be regarded as the same creature whether he be a person or a hamster, exactly like John could be the same creature whether he be a blacksmith or a carpenter.

However, I do not think that you even need to go that far to refute the contemptible non-sense that he has espoused. It can be refuted on purely logical grounds, as after all, his argument was about logic and not epistemology. He merely attempted to introduce an epistemic argument to conceal his sinister infractions against basic logic. (After all, he tried to cite an example of a deductively valid argument with true premises that entailed a false conclusion. He failed because his argument was either invalid or contained a fallacy of equivocation.)

Below, we may appreciate a very succinct and a trenchant demonstration of the verity of the thesis that reason's argument is either deductively invalid (contains a formal logical fallacy) or is guilty of an equivocation (contains an informal logical fallacy). My argument shows that even if reason did not make an error that you have argued he did, his argument would still be refutable on purely logical grounds.

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

In recapitulation, reason's epistemic error consists in this; if provoker can be defined as either a hamster or a human, the two could be used interchangeably, as after all provoker's not change enough for us to regard him as a completely different creature.

Suppose he said that provoker could be either a student of law or a student of sociology. In this case, the name of provoker as a student of sociology can be used interchangeably with the name of provoker as a student of law. In this case, if you, provoker were to change your major from law to sociology, we could still regard you as the same person.

Suppose we are trying to prove a conclusion that is quite similar to that of reason (in this case we are not guilty of the epistemic howler that he has committed earlier), the conclusion is that provoker (the student of law) is a member of typologycentral.

The following are our premises;

1. Provoker is a student of sociology (False premise)
2. All students of sociology are members of typologycentral (False premise)

Conclusion: Provoker (the student of law) is a member of typologycentral. (True conclusion).

This argument is invalid because the truth of the two premises does not guarantee that the conclusion is true. The premises imply that any student of sociology is a member of typologycentral. Even if that is true, the conclusion that provoker the student of law is a member of typologycentral is not vindicated because the premises can only entail the conclusion that only students of sociology may be members of typologycentral.

Suppose we make a deductively valid argument, but in this case it would be guilty of the informal fallacy of equivocation and therefore the conclusion would be false.

1. Provoker is a student of sociology (False premise)
2. All students of sociology are members of typologycentral (False premise)

Conclusion: Provoker (the student of sociology) is a member of typologycentral. (False conclusion).

The bottom line is that if reason's argument is valid, it entails a false conclusion and entails a true conclusion only if it is deductively invalid.



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

One may retort that we should not be defining our terms that way! After all, Provoker can be the same person regardless of whether he is a sociology student or a law student! That is true, but then the problem is not with the syllogism but with the way the terms are defined here, the universe of discourse in this argument offers two different definitions of provoker (one is a student of law and the other is the student of sociology). If an argument is to be left unedited, it should be evidence of an espistemic error as provoker can be the same entity despite the change of his academic course of action.

True this may be, however, Logic and epistemology are different disciplines. Logic is purely concerned with relation of propositions, not with the definitions of terms, epistemology is concerned with the definitions of terms, logic merely deals with the terms as they have been defined by epistemology.

Lets try to define the terms of our discourse in logic in accordance to the epistemic model that we have devised above (one that is devoid of reason's ignominous howler). In this case, the distinction between provoker as a law student and a sociology will be insignificant and we may use the terms interchangeably.

1. Provoker is either a law a student or a sociology student. (He can be either one and legitimately claim the identity of provoker) ----True premise
2. Provoker is a sociology student.---False premise
3.All sociology students are members of typologycentral.--False premise

Conclusion---Provoker (the law student) is a member of typologycentral. (True conclusion)

This argument is deductively invalid as the premises cited above do not guarantee the verity of the conclusion. Again, that is the case because the membership of the forum of provoker the law student is not entailed by anything in the premises. The premises only entail that provoker the sociology student could be a member of typologycentral. Premise one says that he could be a sociology student, but it does neither that premise nor any other supposition of the argument evinces that he is. The or statement means that he is at least one of the two or both, but alone, the 'or' statement does not specify which of the two he is or that he is both.

This version of the argument could also be rendered valid, but only at the price of committing the equivocation fallacy and subsequently the entailment of a false conclusion.

1. Provoker is either a law a student or a sociology student. (He can be either one and legitimately claim the identity of provoker) ----True premise
2. Provoker is a sociology student.---False premise
3.All sociology students are members of typologycentral.--False premise

Conclusion---Provoker (sociology student) is a member of typologycentral. (False conclusion)

The true conclusion is that provoker the law student is a member, not provoker the sociology student is. If one is to assume that the true conclusion that provoker the law student is a member of the forum is entailed by this argument, one is to committ the equivocation fallacy. Whenever you define terms in any way that is different from how you have defined them in your universe of discourse, you do so without a logical justification and are therefore guilty of the equivocation error in reasoning.

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

Alas, not all is lost. Reason's argument could have been salvaged! Perhaps we did not define his conclusion adequately! Lets give this another try.

1. Provoker is either a sociology student or a law student.

Premise 2. Provoker is a sociology student. (False premise)

Premise 3. All sociology students are members of the forum.

Conclusion: Provoker who is either a law student or a sociology student is a member of the forum.

This argument is perfectly valid.

Below is the symbolization of the argument that proves the deductive validity of the reasoning chain in question.

Mx-x is a member of the forum.
Lx-x is a law student
Sx-x is a sociology student
a-provoker

1. Sa V La (Provoker is either a sociology student or a law student. The implications of this is that he could be either one or both without changing his identity)---True premise
2. Sa (Provoker is a sociology student)---False premise
3. (x) (Sx-Mx) All sociology students are members of the forum. ---False premise
4. Sa-Ma (3, universal elimination. If provoker is a sociology student, he is a member of the forum.)
5. Sa ( 2 Reiteration, provoker is a sociology student)
6. Ma (provoker is a member of the forum, 4,5 modus ponens, provoker is a member of the forum because he is a sociology student)
7. Sa V La (Remember, once I prove X, I can say X or anything else true. (X---X V or anything else)
(Statement 7 reads that provoker is either a law student or a sociology student)
8. (Sa V La ) * Ma (6, 7 conjunction) Since I have proved that both items 6 and 7 are true, I am free to say 6 and 7 is true. (Statement reads that provoker is either a sociology student or a law student and is a member of the forum).

In this case we have an unremarkable conclusion of a provoker being a member of the forum regardless of whether he is a sociology student or a law student.

This is a true conclusion that was derived by deductively valid means from a set of premises where at least one of the premises was false. If reason was not guilty of the error you cited in the previous post, he could have corrected his flagrant error by reconstructing his original argument in a way that it is valid and devoid of the informal fallacy of equivocation. As it stands, it is completely hopeless. In fact, if we still have any doubt about this, why don't we plug it into the syllogism so we may be amused at the preposterous results it yields.

1. Provoker is either a hamster or a person. (True premise)

Premise 2. Provoker is a hamster. (False premise)

Premise 3. All hamsters are members of the forum.

Conclusion: Provoker who is either a hamster or human is a member of the forum.

Hx-x is a hamster
Dx-x is a human
Mx-x is a member of the forum
a-provoker

1. Hx V Dx (Provoker is either a hamster or a human)
2. (x) (Hx----Mx) (All hamsters are members of the forum)
3. Ha (provoker is a hamster)
4. Ha-Ma (If provoker is a hamster then he is a member, universal elimination)
5. Ma (3,4 modus ponens. Provoker is a member of the forum because he is a hamster.)
6. Ha V Da ( 3 Vedge introduction)
7. (Ha V Da) * Ma (provoker is either a hamster or a human and is a member of the forum.)

The conclusion is merely saying that either the human named provoker is a member of the forum or a hamster is. It is true, but useless and overly general. It does not cut to the main point that the human named provoker is a member of the forum. Although this syllogistic structure redesigns reason's argument to be both deductively valid and devoid of informal fallacies, most notably equivocation, its epistemically worthless exactly for the reason that you have cited--the category mistake. Knowing that provoker could be either a human or a hamster does not at all help us prove that a human named provoker is a forum member, as here we are dealing with radically different identities, one of an animal and the other of a human. On the other hand, if we used the argument that I have cited where we used two properties of provoker's possible identity (namely, law student or a sociology student). It is conceivable that when we have these propositions, we are talking about the same person (who could either be a law student or a sociology student), yet it is inconceivable that we are talking of the same entity who could either be a human who posts here or a hamster called provoker that runs in the wild.

 

[reason]

That is an awful lot of words, but you're still labouring under a misunderstanding of the original argument.

Against my better judgement:

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.

 

[SolitaryWalker]

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)

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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.


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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.




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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.

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Additional points and recapitulations.

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.