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[NT] Probability Relations and Induction

redacted

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If the argument is deductively valid, then IF the premises are true, the conclusion is necessarily true. A deductively valid argument is false if and only if the premises are false, as by definition, all truth-preserving premises in a deductively valid argument entail a truthful conclusion.

I assume you mean that the conclusion of a deductively valid argument can only be false if some non-zero subset of the premises are false.

I agree.

But that doesn't mean the conclusion can't be true if some of the premises are false.

For example:
Evan is a person.
2+2=4
Green is not a color.
Therefore
Evan is a person.

Just because a premise is false doesn't mean the conclusion is false.

The premises of any deductive argument always entail a set of propositions.

The conclusion of any deductive argument always entails some subset (possibly including the whole set) of those propositions.

If a premise is false, it means some of the entailed propositions are false. If you just avoid those and pick out others that are true as your conclusion, you have an argument where some premise is false and the conclusion is true.

So, is provoker a hamster? No.
Are all hamsters members of typeC? No.

Is every single entailment of those two premises false? Not necessarily. There are plenty of true ones. Like provoker being a member of typeC.
 

SolitaryWalker

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EvanHowever, this reminds me of people who say they have solved the problem of induction, because swans are by definition white, and if we ever found a "black swan," it wouldn't really be a swan at all..


I never said such a thing. My post was not at all concerned with induction. All I maintained was that the argument you have cited is not an instance of a deductively valid argument that contains false premise (s) and a true conclusion.

The person who once induced the false conclusion that all swans are white and later discovers a black swan and claims it is not a swan is making an equivocation fallacy. Feather color is not the defining feature of the swan, hence, by definition it could be either black or white. If the same person in question claims that a black swan is not a swan at all, he subtly changes the definition of the swan. Or simply, he uses the same word to describe two different phenomena. (1) A creature that is defined by its bodily features and (2) a creature that must be necessarily white.

That error was yours, not mine. You've used the term provoker as a human being (if you have asserted that the conclusion of your argument is true) and you've used it to define a creature that is a hamster.

If you did not commit the fallacy of equivocation, your argument has a false conclusion. If your argument has a true conclusion (as it is indeed true that there is a human being who is a member of this message board), then your argument is guilty of the aforementioned fallacy as one of your premises defines provoker as something other than a human being.


I assume you mean that the conclusion of a deductively valid argument can only be false if some non-zero subset of the premises are false.

I agree.

But that doesn't mean the conclusion can't be true if some of the premises are false..

The conclusion of a valid argument can be true if some of the premises are false. An argument with contradictory premises is the only example of a deductively valid argument that contains false premise (s) and a true conclusion. Most arguments with false premises and a true conclusion are deductively invalid.

There are valid arguments containing false premises and a true conclusion. It looks like you've cited another example of such an argument. It is the kind of a reasoning process where one of the premises is identical to the conclusion. (Hence, now we know the contradictory premise and the circular reasoning argument forms can have a true conclusion with some false premises).

Bottom line is, the argument cited by reason wasn't one of those reasoning chains where there is a true conclusion following from false premises.


For example:
Evan is a person.
2+2=4
Green is not a color.
Therefore
Evan is a person...

I think that here, you have unintentionally informed me of something that I overlooked. Namely that the circular reasoning argument is an instance of a deductively valid argument where the conclusion may be true, yet some of the premises may be false. Although, it appears as if you were trying to prove that some arguments (whether valid or invalid) can have false premise(s) and a true conclusion.

Granted, circular reasoning, although containing an informal logic fallacy (question begging) is devoid of formal logical fallacies, hence it is necessarily valid. ...

It's valid, circular reasoning arguments always are.

Just because a premise is false doesn't mean the conclusion is false....
No, it does not. As I have maintained, in contradictory arguments and now we know, in circular reasoning arguments, the conclusion may be true although the premise (s) are false.

In the case of circular reasoning, at least one premise must be true. (As the conclusion is true and its identical to one of the premises.) In the argument of contradictory premises, there is one true premise by definition. It has the form of A, not A. Necessarily it is true that either A is true or not A is true.



The premises of any deductive argument always entail a set of propositions.

The conclusion of any deductive argument always entails some subset (possibly including the whole set) of those propositions.

If a premise is false, it means some of the entailed propositions are false. If you just avoid those and pick out others that are true as your conclusion, you have an argument where some premise is false and the conclusion is true.

So, is provoker a hamster? No.
Are all hamsters members of typeC? No.

Is every single entailment of those two premises false? Not necessarily. There are plenty of true ones. Like provoker being a member of typeC.

These premises do not entail the conclusion that provoker(person named colin) is a member of typeC as the argument is invalid. They do, however, entail the false conclusion that there is a hamster member of the forum who has the name of provoker.
 

reason

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However, the fact that you are utterly unresponsive to carefully constructed arguments and frequently neglect to support your views lead me to guess that your experience with formal logic is limited. Either that is the case or you are inordinately intellectually lazy or terribly obstinate.
Oww, why must you cut so deep? My fragile NT ego cannot take such an assault.

Oops! I mean ... it's not me who doesn't understand logic, but you! I bet you don't even know the difference between a disjunction and a conjunction, or know what the Sheffer Stroke is. It's so sad that an NT can't make one non-fallicious argument! I'd sure hate to be you.
 

reason

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SolitaryWalker,

To not define terms according to your preference is not to commit a fallacy.

If I choose not to define "Provoker" as being, in part, a non-hamster, then that is my prerogative in a logical argument. (I wasn't really defining "Provoker" at all, but using it like a logical variable -- some interpretations of its meaning may lead to contradiction, but others will not.)

If it turned out that Provoker was a girl (with an odd voice and a habit of confusing pronouns), then would you say "my concept of Provoker is, in part, a male, so it turns out that Provoker does not exist. Instead, this person like Provoker in every way except being female exists who I shall designate 'Provoker-1'"?

In any case, this is all really besides the point; namely, false premises can have true consequences.
 

matmos

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In any case, this is all really besides the point; namely, false premises can have true consequences.

If the fairies at the bottom of the garden blow you magic kisses.
 

JustHer

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Just thought this was kinda funny.

The NF Idyllic - last thread posted in : What is your favourite Virtue?
The NT Rationale - last thread posted in : Probability Relations and Induction
 

SolitaryWalker

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If I choose not to define "Provoker" as being, in part, a non-hamster, then that is my prerogative in a logical argument..

No, you did not. You have defined provoker as a hamster only. Your premise states that provoker is a hamster. It does not state that provoker is a hamster and a human being. If you truly have defined the premise as you say you did, your premise would state that provoker is a hamster and a human being, not only that provoker is a hamster. The inference that provoker is a hamster and a human being from the written statement that provoker is a hamster is illegitimate. It is purely psychological and not logical, as there is nothing in the premise stating that provoker is a human being.

If it turned out that Provoker was a girl (with a really deep voice and a habit of confusing pronouns), then would you say "my concept of Provoker is, in part, a male, so it turns out that Provoker does not exist. Instead, this person like Provoker in every way except being female exists who I shall designate 'Provoker-1'"
..

Again, with respect to however you define provoker, the onus is on you to prominently indicate this in the premise. Otherwise, the only legitimate inference regarding the definition of provoker is one that is explicitly stated in the argument and nothing else. All you have stated is that provoker is a hamster.

Bottom line is, if in your argument, you use a word that is present in the premises and you attribute a different definition to the word than the one it had in the premises, you are indeed committing an equivocation fallacy.

In any case, this is all really besides the point; namely, false premises can have true consequences.

Indeed, only in the case of contradictory premises and in circular reasoning. Not in the argument you have cited.
 

Provoker

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The conclusion of a deductive argument may be true even when the premises are false. For example,

Every hamster is a member of TypologyCentral
Provoker is a hamster
Therefore,
Provoker is a member of TypologyCentral​

The deductive relation between premises and conclusion is merely that if the premises are true, then the conclusion is also true.

Basically, on this point I am with SolitaryWalker in thinking that while the conclusion of a deductive argument can be true whilst its premises false, it is not for the reason you concede, which is couched in the dubious example you have provided. Your example commits the fallacy of equivocation and a basic categorical error. It is comparable to the following:

Every mouse is ESFJ
Reason is a mouse
Therefore,
Reason is an ESFJ

Here, the conclusion is true (for sake of argument we will assume you actually are ESFJ as it could be the case that that is a misnomer) but true only insofar as we take Reason to identify with the category ‘human’ and not the category ‘mouse’ (as we are in agreement that the premises are false). It follows, therefore, that if we take Reason to remain in the same category (mouse) as specified in the premises, then in it is impossible for the premises to be false and the conclusion true as you hoped would be accomplished by your example. On the contrary, the conclusion would also be false. It follows then that the only way to have false premises and a true conclusion, which can be deduced from those premises in the type of syllogism you are proposing, requires that you equivocate by taking Reason in the conclusion to apply to a different category than Reason in the premise. That is misleading and a logical fallacy--namely, equivocation. It follows, therefore, that this cannot be used as an example to support your initial claim.

From the point of view of a logic textbook, to have a case where the premises are false and the conclusion is true requires a different kind of argument—namely, an argument that contains contradictory premises. An argument that contains contradictory premises is automatically valid, since its premises cannot be true together, and thus we can never have a situation where all the premises are true (the premises can never be true at the same time), and the conclusion false. The following is an example:

All snails are slimy. Some snail is not slimy. Thus, the President of the United States is a Democrat!

Here, the argument is considered deductively valid as anything follows from a contradiction, and the conclusion is true, but it is very counterintuitive and I doubt it yields much utility outside of logic books.
 

reason

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Provoker,

I understand SolitaryWalker's objection, but its just silly talk.

In short, the claim is that my conclusion, "Provoker is a member of TypologyCentral," is not actually true, is because the "Provoker" in that proposition refers to a hamster. However, this merely follows from including "is a hamster" as a necessary property of being "Provoker," but obviously I would never have meant to imply that when presenting the argument.

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."
 

reason

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One more time, just to clear up the confusion. Here is the argument:

Every hamster is a member of TypologyCentral
Provoker is a hamster
Therefore,
Provoker is a member of TypologyCentral​

Now, if we include among a list of properties of Provoker, "is either a human or a hamster," then the conclusion can still be true even when the premise "Provoker is a hamster" is false. It's that simple: one little disjunction.

This should be quite obvious.
 

SolitaryWalker

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One more time, just to clear up the confusion. Here is the argument:

Every hamster is a member of TypologyCentral
Provoker is a hamster
Therefore,
Provoker is a member of TypologyCentral​

Now, if we include among a list of properties of Provoker, "is either a human or a hamster," then the conclusion can still be true even when the premise "Provoker is a hamster" is false. It's that simple: one little disjunction.

This should be quite obvious.

The conclusion is true only if the argument is deductively invalid. Invalid arguments with false premises and true conclusions are quite common and only a little interesting.

This is an example of such an argument.

The sky is red.
All trains run faster than the speed of light.

In conclusion we know that Barrack Obama is the president of the United States.

Lets try your argument.

1. There is a hamster called provoker. (True premise)
2. All hamsters are members of typology central. (False premise)

Provoker/Colin/author of this thread is a member of typology central. ( True conclusion). If you do not understand how this argument is invalid, may as well give up here or at the very least, since you are now a freshman in your undegraduate program enroll in an introduction to logic or a symbolic logic course. (They are usually titled as PHL 102, 103, sometimes 105 or 107 and so on...)

----------------------------------------------
Lets look at another way your argument can be constructed.

1. Provoker is a human and a hamster. (Premise 1)
2. All hamsters are members of typologycentral. (Premise 2)

True conclusion: There is a human named provoker who is a member of typologycentral.


Step 1 of proof: Provoker is a hamster (conjunction elimination).
2. If provoker is a hamster then he is a member of typologycentral. (Universal instantiation)
3. Provoker is a member of typologycentral. (2,3 modus ponens)

Note, here we have proved that there is a hamster called provoker who is a member of typologycentral. The case is such because this conclusion follows from step 2 of proof which states that provoker is a member if he is a hamster. Since he is a hamster, we know that he is a member.

Your conclusion is false because it states that there is a hamster who is a member of typologycentral. The true conclusion is that there is a human called provoker who is a member of typology central. Therefore this argument is not an instance of a deductively valid argument where there is a true conclusion entailed by false premises.

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

Lets try another derivation.

Same argument.

1. Provoker is a human and a hamster. (Premise 1)
2. All hamsters are members of typologycentral. (Premise 2)

True Conclusion: A human named Provoker is a member of typologycentral

Step 1 of proof: Provoker is a human ( 1, conjunction elimination)
Step 2 of proof: If provoker is a hamster then he is a member of typology central. (2 Universal instantiation).

From here we cannot derive the conclusion that there is a human named provoker who is a member of typologycentral. In order to prove that provoker is a member of typologycentral, he needs to be a hamster as step 2 states that if he is a hamster he is a member of typology central.

Now, suppose I did this, Step 3: Provoker is a member of typologycentral. (Invalid inference! This is affirming the consequent which is a fallacy.)

Again, this underlines that you cannot prove that provoker is a member of typology central without maintaing that provoker is a hamster first. This proof must either remain incomplete, and if it is to be completed, it must be done by virtue of modus ponens where we start with the antecedent of provoker is a hamster (which we derive by conjunction elimination of premise 1). Thus by modus ponens we deduce that provoker is a member of typologycentral because he is a hamster.

The bottom line is that the conclusion that provoker who has a property of being a hamster is false, and by this argument, its impossible to prove that provoker is a member of typologycentral without maintaining that provoker is a hamster. Thus, the true conclusion that there is a human being named provoker who is a member of typologycentral is unattainable.



Provoker,

I understand SolitaryWalker's objection, but its just silly talk.

In short, the claim is that my conclusion, "Provoker is a member of TypologyCentral," is not actually true, is because the "Provoker" in that proposition refers to a hamster. However, this merely follows from including "is a hamster" as a necessary property of being "Provoker," but obviously I would never have meant to imply that when presenting the argument.

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."

Oh, I bet you do. Yet it is really hard to make sense of this mess when you haven't done a single proof in your entire life. Try again next year. When you do enroll in your logic class; don't just look over the nitty-gritty proofs assuming you know all of that stuff, take a pencil and paper out and write out your solution step by step.
 

reason

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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
 

Orangey

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^^I think that about did it for this argument. That is uncontroversially correct.
 

SolitaryWalker

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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

Good work, this proof shows that Provoker is a member of typology central. (Step 2 of your proof indicates that provoker is a hamster. F=is a hamster. a: provoker. Step 3 indicates that if provoker is a hamster (Fa) then he is a member of typologycentral (Ga). Step 4 indicates that Ga is derived by modus ponens of 2 and 3, Ga is provoker is a member of typologycentral. )

This argument is valid but the conclusion is false. Provoker has been defined as a hamster and the conclusion that provoker is a member of typologycentral is tantamount to the claim that a hamster is a member of typologycentral.

There is no hamster on typologycentral and this argument does not prove that a human named provoker is a member of typologycentral. You would need a different set of premises to arrive at the conclusion that a human named provoker is a member of typologycentral.

A-provoker
M- is a human
G- is a member of typologycentral

Premise 1 should say: Ax (Mx-Gx) (If all things are human, then all things are members of typologycentral or all humans are members of typologycentral.)


Premise 2: Ma
Premise 3: Ma-Ga (Universal instantiation or universal elimination by Bergman's notations in the 'Logic Book')
Premise 4: Ga

Within this context the argument proves that a human named provoker is a member of typologycentral. That is a true conclusion.

In addition to this, this argument is the example that you were looking for. Here we have a deductively valid argument with a false premise and a true conclusion. The false premise is that all humans are members of typologycentral.



^^I think that about did it for this argument. That is uncontroversially correct.

Certainly is, but it only proves that provoker is a hamster as in this argument as he is defined as a hamster. The conclusion in this proof could not have been deduced without premise 2 which states that provoker is a hamster (Fa). Because the second to last premise reads that if provoker is a hamster than he is a member and premise 2 states that he is a hamster, the conclusion that he is a member is merely tantamount that a hamster is a member. This is false because a person who is called provoker is a member of typologycentral, a hamster called provoker is not.
 

reason

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Provoker has been defined as a hamster
Here is where you've been going wrong.

The premise "Provoker is a hamster" does not define "Provoker" as a hamster.

That was my point about John the blacksmith: just because John is a blacksmith does not mean John is defined as a blacksmith; otherwise John would cease to exist if he ever changed profession.

To resolve your problem, just define "Provoker" as being either a hamster or a human. Thus:

Every hamster is a member of TypologyCentral
Provoker -- defined as being either a hamster or a human -- is a hamster
Therefore,
Provoker -- defined as being either a hamster or a human -- is a member of TypologyCentral​

That you can't figure this out astounds me.
 

Orangey

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Maybe a better example of a deductively valid argument with false premises and a true conclusion would be as follows:

No Americans are purely Caucasian (False, some Americans are purely Caucasian)
All black people are Americans (False, there are black people in Africa)
===============================
No black people are purely Caucasian (True, because being black means that you are by definition at least partially non-Caucasian)

Here's the proof.

1. (x)[Ax->~Px]...................Premise
2. (x)[Bx->Ax].....................Premise
3. Ax->~Px.........................Universal Instantiation
4. Bx->Ax...........................UI
5. Bx->~Px.........................3, 4 Hypothetical Syllogism
6. (x)[Bx->~Px]...................5 Universal Generalization

A = American
P = purely Caucasian
B = black
 

SolitaryWalker

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Here is where you've been going wrong.

The premise "Provoker is a hamster" does not define "Provoker" as a hamster.

That was my point about John the blacksmith: just because John is a blacksmith does not mean John is defined as a blacksmith; otherwise John would cease to exist if he ever changed profession.

To resolve your problem, just define "Provoker" as either a hamster or a human. Thus, the premise "Provoker -- who is either a hamster or a human -- is a hamster" states that Provoker is a hamster, while the conclusion states "Provoker -- who is either a hamster or a human -- is a member of TypologyCentral."

You may define provoker is either a hamster or as a human, yet this would entail a different proof from the one that you had.

Fx-Is a hamster.
Gx-Member of typologycentral
Mx-Is a human.
a-provoker

Premise 1 -Ax (Fx-Gx) (All hamsters are members of typologycentral)
Premise 2- Ax (Mx-Gx) (All humans are members of typologycentral)


(Note, here I have the liberty to define provoker as either a hamster or a human)

Premise 3: Ma (Provoker is a human)
Premise 4: Fa (Provoker is a hamster)

Step 5- Fa-Ga (1, Universal elimination)
Step 6- Ma-Ga (2, Universal elimination--if provoker is a human then he is a member of typologycentral)

Step 7- Ga ( 4, 5 Modus ponens)
Step 8- Ga (4,6 Modus ponens)


The proof that you posted defined provoker only as a hamster, this proof gives one the liberty to define provoker is either a human or a hamster.
 

reason

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

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

reason

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SW, in other words,

F : ... is a human
G : ... is a hamster
a : Provoker

The term "Provoker" is defined as being "Fa v Ga," but isn't defined as being "Fa" or "Ga" in both the premise and conclusion.

The premise may state that Provoker is a hamster, but the term "Provoker" is not defined as a hamster, because Provoker would still be provoker if he were a human, hence "Fa v Ga."
 
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