# Thread: Probability Relations and Induction

1. 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.

2. Originally Posted by reason
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.

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...)

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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.

Originally Posted by reason
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.

3. 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```

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

5. Originally Posted by reason
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.

Originally Posted by Orangey
^^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.

6. 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.

7. 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

8. Originally Posted by reason
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.

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

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

10. 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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