[NT] Probability Relations and Induction

[Provoker]

This contribution addresses probability relations and hopes to establish a rational component to induction and partial beliefs.

In deductive logic, a conclusion is true just in case the premises are true. An example of this is a categorical syllogism: if all As are Bs, and all Bs are Cs, then it can be deduced that all As are Cs. In deductive logic the movement of reasoning is from general to specific.

With inductive reasoning, empirical observations are made and from these generalizations are induced. For example, if a person throws a brick at a person's head at 10 second intervals, eventually the person may get into the rhythm of ducking at the interval to avoid being hit by the brick. Maybe this goes on for 19 bricks. Maybe at the 20th interval the person ducks in anticipation but is surprised that no brick is thrown. Point: it does not follow that because the brick was thrown 19 times that it will be thrown on the 20th. Similarly, a person can have an intuition that the world is flat, and from this make the claim that the world is flat. Yet we know that this would be a fallacious belief as scientific observations, satellite imaging, and thought have given us much more evidence to think the earth is a sphere. Yet, inductive arguments can be better or worse depending on the evidence in its favor. Indeed, based on a series of precedents one can have a highly educated guess that the brick will be thrown at the 20th interval, that there will not be a tornado in Toronto tomorrow, and that the Sun will rise tomorrow, but these are not and are not supposed to be deductively valid. As a result, the logical thinker can never be certain that the sun will rise tomorrow the way he can be certain that a categorical or disjunctive syllogism is logically valid.

Yet, to say that inductive arguments are non-rational would be misleading. In order to give a rational component to induction we will need to bring in probability. In short, we will need to extend our notion of a logical relation to include probability relations in order to establish how intuitive knowledge can form the basis of rational beliefs that fall short of knowledge. As Keynes has pointed out, a probability statement expresses a logical relationship between proposition ‘p’ and proposition ‘h’ (h is usually a conjunction of propositions). A man who knows ‘h’ and perceives a logical relationship between p and h is justified in believing p with a degree of belief which corresponds to that of the logical relationship. As such, probability is a logical relation holding between propositions which are similar to, although weaker than, logical consequences.

A proposition may be probable to a certain set of data and highly improbable to another set of data which includes the first set as a part. For example, if the only fact you know about a boxer is that his fight ended 30 seconds into the first round, it is quite probable with respect to these data that someone got knocked out or technically knocked out. If you find out after there was a power outage at ringside during his fight, then the probability that someone got knocked out, on combined data, is smaller. We can also think through the following example: given that X is an inhabitant of Europe it follows that he lives either in Britain or in France or in Germany or…Here, one can assign probabilities to person X living in each state based on certain conditions—perhaps relative populations being one of those conditions. Indeed, conditions are critical. To every event there is a finite set of conditions relative to which the event is certain to happen or certain not to happen. If the evidence reveals more of these conditions, we shall say that we have a stronger case, and we can extract probabilities and assign relative values to consequences. Some have argued that one way of quantifying probability relations in a real sense, could be to have the degree of belief in a proposition 'p', which a particular man has at a particular time, by the rate at which he would be prepared to bet upon 'p' being true. In this sense, there is a continuum: falsehood<--l--l--l--l--> truth with various probabilities in between. It is argued, therefore, that the test of relative strength of belief can be quantified and thus measured by a bet a person is willing to stake on their belief.

All in on this point: in matters of induction we cannot hope for anything more than probable conclusions and, therefore, the logical principles of induction must be the laws of probability. In this way, we are able to link intuitions and partial beliefs to a rational base grounded in probabilities.

 

[Wonkavision]

^^^^^

That's a great explanation.

Thank you very much.

I wonder if there's a more concise and down to earth way of putting it.

Could we say that "Intuition alone only perceives the possibilities, but when combined with a rational base, it can determine logical probabilities?"

 

[reason]

In deductive logic, a conclusion is true just in case the premises are true.

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.

In deductive logic the movement of reasoning is from general to specific.

A deduction can be from a generalisation to a specific, from a generalisation to another generalisation, or a specific to another specific. In any case, this characterisation -- an attempt to make deduction appear the opposition of induction -- is rather misleading outside of basic syllogisms.

With inductive reasoning, empirical observations are made and from these generalizations are induced.

Unless we have stopped talking about logic and have started talking about psychology, inductions are never from observations. Premises are propositions, and while we may use them to describe some experience we had, there is no logical connection.

Yet we know that this would be a fallicious belief as scientific observations, satellite imaging, and thought have given us much more evidence to think the earth is a sphere.

A fallacy is an error of reasoning, not fact. A belief, by itself, cannot be fallicious, because a belief is not a kind of reasoning, but merely a claim, assertion, or proposition.

Yet, inductive arguments can be better or worse depending on the evidence in its favor. Indeed, based on a series of precedents one can have a highly educated guess that the brick will be thrown at the 20th interval, that there will not be a tornado in Toronto tomorrow, and that the Sun will rise tomorrow, but these are not and are not supposed to be deductively valid. As a result, the logical thinker can never be certain that the sun will rise tomorrow the way he can be certain that a categorical or disjunctive syllogism is logically valid.

Are you the arbiter of what people "can" and "can't" be certain about?

In any case, if it's just educated guessing, then why do you need induction at all? Why not just educatedly guess that the sun will rise tomorrow in the premises? Or more on point, what does putting such beliefs in a logical argument achieve in the first place?

Yet, to say that inductive arguments are non-rational would be misleading. In order to give a rational component to induction we will need to bring in probability. In short, we will need to extend our notion of a logical relation to include probability relations in order to establish how intuitive knowledge can form the basis of rational beliefs that fall short of knowledge. As Keynes has pointed out, a probability statement expresses a logical relationship between proposition ‘p’ and proposition ‘h’ (h is usually a conjunction of propositions). A man who knows ‘h’ and perceives a logical relationship between p and h is justified in believing p with a degree of belief which corresponds to that of the logical relationship. As such, probability is a logical relation holding between propositions which are similar to, although weaker than, logical consequences.

But evidence can provide no probabilistic support for a hypothesis beyond that part of the hypothesis which is equal to the evidence. In other words, attempting to attribute the increased probability to all parts of the hypothesis equally commits the fallacy of composition.

All in on this point: in matters of induction we cannot hope for anything more than probable conclusions and, therefore, the logical principles of induction must be the laws of probability. In this way, we are able to link intuitions and partial beliefs to a rational base grounded in probabilities.

The statement "there are at most a, b, and c ravens" is equivalent to "all ravens are not d, e, f, ...". In other words, unless the number of something is limited by pure logic, then positing such a limit in order to enable induction assumes a universal, and thus the underlying problem isn't solved.

 

[Provoker]

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​

.

What you want to say is that the syllogism is logically valid, which doesn't necessitate the conclusion's truth. Here, if one falsifies the premises, then the conclusion is not true because of its relation to the premises. In this particular case, the conclusion happens to be true for reasons outside of the syllogism itself.

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

Here, you've managed to conclude this part with basically restating what I said in other words: (Provoker "In deductive logic, a conclusion is true just in case the premises are true").

A deduction can be from a generalisation to a specific, from a generalisation to another generalisation, or a specific to another specific. In any case, this characterisation -- an attempt to make deduction appear the opposition of induction -- is rather misleading outside of basic syllogisms.

Aha! I think if we look at the bigger picture afforded by history deductive reasoning, if it is to yield fruit, tends to move from general to specific. For example, Newton induced his theory of gravity.Then in the 19th century other thinkers took Newton's theory (general principle) to deduce the existence, mass, position, and orbit of Neptune (specific conclusions) from perturbations in the observed orbit of Uranus (specific data). So, if your point is to be relevant, you would have to show that the movement from generalization to generalization and spefic to specific is not merely contained in a broader movement from general to specific. So long as these microdeductions are contained in a broader movement from general to specific, my statement remains true.

inductions are never from observations.

You don't really mean that do you? When a child induces the general claim that the world is flat because it appears that way, where do you think this comes from if not an observation?

A belief, by itself, cannot be fallicious, because a belief is not a kind of reasoning, but merely a claim, assertion, or proposition.

Ok this is a rather moot point, and you've isolated one point rather than considering the context in which it was used. By saying 'fallacious belief', it is taken to mean a belief that that is arrived at fallaciously--therefore a misleading belief. I suppose I should have explained myself more thoroughly in that the inductive reasoning is the part that is fallacious and that the belief that came out of that reasoning is false. In leaving this out, my assumption was that an intelligent audience would make these connections automatically.

In any case, if it's just educated guessing, then why do you need induction at all?.

Induction, experimentation, and a more empirical approach has historically been the source of many discoveries. The trouble with deductive reasoning is it is very limiting in that the conclusions are never beyond what is already contained in the premises. If inductive reasoning was never used, the human species would miss out on a lot of innovations, discoveries, and so forth. Orson Welles would have been disabled from experimenting during Citizen Kane, which revolutionized cinema. Newton would have been disabled from discovering gravity. In one sentence, if induction was abandoned many great discoveries would be foregone. Therefore given a weighty opportunity cost, perhaps you can detail why this would be a trade-off worth making?

 

[redacted]

Here, you've managed to conclude this part with basically restating what I said in other words: (Provoker "In deductive logic, a conclusion is true just in case the premises are true").

He's saying the conclusion is definitely true if the premises are true, but can also be true if the premises are false.

Aha! I think if we look at the bigger picture afforded by history deductive reasoning, if it is to yield fruit, tends to move from general to specific. For example, Newton induced his theory of gravity.Then in the 19th century other thinkers took Newton's theory (general principle) to deduce the existence, mass, position, and orbit of Neptune (specific conclusions) from perturbations in the observed orbit of Uranus (specific data). So, if your point is to be relevant, you would have to show that the movement from generalization to generalization and spefic to specific is not merely contained in a broader movement from general to specific. So long as these microdeductions are contained in a broader movement from general to specific, my statement remains true.

Here's a deductive argument.

A
B
therefore
A
B

Your conclusion is always going to be a subset of your premises (as long as you rewrite them accordingly), but it can also be a restatement of all of them.

So it can go from general to general, specific to specific, or general to specific, but never specific to general (that's induction).

Induction, experimentation, and a more empirical approach has historically been the source of many discoveries. The trouble with deductive reasoning is it is very limiting in that the conclusions are never beyond what is already contained in the premises. If inductive reasoning was never used, the human species would miss out on a lot of innovations, discoveries, and so forth. Orson Welles would have been disabled from experimenting during Citizen Kane, which revolutionized cinema. Newton would have been disabled from discovering gravity. In one sentence, if induction was abandoned many great discoveries would be foregone. Therefore given a weighty opportunity cost, perhaps you can detail why this would be a trade-off worth making?

I don't think he's saying induction, in your definition, shouldn't be used. He's just saying it's not logically justified (think Hume).




Here's my question...I see your whole thing about probabilities changing based on new inputs. But what if your inputs equally support two theories? You've provided no explanation for how you would choose between them.

It's likely you've heard of the "grue" example, but I'll go over it anyway. Suppose the term "grue" means green before 2012 and blue after 2012. So if you look at a leaf, you could call that green or grue, and be equally correct. No matter how many things you've seen in the world, there's no way to distinguish between those two labels. So if you make the conclusion, "in february of 2012, leaves will be blue", that's just as likely based on the data you've seen as "in february of 2012, leaves will be green".

This is meant to illustrate the point that there are an infinite number of hypotheses that are consistent with a set of data. So not only do we need a way to modify the strength of our beliefs based on data we see, we also need to be able to explain why we prefer certain kinds of hypotheses when an infinite number are all consistent with the data we've gotten.

 

[reason]

Two comments:

First, it is possible to validly infer a universal statement from a set of premises including only singular statements. For example,

a is y |= All x are y or not-y​

This only works when the conclusion is a tautology (and, therefore, even follows from the empty set of premises), but should be noted as a caveat to previous comments.

Second, here is a simple explanation of why inductive logic does not exist. Take this simple induction:

a is y, b is y, c is y, d is y |- every x is y​

Then define a new predicate z:

if x equals a, b, c, or d, and is y, then x is z; otherwise, if x is not-y, then x is z.​

Therefore, "every x is z" is true if:

a is y, b is y, c is y, d is y, e is not-y, f is not-y, g is not-y, ad infinitum.​

Since a, b, c, and d, are also z if y, then:

a is y, b is y, c is y, d is y |= a is z, b is z, c is z, d is z​

Therefore,

a is y, b is y, c is y, d is y |- every x is z​

However, "every x is y" and "every x is z" contradict each other. Therefore, two mutually inconsistent universals can be induced from the same set of premises. Using this procedure, we can define infinitely many predicates like z, all of which lead to mutually incompatible inductions. The only limit on what can be "induced" comes from deductive truth relations.

This argument is a pure logical analogue of the problem of theory-ladenness; it demonstrates the arbitrariness of inductive inference, and, therefore, the absence of an inductive logic -- which, by definition, is non-arbitrary. In other words, there is no way to define "inductive validity."

NOTE TO EVAN: The above is a formalised -- and stronger version -- of the grue problem of induction, but one which does not muddy the waters with issues of time, gems, and colours. Since it is purely formal, all it demonstrates is that induction is no logic -- it doesn't mean that people do not make generalisations.

 

[SolitaryWalker]

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

This argument is guilty of the informal logical fallacy of equivocation. Logic has two components, extensions and intensions. Extensions are concerned with how entities of discourse relate to each other. (In accordance to laws of reasoning, for instance, demorgan's law, modus ponens, modus tollens and so on.) Intensions are concerned with how the terms of discourse are defined.

When I say that provoker is a member of typologycentral, I define provoker as a person who made several posts on this forum. If I say that provoker is a hamster, I mean something entirely different. Hence, in the argument above, the conclusion is false. A hamster named provoker is not a member of typologycentral.

For this reason, the argument cited above is not an example of an argument with false conclusions leading to true premises and therefore does not prove that it is possible for a deductively valid argument to have false premises and a true conclusion. What we have here is an argument with false premises leading to a false conclusion. However, because the reasoning in question is deductively valid, the conclusion would have been true if the premises were also.

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

However, an argument with contradictory premises may be deductively valid and at the same time entail a false conclusion. Since anything follows from a contradiction, it is possible to derive a true conclusion from an argument with contradictory premises. If you construct the truth table for the following argument, you will see that not a single instance is documented where both premises are true and the conclusion is false. (You will also notice that not a single a instance is documented where both premises are true, hence in this case, all arguments with contradictory premises are valid by definition as it is impossible for such arguments to contain all true premises and a false conclusion. An argument is invalid if and only if it contains a single instance of all true premises and a false conclusion.)

Premise one: A
Premise two: Not A.

Conclusion: B

In the argument with contradictory premises, it is possible to have one or more false premises that lead to a true conclusion.

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

In short, I endorse reason's view that it is possible to have an argument with false premises that lead to a contradiction, yet I think the argument he cited does not adequately exemplify this concept. An argument with contradictory premises is the only example of this phenomen that I can think of. Such an argument is deductively valid, but always epistemically unacceptable as such a reasoning process is ostensibly preposterous whether it yields the true conclusion or not. This is so because a reasoning process of this kind can entail any conclusion.

So it can go from general to general, specific to specific, or general to specific, but never specific to general (that's induction).

Although you are onto something with the supposition that induction is typically more general than deduction and in most cases deductive reasoning goes from general to the specific, your claim that a deductive reasoning process never leads from specific to general is false.

A general proposition is one that contain many components and is typically opaque in meaning. A specific proposition is one that contains very few components and tends to be specific in meaning.

Pay careful attention to this reasoning process.

Premise 1: A
Premise 2: If A then B (A horseshoe B in the language of Symbolic Logic)
Step 1: B (1,2 Modus Ponens)
Conclusion: B or A ( Or is used in an inclusive sense in symbolic logic. Such a statement is merely saying that at least one of the two terms is true. Could be just one, it also could be both. Hence, when I prove that B is true, I can infer the proposition of B or Anything. I could have as justifiably stated B or not A, or B or Z)

Another example.

Premise 1: If A and B.
Step 1: A
Step 2: B
Conclusion: if A then B. (Once you have proved B, you can state that if anything then B, as in material implication the antecedent serves as just one possible reason for the verity of the consequence. If the consequence is true, the truthfulness of the antecedent is no longer relevant as the consequent may be true for reasons other than the verity of the antecedent. This is not to be confused with the triple bar symbol, or logical equivalence. In that case the antecedent serves as the only reason for the truthfulness or falsity of the conclusion. In the expression of A if and only if B (triple Bar), B is true only if A is and A is true only if B is. )

In both examples we start with very specific premises (straightforward and simple) and arrive at a conclusion that is very general. Because the examples above evince that it is possible for deduction to proceed from specific to general, we know that it is possible for deduction to proceed from specific to general and vice versa. (Its a truism that deduction can go from general to specific.)

 

[reason]

SW,

You really need to go and learn something about logic.

 

[reason]

By the way, this is a valid deduction,

The table is white and the sky is blue
Therefore,
The table is white​

Or,

The table is white
Therefore,
The table is white or the sky is blue​

Deduction can't be adequately defined as a mode of inference from the general to the specific, because only in some cases does this characterisation hold.

The matter is one of logical strength or content. In a deduction, the premises must have greater logical strength than the conclusion, i.e. the conclusion cannot entail anything which the premises cannot also entail. Or in other words, the logical content of the conclusion must be a subset of the premises. In some deductive arguments (quantitive relations), this can manifest itself as inference from the general to specific, but it need not for all cases of deduction.

NOTE: Regarding my former comments, this is why the relation between induction and deduction is not like addition and substraction. For example, consider trying to induce the premises of the first argument above from its conclusion. I don't think even the most ardent inductivist considers that an "inductively valid" inference, and yet its reverse is valid deductively.

 

[redacted]

Two comments:

First, it is possible to validly infer a universal statement from a set of premises including only singular statements. For example,

x is y |= All x are y or not-y​

This only works when the conclusion is a tautology (and, therefore, even follows from the empty set of premises), but should be noted as a caveat to previous comments.

Ah. Sure. But then you could rewrite the premises to be general too. I guess that means my previous statement is kinda useless. I probably should have specified non-tautologies.

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

This argument is guilty of the informal logical fallacy of equivocation. Logic has two components, extensions and intensions. Extensions are concerned with how entities of discourse relate to each other. (In accordance to laws of reasoning, for instance, demorgan's law, modus ponens, modus tollens and so on.) Intensions are concerned with how the terms of discourse are defined.

When I say that provoker is a member of typologycentral, I define provoker as a person who made several posts on this forum. If I say that provoker is a hamster, I mean something entirely different. Hence, in the argument above, the conclusion is false. A hamster named provoker is not a member of typologycentral.

Huh?

You seem to think his conclusion is "provoker is a hamster and a member of typecentral" instead of just what it says.

 

[SolitaryWalker]

SW,

You really need to go and learn something about logic.

You've got it backwards buddy, you're the ones propounding conclusions without stating arguments to support them. On the few occassions you do have arguments, they are subtly fallacious. And of course, the way you responded to this charge shows that you're not knowledgeable or skilled enough to see the obvious howlers. (E.G equivocation fallacy where you define provoker as a hamster and at the same time maintain that provoker as a person who posts on this message board.)

I bet you haven't done a single SL proof in your entire life.

 

[SolitaryWalker]

Ah. Sure. But then you could rewrite the premises to be general too. I guess that means my previous statement is kinda useless. I probably should have specified non-tautologies.



Huh?

You seem to think his conclusion is "provoker is a hamster and a member of typecentral" instead of just what it says.

The premise of that argument defines provoker as a hamster. If I say Evan is a hamster, I define Evan as a hamster. (In this context it is not implied that Evan is a human being, as nothing of the like is stated specifically. You are the one assuming that Provoker/Evan (substitute it with any name), is a person instead of what it says. All it says is that Provoker/Evan/any name is a hamster. Nothing else.)

In other words, what is meant by my proposition is not 'provoker=person who posts at this forum AND provoker=hamster who is a member of this forum' but simply this: provoker=hamster is a member of the board, and only this.

The expression provoker=human being who posts on this board is outside of the context of this argument because the premises defined provoker as a hamster only.

 

[reason]

You've got it backwards buddy, you're the ones propounding conclusions without stating arguments and on the few occassions you do have arguments, they are subtly fallacious. And of course, the way you carry your discussions shows that you're not knowledgeable and not skilled enough to see the obvious howlers.

Backwards? I didn't write anything about me.

So if I have it backwards, then you still need to go and learn some logic, since I need to go and learn more logic, too.

 

[redacted]

The premise of that argument defines provoker as a hamster. If I say Evan is a hamster, I define Evan as a hamster. (In this context it is not implied that Evan is a human being, as nothing of the like is stated specifically. You are the one assuming that Provoker/Evan (substitute it with any name), is a person instead of what it says. All it says is that Provoker/Evan/any name is a hamster. Nothing else.)

Uh....

I don't really know how to respond here.

The point was that if the premises are false (so assume that provoker is not a hamster), the conclusion can still be true.

 

[Orangey]

This is meant to illustrate the point that there are an infinite number of hypotheses that are consistent with a set of data. So not only do we need a way to modify the strength of our beliefs based on data we see, we also need to be able to explain why we prefer certain kinds of hypotheses when an infinite number are all consistent with the data we've gotten.

Become a constructive empiricist a la Bas von Fraassen.

 

[SolitaryWalker]

Uh....

I don't really know how to respond here.

The point was that if the premises are false (so assume that provoker is not a hamster), the conclusion can still be true.

Let me try and walk you through this reasoning process more carefully and thoroughly.

Premise 1: Provoker is a hamster. (What I mean here is that the identity of a provoker is a hamster. )
Premise 2: All hamsters are members of the message board.

Conclusion: Provoker (a person who is the author of this thread) is a member of the message board.

This argument is deductively invalid. The premise that there is a hamster named provoker and the premise that all hamsters are members of the message boards do not guarantee the conclusion that the person who authored this thread is a member of this message board.

Plug it into the truth table. Have statement H represent provoker is a hamster. Statement M represent that all hamsters are members of the message board. Statement P represent that provoker is a member of this message board. You will find one instance where both of the premises are true and the conclusion is false.

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

I shall expound upon the rationale behind the views expressed above once more.

The conclusion that we are trying to prove is this: A person named Colin who uses the account of Provoker, the author of this thread is a member of the message board.

This is one of our starting points: Provoker is a hamster. Another way of saying this is that there is a hamster who is called provoker.

This is another one of our starting points: All hamsters are members of the message board.


The above starting points (premises) do not prove the conclusion. They have nothing to do with the person named Colin who is the author of the thread. They only prove the conclusion that there is a hamster who is called provoker and is a member of the message board. This, however, is not the conclusion we intended to reach. The conclusion we intended to reach was that there is a person named Colin who uses the account of Provoker and is the author of this thread. Proving this is a radically different result from proving that there is a hamster who is called provoker.

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

What you seem to be confused by is the following phenomenon, the definition of provoker as the person who interacts with us on the message board and the definition of provoker as a hamster (as stated in premise one). This is merely a play on words, as two different creatures seem to have the same name.

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

Finally, what is my point?

The point was that if the premises are false (so assume that provoker is not a hamster), the conclusion can still be true.

This is true, but the argument selected by reason does not provide an adequate example to support this point. The adequate example to support this point is the argument with contradictory premises that I have cited.

 

[SolitaryWalker]

Backwards? I didn't write anything about me.

So if I have it backwards, then you still need to go and learn some logic, since I need to go and learn more logic, too.

That is true. The conversation can be summarized as the following: X said that Y needs to learn logic. The reversal of this would be this: Y said that X needs to learn logic.

Indeed, you did not say that you knew logic and for this reason, the reversal of my proposition does not entail the conclusion that I do not need to learn logic and you do. 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.

In any case, I stand by my claim that you've committed the equivocation fallacy as you have failed to construct an argument to refute this view.

 

[redacted]

Let me walk through the rationale behind the argument again.

The conclusion that we are trying to prove is this: A person named Colin who uses the screenname of Provoker, the author of this thread is a member of the message board.

This is one of our starting points: Provoker is a hamster. Another way of saying this is that there is a hamster who is called provoker.

This is another one of our starting points: All hamsters are members of the message board.


The above starting points (premises) do not prove the conclusion. They have nothing to do with the person named Colin who is the author of the thread.

When you use one term twice in an argument, it seems like a good assumption that you're referring to the same thing.

The argument can be rewritten as this:
All As are Bs
x is an A
therefore
x is a B

Another way of thinking about this is that IF the premises are true, the conclusion is true. It says nothing about IF the premises are false. If the premises are false, we know nothing about the conclusion. Therefore the conclusion could be true or false. In this case, the conclusion is true.

It's like this. If you know A->B, and ~A, you know nothing about B.

 

[SolitaryWalker]

When you use one term twice in an argument, it seems like a good assumption that you're referring to the same thing.

The argument can be rewritten as this:
All As are Bs
x is an A
therefore
x is a B

Another way of thinking about this is that IF the premises are true, the conclusion is true. It says nothing about IF the premises are false. If the premises are false, we know nothing about the conclusion. Therefore the conclusion could be true or false. In this case, the conclusion is true.

It's like this. If you know A->B, and ~A, you know nothing about B.

Lets see if we are on the same page.

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.

In this regard, I agree with your statement that in a deductively valid argument, the conclusion could be either true or false.

However, there appears to be a communication gap that has not yet been bridged.

Lets consider the following argument.

Premise 1: Provoker is a hamster. (There is a hamster who is called provoker)
Premise 2: All hamsters are members of the message board.
Conclusionrovoker is a member of the message. (A hamster who is called provoker is a member of the message board.)

This argument is deductively valid, however the conclusion is false because the premises are false. If I was to make a logical fallacy and at the conclusion define provoker as a human being, my argument would be invalid. (As I said in my previous post, the fact that there is a hamster who is called provoker and the fact that all hamsters are members of the message board has nothing at all to do with a human being named provoker possessing membership at a message board. In other words, this argument is deductively invalid because EVEN IF the premises are true, there is no guarantee that the conclusion will be true. Remember, the definition of the deductively valid argument is that the truthfulness of the premises guarantees the truthfulness of the conclusion. If there is no such guarantee, then the argument in question is necessarily invalid. In this case, as I have explained, even if both premises are true (that there is a hamster called provoker and all hamsters are message board members) there is no guarantee that the conclusion( that a human being named provoker) is true. Therefore this argument is invalid.

Here, I have demonstrated why equivocation is a fallacy: in this instance it directly leads to a deductively invalid chain of reasoning.

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Again, what is my point.

When you use one term twice in an argument, it seems like a good assumption that you're referring to the same thing.

The argument can be rewritten as this:
All As are Bs
x is an A
therefore
x is a B.

In the context of formal logic, what you have said here is false. The assumption precludes the possibility of people making the equivocation fallacy. At times (as it has occured in this case) people mean different things when they use the same word twice. This is known as a play on words. It has value in poetry and literature, but does not carry any weight in formal logic. It is known as the equivocation fallacy.

As aforementioned, if there is no equivocation fallacy in the argument above(the word means the exact same thing both times it is used), the result is the following: the conclusion is false, but the argument is deductively valid. Reason claimed that this example proves that an argument can have false premise(s) and a true conclusion. I have shown that this argument has false premises and a false conclusion.

See the figure below.

Premise 1: Provoker is a hamster. (There is a hamster called provoker)--Lets say that this is true. There may be a hamster somewhere in the world who is called provoker.

Premise 2: All hamsters are members of the message board. (This is our false premise, no hamsters are members of the message board.)

Conclusion: Provoker is a member of the message board. (False, there is no member of the message board who is a hamster and is called provoker.) This statement, however, would be true if premise 2 was true.

 

[reason]

Evan,

All SolitaryWalker is saying that the concept of "Provoker" includes the property "non-hamster," so my premise "Provoker is a hamster" is like saying "A non-hamster is a hamster" i.e. a contradiction. (It's still not an equivocation, though).

However, 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.

To say that such an objection is besides the point is an understatement, and I would advise anyone who said such a thing to go and learn some logic.