Inference rules

[vano]

Regarding an argument, if premises are true and if the rules of inference are valid then conclusion can only be true. I have a question about this: is this just an assumption or something that has been proven?

 

To be more clear, if in an argument premises are false and inferense is valid or invalid conclusion still can be true and that can be shown by examples (I want to emphasize here "by example"). Or, if premises are true and inference is invalid the conclusion can still be true. And, again, that can be shown by examples.

 

Examples:

 

1) false premise and false inferense:

 

humans don't talk

therefore my first name is Ivan (and it is I promiss you)

 

2) false premise and true infernece:

 

The earth is a flat object

All flat objects are planets

Therefor earthis a planet

 

3) true premise and false inferense

 

humans can talk

therefore my first name is not Ivan

 

What about the question above? Clearly, you cannot use examples here; you would have to give examples for infinite number of arguments and check conclusion's truth-value infinitely. So, why we accept above thing?

[HydrogenBond]

In your examples you are starting and concluding things you already assumed to be true or false. Your examples skip steps within logic that fully support the conclusions. If you were trying to infer something new or unproven one would need to start with true premises or premises that are logical conclusion that follow from already established premises. One can extrapolate quite far as long as you start with solid base premises, use all the needed reasonable logic steps and draw reasonable conclusions.

 

Some human's don't talk. Some humans are mute. Names are given at birth. Humans who are mute are also given names at birth. This person who does not talk is called John. John may not be able to talk but he can hear. Any person can say their name in their mind's ear (audio imagination) without moving their lips or uttering a sound. John can say his name without talking, even though he is mute.

[vano]

first of all you did not answer the question i asked. and i do really want to know the answer.

 

secondly, it is a fact in logic that false promises and valid or invalid inferense will not guarantee that conclusion is false, whether my examples are inappropriate or not; same holds true for arguments with true promises and invalid rules of inference.

 

thirdly, in the example "humans do not talk" i meant that none of the human beings can vocalize or talk and this makes the statement false(sorry if the premise was not unambiguous). i completely do not understand second paragraph: if you mean "my name is ivan" is not true statement you are not wright, since my name is ivan. if you mean there's no logical connection between premise and conclusion, then that is the idea of that example; namely i used invalid inferense.

 

and lastly, there's nothing in the logic that says one cannot start with premises that are asumed to be true or come to conclusion that we asume to be true. if you imply that premise/conclusion statements might not be true then tell me in which example i stated premise's or conclusion's truth value was not the way i stated in the begining of the examples.

[CraigD]

Regarding an argument, if premises are true and if the rules of inference are valid then conclusion can only be true. I have a question about this: is this just an assumption or something that has been proven?

Within a fairly esoteric, mathematical relm typically known as “formal logic”, this has been proven. The proof is trivial – a formal system is defined as a collection of algorithms to generate true statements, a collection of statements assumed to be true – premises – and the true statements (theorems) generated from them. By definition, if the algorithms are followed without error, they generate true statements.

 

Applying formal systems to practical, perceived reality, is less trivial, and highly counterintuitive. Formal systems readily contain contradictions – a statement and its negation, both formally true. They can also be shown to be incomplete – unable to generate statements that can be useful and consistent (not contradictory) postulates of the system. This has been known and proven by example for over a century. For much of that time, it was assumed that physical reality, or at least arithmetic, could be mapped to some formal systems that was not contradictory, and perhaps, even to one that was both consistent and complete. The search for these formal systems was a major and grand enterprise of Mathematics and Mathematical Physics, such as the encyclopedic Principia Mathematica and David Hilbert’s program for the future of Mathematics.

 

In 1930, Kurt Gödel published his “incompleteness theorems”, showing that any formal system capable of describing arithmetic on the natural numbers (counting numbers), cannot be both consistent and complete, pretty definitively putting an end to this grand search. It’s fair to say that the mathematical community is still reeling from Gödel’s theorems. To this day, they’re still shocking to mathematicians and Math enthusiasts at the point in their education that they’re able to appreciate them.

To be more clear, if in an argument premises are false and inferense is valid or invalid conclusion still can be true and that can be shown by examples (I want to emphasize here "by example"). Or, if premises are true and inference is invalid the conclusion can still be true.

Yes. The examples you give are “informal”. In a formal system, theorems generated by “false premises” are as valid as those generated from “true” ones, while conclusions generated by “invalid inference” are not therems. Your examples resembles the “inappropriate” choices of a formal system, in which the system generates some of the theorems you want it to (the “true conclusions”), but also generates many you don’t (eg: “a book is a flat object”, “all flat objects are planets”, “therefore a book is a planet”).

[Turtle]

Applying formal systems to practical, perceived reality, is less trivial, and highly counterintuitive. Formal systems readily contain contradictions – a statement and its negation, both formally true.

 

That sounds agreeable Craig.:shrug: This link from ughaibu in another thread expounds on some formal logical that surmounts apparent paradoxes:

http://www.cs.bham.ac.uk/~mmk/papers/TR/CSRP-03-01.html

 

This is the thread where it came up:

http://hypography.com/forums/books-movies-games/5865-diamond-2.html?highlight=fuzzy+logic

 

 

 

 

 

:shrug:

[Qfwfq]

Regarding an argument, if premises are true and if the rules of inference are valid then conclusion can only be true. I have a question about this: is this just an assumption or something that has been proven?

It is one of the inference rules of logic, called modus ponens, it can only be considered obvious.

 

If A => B

And A

Then B

 

Lewis Carrol wrote a well known and entertaining spoof of it, as an argument between Achilles and the Tortoise, leading to an infinite regress. You might like to read it and see what you think of it. Would you say the Tortoise is being reasonable?

[Doctordick]

Regarding an argument ... is this just an assumption or something that has been proven?

I don't know; do you think it's true? In fact, how do you decide something is true? Or, more importantly, what do you mean by the word "true"? :cup:

 

In order to discuss the subject of logic, we must first establish what we mean by "truth". Notice that I did not say, "how we determine what is true"; which is the subject of logic. If you read Manfred Kerber's presentation, you will find that the presentation contains a lot of words. One of the problems this presents is that we must figure out what these words mean before we can understand what he is saying. How do we approach such a problem? In essence, how do we "know" when we understand what he means by any given word? :)

 

Could I suggest that the central issue there is that we should not surprised by his usage of those words? That is, when a collection of words he uses constitute a collection of words we would expect in the circumstances where he uses them we can begin to conclude that we understand him. What I am saying is that our expectations are the central determining factor when it comes to the issue of understanding anything. Until we are no longer surprised by anything he says, we must presume the possibility exists that we are misunderstanding what he is saying. (Either that or we can conclude he is a nut case; which should not really be our goal.) :)

 

I found it interesting that Kerber makes the comment,"This way it was possible to build a safe area, which is free of paradoxes." It seems that being free of paradoxes can be seen as central in the definition of truth! In fact, it appears to me that finding our interpretation of Mr. Kerber's words to be "free of paradoxes" can be seen as an issue central to concluding that we understand what Mr. Kerber is talking about. So "paradoxes", what ever they are, seem to be a very important issue to understand. :D

 

Thus it seems to me that we need to understand what is meant by "a paradox". All I really have to go on there is all the examples I have found in my life with the usage of the word "paradox". It seems, at least on the surface, that a paradox is a case which yields "true" and "not true" for the interpretation of the same expression. If that is indeed what is meant by the word "paradox" (and it should be clear that the correct meaning is something I must leave to after I understand everything so I must only use this as a hypothesis) then it becomes very clear as to why paradoxes are to be avoided in our logic. They are to be avoided because the fail to provide us with any understanding; they fail to yield those "expectations" we use to determine if we understand. ;)

 

Whoa, the thing has just gotten circular: we can't know what he means if what he says is paradoxical! That seems to be a good reason to conclude paradoxes are not acceptable in our interpretations. How about the other side of the coin? What happens if we say, "anything totally without paradox is an acceptable explanation"? Would anyone like to discuss that hypothesis? :)

 

What I am getting at is the fact that everything comes down to explaining things and our problem is to understand the explanation. The only possible way of judging our understanding of any explanation is by our ability to make accurate judgments of expectations regarding further discussion. A decent prediction of the future is the central issue of understanding anything. Anyone who can't comprehend that can't comprehend very much. :)

 

Have fun -- Dick

 

"The simplest and most necessary truths are the very last to be believed."

by Anonymous

[vano]

Within a fairly esoteric, mathematical relm typically known as “formal logic”, this has been proven. The proof is trivial – a formal system is defined as a collection of algorithms to generate true statements, a collection of statements assumed to be true – premises – and the true statements (theorems) generated from them. By definition, if the algorithms are followed without error, they generate true statements.

 

Maybe I am wrong, but I don't see a proof. I can't see how definition can claim a proof of something. Just because we define a formal system that way it does not mean that such a system can be constructed.

 

It’s fair to say that the mathematical community is still reeling from Gödel’s theorems. To this day, they’re still shocking to mathematicians and Math enthusiasts at the point in their education that they’re able to appreciate them.

 

One of the reasons I asked this question is many mathematicians still think that aplies only to a small number of statements. But, what if, say, within a number theory there's a chanse that you could prove that 2a(ac+:rolleyes: does not equal 2ab+2a^2c. I now you can prove otherwise using basic rules of the system, but that does not mean there's no way proving above statement is not true. If there's a way of proving it false outside the system then you come with a probelm of proving consistensy within the new formal system and so on.

[vano]

It is one of the inference rules of logic, called modus ponens, it can only be considered obvious.

 

If A => B

And A

Then B

 

Thank you very much. That is most sincerey answer that I had. And with that nice example it's clear that it's a rule only which unavoidably leads to a problem of infinite regresion.

 

What I want to know is this though. Let's assume one comes up with a formal system as "clean" as possible: you have basic definitions without any reference to any model; you have starting postulates and and as carefully selected rules of inferense(I don't know though what that could mean) as posible. Now you procede with manipulating symbols and come up with a string. Asume this string is {a}>>c and this is our theorem. Is there any way of proof that there's no way within this system you could show ({a}>>c) where (string) would mean negation of a string. I asked that question to CraigD but I will appreciate different opinions.

[Qfwfq]

Asume this string is {a}>>c and this is our theorem. Is there any way of proof that there's no way within this system you could show ({a}>>c) where (string) would mean negation of a string.

First, it's important to point out that not any formal system must have negation in its syntax. Negation is one feature of formal logic.

 

If your formal system does comprise the inferemce rules of logic, then it is an assumption that any assert is either true or (aut, not vel) false, tertium non datur. This is called logical consistency and isn't strictly an axiom of the formal system but one about it. False, of course, means that the negation of A is true. This does not mean that for any well formed assert there must be a proof of either it or its negation.

 

To see why the consistency assumption is fundamental it can be shown that logic would be no use without it. It is easy to prove that if A is true and its negation also is, where A is any assert, then given any assert B one can prove it true as well by using the basic inference rules of logic. It may seem amazing, but it simply shows that the inference rules are such that negation represents what we mean by saying "true" and "false".