You guys need to find a bedroom. I'm surprised we others are allowed to witness the proceedings. — jgill

Was just trying to be a help for the request from FJ for clarification.

Yes, or in other words: denying the “antecedent” of a biconditional is not a fallacy. Yet denying the antecedent of a conditional is a well-known fallacy. — Leontiskos

The post seem to be in an obvious case of internet info. snack gone down into wrong pipe.
In deductive syllogism it is fallacy, but in inductive case, it is not fallacy. Because the real life case can be contradiction to the premise.

Yeah I am busy and in the middle of doing other stuff, and will have a look at it when I am free.

That's from the truth table.

Of course there are general rules for truth table and syntax rules, but for proof process, you must reason yourself for brining in the relevant inferences to the process.

(A implies B) is true, but (~B implies ~A) — flannel jesus

Even if it says True, when you compared with the real events, if they are false, then it is false. Also the premise is false.

There is no hard coded laws here. It depends on your inference and the case, and also checking with the real events in the world. The rules only says the principle. The application for finding out truth is flexible with other factors related.

If the truth table says so, yes. But compared with the reality event, it could be denied, in which case, the premise would be denied too. Proof process is for using the truth table rules, your inferences and also the reality events trying to prove that your argument was true. Some folks only cite truth table, and never allow you to make any inferences. Or they only check validity of the formulas which are not relevant. That is not logical proof.

n classical logic, A -> B being true always means ~B -> ~A is also true.

They have the same truth tables as each other. — flannel jesus

Sure, if the truth table says so, then it must be it. But in the empirical cases, you can also compare the TF values with the reality events, in which case if it is FALSE, then it proves the starting premise were FALSE.

If it rains, then ground is wet.
The ground is not wet, so it doesn't rain.

This is the case of contraposition isn't it?
It looks like it is sometimes true or sometimes not true.
Would it not depend on the checking over with the reality outside?
Did you check the truth table for it? I am not sure off hand without making up the truth table for it.

Our preferences and the resultant choices are products of hard determinism. — Truth Seeker

Could it not be your judgement which takes place always after your decisions on the choices? You are thinking them all as hard determinism, because you always reflect on them after the events, at which you cannot change them anymore from the view point of your reflections on them? Surely before the decisions, you had freewill to choose?

then accessing God would be by being that nature, your true nature, that living breathing organism. Be-ing a human creature, accepting that reality, as opposed to what we spend the vast majority of our time doing, creating our own reality becoming a human "god." — ENOAH

That sounds like Buddhism. Could it be right?

To incorporate the Abrahamic tradition, the former is the tree of life, the latter is the tree of knowledge. — ENOAH

Which sphere in the Tree of Life depicts morality?

A person sees it has not rained (~A), but then goes out to find the lawn is wet (B). This is possible because there are many ways for the lawn to get wet (B). If it rains, the lawn will be wet, but the lawn might also be wet for other reasons. — Count Timothy von Icarus

The argument was meant to prove R -> W, not (R or H) -> W.
If you wanted to prove (R or H) -> W, then yes of course, the premise should have begun with it.

For (P1 ..... Pn)-> W, you would need to go into Higher Order logic, wouldn't you?
We were trying to probe P1 -> W, but if you bringing in P31-> W, then you would be brining in irrelevant inference into the argument.

IFF must be regarded as presumed in the argument looking at the starting premise from the number of variables in the premise i.e. 1.

That would depend on the case under the proof. I will get back on that later. cheers.

Yes, it is correct. You don't need to always bring in Not A -> Not B for your inference. It is totally under your discretion of your inference for the case. It could be any case in the truth table you can bring in as your inference.

Every case in the truth value can be applied for your inference. If it helps to contrapositive for your argument, and it matches the reality event, then yes it can be adopted.

If you look closely the truth table, it only ever contains P and Q cases with various connectives and extensions. You never see in the table H -> W, once it started with P, Q. It tells you that in an argument, you don't bring in some unrelated cases and argue, the logic is incorrect. That is just sheer nonsense.

If you started with H, G in an argument, then it only have cases with H and G in the truth table. That is the rule No.1.

Ok, we are going out for a meal here. I am not sure if it is under determinism or freewill. Have a good day, and talk to you later for any other points.

In the case of If it rains the the ground will be wet. You think and infer, which will be most relevant and good to test, if it is true? Ah... how about if it doesn't rain, and the will not be wet? NotR -> NotW. And check it with the reality.

But you can bring other assumptions as counter reasonings and inferences for the proof process. If not sure, make a truth table, insert all the TF values according to the rules. Ok, Later~

It is not "lead to". It is an assumption, which you bring into the argument that you think most relevant and reasonable for the premise that you want to prove. But the truth table contains all the cases and TF values. If it is an empirical case, you must check with the reality for the TF values for the real TF values.

do you have examples where it doesn't apply? — flannel jesus

If not sure, make up an extended truth table for all the possible scenarios, and see all the cases for the TF values. It gets apparent.

I will get back on that, as I must go out now. cheers.

It would also depend on the case by case.

I think you are correct. But you don't have to always apply the contradiction. It depends on the case, and your inference.

The truth table value is from the rule. But you also check with the reality observations here too, because it is an empirical argument, not analytic.

Logic is a science of inference. That is one of my old logic book says. To prove a statement for true or false, you must start with a premise to prove, then make the relevant inference. (Don't make irrelevant nonsensical inferences such as hosing a garden into this argument, and you get told, are you having a laugh mate?)

If it rains, then the ground will be wet. (Premise to prove)
R -> W

If it doesn't rain, then the ground will not be wet. (Inference = WHY NOT??)
Not R -> Not W

At this moment, you check to see if it rains, and if the ground is wet in reality.
It doesn't rain, and the ground is not wet. (TRUE from the inspection and observation)
Not R -> Not W = True

Therefore If it rains, then the ground will be wet. (True)
R -> W

Therefore for proving A->B is true or false, Not A -> Not B premise works fine.

You can use a proof based on the Truth table too.
R | W | R and W | R or W | R-> W | NotR | NotW | NotR -> NotW
T | T | T | T | T | F | F | T
T | F | F | T | F | F | T | T
F | T | F | T | T | T | F | F
F | F | F | F | T | T | T | T

:worry: :fear: No need for apology my friend.
There seem to be definitely element of determinism in life. In fact every events in the past are under the hard determinism. No one can change the past.

Some events in the future are also under the HD e.g. old age, deaths - we know them for sure, but cannot change them. Only some actions and events at present and future are in the realm of freewill e.g. drinking coffee instead of tea, reading instead of going for a walk etc.

Regardless of what the hard determinism or constraints were, if someone came into your house, stole everything of your valuables, then you will morally accuse and legally punish the wrong doer, even if he says to you, that he was programmed to steal your valuables by his DNA, and he had no choice, and the wrong doings were the results of determinism and inherent constraint. Would you not? — Corvus

I am not sure if you answered this question.

https://en.wikipedia.org/wiki/Peano_axioms PA = (Peano Arithmetic) — PL Olcott

Ok, Peano. :up:

The HOL book by Bacon must be an introduction to the subject, and doesn't seem to discuss anything about Liars paradox or Tarski's truth. I will try to finish it first, and then look at the Tarski's truth and Godel numbers.

The full actual proof: https://liarparadox.org/Tarski_275_276.pdf

Is the actual Liar Paradox in PA ? — PL Olcott

What is PA?

https://andrew-bacon.github.io/papers/Front%20matter.pdf — PL Olcott

Strange, the book doesn't say anything about Godel, Tarski, undefinability, or Paradox in the whole content. It talks a lot about Labmda Calculus, Frege, Hilbert and High-Order Model Theory.

https://andrew-bacon.github.io/papers/Front%20matter.pdf — PL Olcott

That's the book.

s the whole paper. I cannot copy and paste from it.

I think you will find on p. 254 the requisite qualifications. — tim wood

Downloaded and saved into my iPad, thanks. Great reading material. :up: :pray:

Sure, I am currently reading on "High-Order Logics" by Andrew Bacon, and this is a really nice supporting thread for the reading. Thanks.

:blush: :pray:

My copy of " A Dictionary of Philosophical Logic" by R. T. Cook says,

"Tarski's indefinability theorem states that arithmetical truth cannot be defined in arithmetic - that is, there is no predicate definable in arithmetic that holds of exactly the Godel numbers of the truths of arithmetic. The indefinability theorem is closely related to Godel's incompleteness theorems, and is also at the heart of much research on the Liar paradox." -pp. 285-286

It sounds like Tarski's indefinability theorem is only applicable to arithmetical truths according to the dictionary.

What is the meaning of the word arithmetical?
/əˈrɪθ.mə.tɪk/ uk. /ˌær.ɪθˈmet.ɪk/) involving adding, subtracting (= removing a number or amount), multiplying, or dividing numbers: arithmetical problems. Figuring the amount is a simple arithmetical calculation.

Is the Indefinability theorem in the dictionary same as "undefinability" in the OP?

I agree with you.

You are very welcome TS. So is it the speech act or language which makes subjectivity to objectivity? We are communicating via writing on the keyboard and internet. But if we were in the same room, we would be conversing in ordinary language for sharing the subjectivity.

I agree with you.

I didn't say that objective moral values exist. I said that morality does not have to be objective to be valid. How can we possibly know about anything objectively? Everything we know is subjective. — Truth Seeker

But how can anyone know about subjectivity apart from one's own? Isn't shared subjectivity objectivity?
Until subjectivity is shared with others, it is private to the owner of subjectivity, which is unknown. When you shared your subjectivity, it then becomes objectivity.

Good points.

OK, you say that it is still good to have objective moral values. I think we do. It is far stronger than one's opinions or wishful thinkings on the moral issues. And some moral issues are also legal issues, and vice versa.

Whatever the case, the world don't care about HD. Regardless of what the hard determinism or constraints were, if someone came into your house, stole everything of your valuables, then you will morally accuse and legally punish the wrong doer, even if he says to you, that he was programmed to steal your valuables by his DNA, and he had no choice. Would you not?

As far as I know, morality is made up by humans. This is why it varies across time and place. Morality is a matter of subjective opinions. — Truth Seeker

Does it mean that there is no such things as morally good or bad to begin with? Does it not invalidate the question "Who is morally culpable?"?

I am trying to see whether or not HOL actually defeats Tarski Undefinability. — PL Olcott

One can formalize the semantics—define truth—of lower order logic in high-order logic. Under that fact, isn't it the case HOL defeats Tarsky Undefinability in the formalization, because TU only applies to the domain of Algebraic statements?