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?

But that doesn't mean the door is closed on God. Only that God cannot be accessed by our Minds. Other means must be employed... — ENOAH

Sure, God could be an abstract object which is not perceivable via sense perception. Or it could be a manifestation in some mundane physical forms such as light (remember, God said, Let there be light, and there was light. in the Genesis?), or as ancient Egyptians believed God could be Sun. Without Sun, all life on earth will be extinguished within days.

As you suggested, God could be contacted in some other way than seeing or hearing him. It proves human reason and thinking is not really the 100% certain criteria for all the knowledge in the world.

What are the other means for employment you suggesting?

So, though I am yet to be a reader or student of Wittgenstein, I've come across enough to say, and I paraphrase, that of which we cannot speak, we must remain silent. — ENOAH

When W. said that of which we cannot speak, we must remain silent, was he not saying something already on something of which we cannot speak?, which implies that it is impossible for a language possessor to remain silent on things one feels inquisitive? Therefore he broke his own code as soon as he uttered the sentence?

I don't know the answers to your questions because I have never met any God or Goddess or Gods or Goddesses. It's possible that they are all fictitious. I am agnostic about it. — Truth Seeker

Fair enough. At least you are honest with you answer on the questions, which deserves respect. Not knowing can be start of new investigative discussions and analysis. Not knowing can be better position than knowing incorrectly.

Anyhow, I have asked the questions regarding Gods, because I believe that morality can have close connection with existence of God. For example, we can ask where does morality come from? Is morality based on human reasons? Or social customs and rules? Or on God?

If morality is based on human reason or social customs, then does it have absolute right or wrong in the judgements? In this case, would it be right to ask who is morally culpable?

For something to be absolute right in judgements, should it not be from something absolute in knowing and existing?

A knowledge ontology https://en.wikipedia.org/wiki/Ontology_(information_science) is essentially an inheritance hierarchy of types from type theory which is apparently the same thing as HOL. — PL Olcott

Great link with much useful info to learn. Thank you PL.

There seems to be a finite limit to the number of orders of logic needed. — PL Olcott

I was under impression that higher than 3rd-order logic would be for the multiple set theories and advanced calculus applications, therefore they wouldn't be used for describing the empirical world cases.

For formalizing the entire body of human knowledge that can be expressed using language we need this: — PL Olcott

High-Order Logic seems to be more flexible and powerful for the real world cases due to its expanded variables availability for the properties and relations.

If God or Goddess or Gods or Goddesses made me all-knowing and all-powerful, I would be convinced that it is possible to be all-knowing and all-powerful and I would then know that God or Goddess or Gods or Goddesses exist and what God or Goddess or Gods or Goddesses are actually like. — Truth Seeker

Would it be the only way that you could know the existence of Gods and Goddesses? No other ways?

Since you have studied the world religions extensively in the past, another questions would be, are Gods always in both sexes? Why are there so many Gods? Should it not be the only one God for the whole universe? If there are so many Gods, then which one is the real God?

I have studied most of the religions on Earth but not all of them. I have tried praying to many gods and goddesses. None of them answered my prayers. — Truth Seeker

Could it be the case you might have had prayed to the unreal or fake Gods, and there were no response for your prayer from them?

In Merlou-Ponty, the body itself is consciousness. According to his view, consciousness in the body would mean the body in the body, which sounds confusing. But the point seems clear. When the body dies, the consciousness also dies too.

Higher-order logic is the union of first-, second-, third-, ..., nth-order logic; i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply. — PL Olcott

Do we need more than first and second order logic in practical uses?

I don't know if any god or goddess exists. I have never met any. I don't know what they are like except for how they are portrayed in religious books. — Truth Seeker

So you studied all the religions in the world, and also the concepts of God extensively. You think about God much of your life. However, you are still agnostic until you actually meet God yourself face to face. Is it correct?

You can carry on with the other interlocuters and I am sure you will have good discussions. cheers.

Good question. It depends on the case you are trying to prove. Some cases will work ok with MP or MT. But the cases like Cogito is awkward with the formalisation. You try different inferences and reasonings, and whatever looks most reasonable should be used, I believe.

The classic syllogism cannot handle more complicated cases well, and it would be better to use Modal, Epistemic or Descriptive Logics. But if you convert the complex sentences into more atomic ones, and formalise them, then it works ok too. I am not a Logic expert, and I will be rereading my old logic books to brush up my knowledge on it.

I am now really bowing out from this thread. I have spoken enough, and learnt a lot myself. Thank you for your engagement with me. Although there were some rough times between us, I respect your strong interest in the subject. I hope to meet you in the other threads for the other discussions later hopefully. All the best.

It has to be one or the other. Either (a -> b) leads to (~a -> ~b) as a general rule, or it's not a general rule. I would like clarity on this. — flannel jesus

Thats an inference.

A→B ↔ ¬A∨B
¬A∨B ↔ B∨¬A
B∨¬A ↔ ¬B→¬A = ¬A -> ¬B ? — Corvus

This is inference from the rule.

If this proof were valid, A→B would always imply ¬A → ¬B - that's what I call a "general rule". — flannel jesus

Yes, that was the general rule. It was to show the logical inference processes in detail from the rule to Lion because he seems having difficulties understanding it.

Dumb troll. — Lionino

It is not. You are wrong again.
¬(a→b) = It is not the case (a→b) = negation. It is not contradiction.
You never admit the truth as truth. That is part of your problem.

The contradiction to a→b is ¬(a→b), it is not ¬a→¬b. — Lionino

This seems to be your problem. ¬(a→b) is negation, not contradiction.
You don't know the difference between negation and contradiction.

It is not, these two are not mutually contradictory. One translates to (a∨¬b) and the other to (¬a∨b). Both are true if a and b are true. — Lionino

A→B ↔ ¬A∨B
¬A∨B ↔ B∨¬A
B∨¬A ↔ ¬B→¬A = ¬A -> ¬B ?

I await to hear your contradiction of "I think therefore I am" in plain English, and will take it from there.

The contradiction to "I think therefore I am" is not "I don't think therefore I am not".

More BS — Lionino

What is the contradiction of it? Tell us exactly what is the contraction of "I think therefore I am" in plain English.

The contradiction to a→b is ¬(a→b), it is not ¬a→¬b. — Lionino

You must reason the contradiction, and check it over with the real events or existence for the truth or falsehood. You don't keep on going on with the set truth table on this cogito case.

~~