@user21820 I'm reviewing things. I want to ask, when truth tables are used (part of curriculum to us), we are considering the possibility of a statement being false. But in natural deduction, there's no such thing as "false", since only a statement is written or it is not, and there is no sense of "true" or "false", right?

@soupless Wrong.

Ok. I think I am misunderstanding things again.

Although there is no notion of truth in the formal system itself, the goal of the system is to make sure that you can only write true statements (in their context).

Otherwise, the system would not be as meaningful as it is.

So you are only half-correct.

To use the system, you do not need to know anything about truth; you just need to know how to follow the rules.

But to be convinced that the system rules are meaningful, and even desirable, you do need the notion of truth.

@soupless Ah ok that is a different question.

And yes, in the system there is no "is true" or "is false", so the former would be syntactically malformed whereas the latter would be allowed.

Because premises are supposed to be true from the start, right?

No that's not the reason.

Not really the reason.

Oh, right. I forgot that contradiction exists.

When you write "If P:", inside the system it doesn't have to have any meaning. The system doesn't care whether you used "If" or "Eeks" or something else.

However!

To convince someone else that the system is useful for reasoning...

You would say something like:

You may want to review that old conversation sometime.

In that 'attempt to convince', you would be talking about whether "P" is true or not.

This talking is outside the system.

I meant that there's no "is false" in premises, since we would use a negation for that. And yes, our teacher did mention that along the lines of "If you start with true premises, you must end up with a true conclusion."

Yes.

But I still want to emphasize something you don't seem to get.

"¬P", viewed purely from inside the system, doesn't have to mean "P is false"!

But!

If we want people to think the system is meaningful, we would tell them that we interpret "¬P" to mean "P is false".

Maybe even "P is false in this world".

It is possible that there are alternative interpretations of the formal system.

But the one that makes it most useful does happen to be the one that interprets "¬P" as "P is false".

Does this make sense?

I think I understand this for some degree since ¬P being "P is false" is defined in classical logic, right? And other formal systems may(?) allow a statement being both true and false, or maybe even allow values aside from true and false. Is this reason correct?

I'm not too sure whether what you have in mind is correct.

What I was saying is that the exact same deductive system that I gave you for (classical) FOL may have alternative interpretations.

You should mentally be clear about the distinction between using the system and interpreting what is being written in the system.

That's what I am trying to say, although I'm not sure if I expressed it correctly.

What you said referred to "other formal systems". I did not.

I was only talking about one single formal system.

You can see a simpler example of this issue in the interpretations of classical PL (just the propositional logic).

The standard (and most useful) interpretation is that every atomic proposition is interpreted to have a boolean truth-value.

And the interpretation of each boolean operation is according to the truth-tables you know.

But that is not the only interpretation that makes the classical PL deductive system meaningful.

Another meaningful interpretation is that there is a fixed set U and each atomic proposition is a subset of U.

I still haven't realized any interpretations of classical PL aside from the boolean truth-value, so that is what I missed.

Letting i(P) denote the interpretation of atomic proposition symbol P, in this interpretation i(P∧Q) = i(P)⋂i(Q) and i(P∨Q) = i(P)⋃i(Q) and i(¬P) = U∖i(P). It can be shown that the system is meaningful under this interpretation. What exactly is the meaning, you can try to figure out if you like.

That's why I said it is important to distinguish between the rules of the system (which are strictly speaking meaningless in themselves), and how we interpret the statements in the system and the meaning of the rules.

Wait. Since the rules of the system, by themselves, are meaningless, then our interpretation determines the meaning?

Exactly!

And there is no way around this. There must be interpretation at some point to imbue meaning to anything.

For beginners, we might omit this discussion, but since you asked about the difference between "P" and "P is true", it's definitely something we need to discuss. =)

So for a beginner, we can say:

> When we (are allowed to) write the statement "P." in the system, it means "P is true."

This isn't wrong. As long as we consistently talk about what the system means to us under the standard (most useful) interpretation, we are fine.

But once you want to separate the system from its meaning in our world, then you need to distinguish the written statement and its interpretation.

Makes sense now?

Beginners do not need to know alternative interpretations. That is why we don't need to complicate things for them.

Kind of. Is it is something like when an author gives the exact definition and rules when using an operator that is defined elsewhere, since the interpretation of the said operator is already given?

@soupless Yes there is a similarity there, but maybe don't take the similarity too far.

Just think of the written statements as being a grammatical string of vocabulary in some language.

The grammar is the rules, and the vocabulary is the symbols.

In the absence of a world, the language is meaningless.

But humans designed languages so that each word/symbol in the language either refers to something in our world or refers to some concept that pertains to our world.

That referring is the interpretation.

For example, in older English "want" can mean "lack". This is part of the older interpretation. But in modern English, this possible meaning is mostly lost (i.e. most native speakers do not know about it) so many native speakers actually misinterpret the sentence "I shall have no want."!

Same sentence, but different interpretation.

So misinterpretations are language barriers, for "languages"?

That's also why it's an important fact that the standard interpretation of FOL is so obviously meaningful that nobody has any trouble getting to the same interpretation.

Otherwise we would be running in circles just trying to explain what the boolean operations mean!

Can you explain "and", for example?

(It's a rhetorical question. You don't have to try.)

@user21820 I think I am getting it. To check, "and" is defined as the boolean operation where the "and" of propositions become true if each proposition is true. Otherwise, it is false. I'm not too sure how to define "true" and "false" if it is needed, which seems like an intuition as to how I understood it.

@soupless You are correct. There is no way to define "true" and "false"; you have to rely on the other person's understanding of truth in the real world.

But it just so happened that we had the same interpretation regarding truth values, so I think it's left as it is.

In mathematics, if you work within a foundational system, you can of course define the truth-value of a PL sentence given the truth-values of its atomic parts, with some arbitrary definition of "truth-value" (say 1 for "true" and 0 for "false"), but you can see that it again has no meaning unless you believed that your foundational system is meaningful in the first place.

And how to believe that? By interpreting the rules and assumptions in the foundational system in such a way that they seem to be true in the real world. (And you cannot define "true in the real world".)

If we didn't have the same interpretation, either I will explain my interpretation for others or I'll rely to their interpretation for us to be able to understand each other's explanations. Is this correct?

@soupless Sort of. You can try to explain in different ways until they get it, just like what I'm trying to do now. =)

There's no 100% guaranteed way.

Okay. I think I got it now.

Great!

Feel free to ask again to clarify anything in the future if you want.

Just to be sure, my interpretation is not necessarily correct, but it could be correct.

And it depends whether I'll accept my interpretation as correct, since I have no means to define how something is correct.

@soupless Correct.

XD

Exactly as what you say, "truth-preserving" is a matter that depends on your interpretation of "true". We could have also said:

> Each inference rule is chosen to be such that, if you start with joyful statements and use the rule you will deduce only joyful statements. We say that these rules are joy-preserving.

As long as you interpret the joyfulness of each statement in a way such that you agree with this claim, then it doesn't matter whether your notion of "joy" matches mine or not, at least for the goal of obtaining joyful statements.

Same with "truth".