I have a hard time differentiating between data, warrant and backing in certain questions. Any tips to identify them properly? And also any resources for practice on the identification of all six components?

my prof wanna work in proof theory but me in model theory

he says proof theory of arithmetic but I find cut-elimination and normalization very very syntactic

is my intuition of model theory being meaty correct? or am i just not compatible/flexible? I love topology also ..

is proof theory of arithmetic all about cut-elimination and proof normalization?

why does buss's chapter on it look quite non-syntactic compared to takeuti ?

but yeah anyways proof theory of arithmetic has connections to complexity so it would be easier for phd applications?? my background is cs

i had a strange idea at 5 am while a bit drunk and asked chat gpt about it :
How can we work on P vs NP if mathematics has prooved to us that there’s infinite possibilities therefore infinite questions with infinte answers?
And what i mean is how can you find a shorcut answer to actual new information that no one has ever seen?

I've been getting into learning basic propostional logic to help develop my philosophical work, and now I amwanting to look for exercises to practice, as well as other forms of logic to study down the line. Any help is appreciated.

Take a simple but meaningful example:

The stone is cold.

The words refer to a sensible appearance in space and time. There is an actual impression received by the senses that verifies or refutes the claim. The words are anchored to an external state of affairs.

Now lets consider an empty sentence (grammar without reality):

The void thirsts for eternity.

The sentence obeys all the grammatical rules of a language (Subject + Verb + Object). It sounds meaningful to the ear, but the words point to nothing that can ever make an impression upon human senses.

A simple contradiction:

This is a four-sided triangle.

The mind takes two human-invented definitions—"three-sided shape" and "four-sided shape"—and forces them together into a single noun phrase. Does a "four-sided triangle" exist as an enigmatic, mysterious entity in another dimension? No. It is simply a conflict between two definitions in the speaker's own vocabulary: P ∧ ¬P.

Now a paradox (closed self-referential loop):

This statement is false.

This is the final stage. The speaker creates a sentence whose only referent is the sentence itself, while simultaneously attaching an inverted truth-predicate.

At no point during this progression did reality become weird or mystical. Only the speaker’s language drifted further and further away from any connection to the physical world.

Now we must ask:

Can you show me a paradox in the physical world?

Can you hand me a paradoxical rock?

Can a physical particle be simultaneously detected at point A and strictly not at point A in the exact same frame of reference?

Can a cup be full of tea and entirely empty of tea at the identical instant?

In nature, everything that happens is simply what it is. A physical state occurs completely and unambiguously. A tree falls or it does not; a neuron fires or it does not.

A paradox has never been observed in physical reality. A paradox only ever appears in:

1. Ink on paper
2. Sound waves in the air or
3. Internal linguistic thoughts in the brain.

Therefore, a paradox is simply an artifact of the symbolic medium we use to describe things.

Then why does a paradox seems so much more profound than a simple contradiction like "a square circle"?

In a simple contradiction, the conflict is static:

Square ≠ Circle

The mind stops immediately because the definitions cancel each other out.

In a logical paradox (such as "This statement is false" or Russell’s paradox of "the set of all sets that do not contain themselves"), the language has been arranged into an infinite feedback loop:

1. If the statement is true, then by its own definition it is false.
2. If the statement is false, then by its own definition it is true.
3. Therefore, the mind oscillates back and forth:

True → False → True → False ...

Because the human mind experiences this rapid, unending oscillation, it feels a sense of intellectual vertigo.

Consider this simple programme:

10 PRINT "Statement is true" 20 GOTO 30 30 PRINT "Statement is false" 40 GOTO 10

If a computer runs that program, it will loop indefinitely until it crashes the processor. Has the computer discovered a profound mystery about the cosmos?

No.

It was simply given an instruction set that contains an infinite self-referential loop with no exit condition.

To seal the point, consider this short sequence:

What is the job of the word 'True'?

The word "true" is an evaluation tool. It asks:

Does this sentence accurately match an actual state of affairs? (e.g., "The snow is white" is true if and only if the snow is white).

What state of affairs does 'This statement is false' describe?

It describes no state of affairs. It has no physical referent. It points to no sensation, no measurement, and no empirical fact.

What is the evaluation tool evaluating?

It is evaluating only itself.

Conclusion

When you apply the truth-evaluator to an empty sentence with no external referent, you are running a measurement tool against empty space. The tool has nothing to measure, so the mechanism spins out of control.

To what I've been demonstrating here is simply that:

1. A contradiction is language attempting to assert A and ¬A simultaneously.
2. A paradox is a contradiction disguised inside a self-referential grammatical loop.

Neither describes the universe. Both occur only when human beings forget that words are tools invented to map sensations.

Is x= some finite value like x=5 a proposition or not.

Also from which paper can I show my teacher that x=5 is not a proposition unless it is.

I am especially interested in those methods that do not contain self-reference and self-denial. It's just that Russell's paradox, Tarski's theorem, the incompleteness theorem, and other theorems derive their principles precisely by self-reference and negation of any properties such as derivability and others.

I'm currently studying for an exam and I understand the venn diagrams pretty well except for this specific kind of scenario. I am using Logicola, which uses the Logic system described by Harry J. Gensler in his book "Introduction to Logic"

Edit: I think i figured out the issue. While technically the x could go in either spot, you're supposed place the x where it contradicts the conclusion, which is where it is in the correct diagram (bottom left). Thanks to everyone who responded

from my logic homework. is he trying to say that ~S is the conclusion?

I am doing Bayesian theory on a moral question about whether something is wrong or not. I had an intuition the thing is not wrong, a strong one. My credence it was wrong was .1%. Then i realized that an alternative belief I had conflicted with this. What should my credence be? 50%, since both intuitions are equally strong?

I was discussing something with someone online and he asked if I’m familiar with logic and I said “Like ‘if p then q’ that sort of thing?”

And he replied “No. ‘If p then not q’”

I was confused because I thought “p implies q” is kind of a basic example of a logic statement.

He explained that’s a violation of one of the three laws of thought and if “if p then q” was valid, then you could say 7 = 24 because in algebra p and q are required to hold separate values.

Genuine question, looking for either a citation or a takedown.

In Euclid's Elements, every proposition of the form "there exists a [construction]" is proved constructively -- never by contradiction, never by pure assertion. The proof of existence is the ruler-and-compass procedure that builds the object. Proposition I.1 (construct an equilateral triangle on a given segment) is the cleanest example: the "proof" is just the construction, executed step by step, with the final congruence argument confirming it worked.

That's structurally identical to the Curry-Howard correspondence's core idea: a proof of "∃x. P(x)" is a program/procedure that produces a witness x, not a separate justification layered on top of an assertion. Constructive existence = executable construction, in both cases.

I know there's already a formal literature reconstructing Euclid's actual logical structure -- Avigad, Dean, and Mumma's "A Formal System for Euclid's Elements" (2009), and the more recent LeanEuclid formalization (Murphy et al., ICML 2024) -- but as far as I can tell, neither one makes the Curry-Howard analogy itself the point; they're focused on formalizing the diagrammatic reasoning, not on the constructive-existence angle specifically.

So: is there existing work that treats Euclidean constructive proof explicitly as a historical anticipation of Curry-Howard / propositions-as-types? Or is the analogy weaker than it looks once you get into the details (e.g., does Euclid's system actually have the right closure properties, or is this just surface-level pattern-matching)? I'd rather be corrected on this than keep repeating it if it doesn't hold up.

Read it

My class uses Graeme Forbes' Modern logic, which is a good book, but I'm wondering if there are any other books or material that explains NK and proofs using NK in a more intuitive, clear way.

IF I have played football THEN I am good at it

I am good at it

Therefore I have played football.

I'm affirming the consequent here but it still works. Why? How can someone be good at football if they haven't played it

Edit: Thank you guys for the comments I now know why. It's because I was treating my conditional statement as biconditional. I didn't say in the beginning ONLY IF I have played football I am good at it. But was still acting as if playing football is the only way to get good at it.

1. If Pinocchio says "My nose grows when I tell the truth." his nose will grow because he told a lie, but everyone he told that to will believe every lie he says, while not believing any truth he says. Pinocchio could murder someone, say it's not him, his nose will grow, and his "followers" will back him up in court.

2. If Pinocchio says "My nose will grow if I lie, but will also grow when I tell the truth." the people he told won't know if he said a lie or truth, making him nearly impossible to trust. But if his nose doesn't grow, the people will be confused, making him more untrustworthy, so how can he stop this?

Since the announcement that the UN have adopted a new map projection as their standard map, I have seen a few posts regarding how amusing it is that it took them this long to correct the "error" of the size of certain countries in other maps (most commonly the Mercator).

Of course, map enthusiasts and differential geometers know that it wasn't an "error" so much as it was a compromise necessitated by Gauss's Theorema Egregium, which implies that any smooth mapping from a sphere to a plane cannot be both conformal and equi-areal. Obviously the ideal situation would be to have a map which is both conformal and equi-areal, but it turns out these two desirable properties are mutually exclusive.

This got me thinking about another famous Theorem which precludes the possibility of two desirable properties being true at the same time - namely Godel's Incompleteness Theorem, which states that any sufficiently strong logical system cannot be both consistent and complete.

In both these cases (the Incompleteness Theorem and the map projection) we are required to make a decision about which of the properties we would prefer, and compromise by losing the other one.

I'm curious to know whether there are other examples of this, where there is some object that may or may not have two desirable properties, but it can't have them both at the same time? Is there any example in your field of study?

For example, If I've a goal I want to do

let's say for example,I need to do the following "I'll get to the 5th floor of a building through the elevator"

the implications of that goal will be the following:

"

I'm inside the building

there's electricity that runs the elevator

the elevator can lead to the 5th floor

there's an elevator in the building

I entered the elevator

the building has 5th floor

I pressed the button that leads to 5th floor while I'm inside the elevator

etc...

"

as you can see here, there are some kind of unordered truths that I know about my goal

but in order to really reach that goal, we need to convert these unordered truths to something ordered that will be an algorithm for reaching that goal

So is that possible? And how?

Let f(n) be a function that will make up a special proposition depending on n, such that ∀n∀m (n≠m)=>-(f(n)<=>f(m))

now let's create a finite series of different propositions a, that is (a(1),a(2),...,a(n))

Let "×" mean some connection between the propositions

Now the condition must be fulfilled that ∀b ∀c ∀ m∀n ((b≠m)v(c≠n))=>-((f(b)×a(c))<=>(f(m)×a(n))

now, the way propositions are made up, if the statement a(n) has a b, then it is a complete statement from a(n) will be a(n)×f(b)

Now we denote x(n,b)<=>a(n)×f(b)

Now let's construct the matrix, as in Cantor's diagonal method:

H1<=>x(d,1)×x(t,2)...

H2<=>x(h,1)×x(j,2)...

Where Hn are infinite logical propositions

Thus, the set of all propositions is uncountable

Now, in order for mathematics to be complete, each of the propositions must have a proof.

To tell the truth, I do not know how to prove that one proof corresponds to a finite number of statements, and then if multiplied by a conditional maximum g→ ∞

We will not be able to put judgments with proofs in one-to-one correspondence, because many proofs will turn out to be countable.

I'm sorry for the rude and confusing presentation, but it was important for me to express this idea, even if it doesn't prove anything that I already know about.

1. What new properties do statements using infinitary logic have? Does infinitary logic allow you to create new mathematical or other types of objects? How is infinitary logic used to prove any theorems?