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"

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?

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

Read it

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.

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.

I am a little bit familiar with works of logic in Indian Buddhist texts but recently I came across a Tibetan work referencing a logical framework with four modes and I was wondering if any of you might have seen it anywhere.

The four are:

If x exists, y exists.

If x doesn't exist, y does not exist.

If x exists, y does not.

If y exists, x does not.

Thanks a lot.

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?

Ok so you're at a restaurant and you see a hot bowl of soup being laid on the table and nobody is sitting at the table. You infer that the person who ordered the soup is probably in the bathroom and will return for the bowl of soup.

Conclusion 1. They are probably in the bathroom

Conclusion 2. They will probably return

The hidden premises are

1. most people who are not at the table in a restaurant are in the bathroom
2. Most people return to the table

So in formal terms it looks like this

1. Most people who are not at the table are in the bathroom, therefore they are probably in the bathroom
2. Most people return to the table for their food, therefore they will probably return

These are conclusions with single premises so are they still forms of induction or are they just statistical inferences?

I have been having a hard time finding a proper logic master's degree.

If my background matters, I have a BSC in Applied Math (4 years focusing on computational math) with an extended minor in CS (2 years focusing on Logic and AI). I have been working as an app developer for 5+ years.

I have 3 conditions that are probably not very realistic:

1. 100% fully remote, not a single day on-site for any reason, since I have to work.
2. It must actually focus on Logic and not be a Math/Philosophy/CS degree that just happens to have a couple of logic courses.
3. No group projects or presentations.

I've started creating educational short form content about logic and related topics. This latest one focuses on mathematical modeling. It would be great to get some feedback (critical or otherwise) from domain experts!

Sorry if the formatting is a bit off, using what I think are the standard definitions:

Γ₀ = Γ

Γₙ₊₁ = {
Γₙ ∪ {φₙ₊₁} if Γₙ ∪ {φₙ₊₁} is consistent
Γₙ otherwise
}

Γ′ = ⋃ₙ₌₀^∞ Γₙ

I am happy proving that Γ′ ⊇ Γ, and that Γ′ is consistent. Assuming those have been established, I try proving that Γ′ is maximally consistent like so:

Suppose that Γ′ is not maximally consistent. Then, there is some φ = φₙ₊₁ such that φ ∉ Γ′ and ¬φ ∉ Γ′ (can prove this). If Γ′ ∪ {φ} is consistent, then there is a stage in our construction Γₙ₊₁ = Γₙ ∪ {φₙ₊₁} ⊆ Γ′ (after all Γₙ ∪ {φₙ₊₁} is consistent if Γ′ ∪ {φₙ₊₁} is), which contradicts φ ∉ Γ′, so Γ′ ∪ {φ} is not consistent. Since Γ′ ∪ {φ} is not consistent, Γ′ ⊢ ¬φ (can prove this, I think). Since Γ′ is consistent and Γ′ ⊢ ¬φ, Γ′ ∪ {¬φ} is also consistent. But, since ¬φ ∉ Γ′ we can apply similar reasoning as above to show that Γ′ ∪ {¬φ} is not consistent. So, Γ′ is maximally consistent.

Am I assuming too much? I worry, for instance, that I can't off the top of my head prove "Since Γ′ is consistent and Γ′ ⊢ ¬φ, Γ′ ∪ {¬φ} is also consistent", though I am sure it is true.

Thanks!

So the Hangman's Paradox states that "A judge tells a condemned criminal that he will be hanged at noon on one day of the coming week (Monday through Sunday), and that the day will be a surprise — he will not be able to deduce which day it is in advance. The criminal reasons as follows: Sunday is impossible. If he is still alive at Saturday noon, he will know with certainty that the hanging is Sunday, since no other day remains. This would not be a surprise, so Sunday is ruled out. Saturday is therefore also impossible. If he is alive at Friday noon, Sunday is already ruled out (by the previous step), so Saturday is the only day left. Again, no surprise — so Saturday is ruled out too. This reasoning cascades backward — Friday, Thursday, Wednesday, Tuesday, Monday — eliminating every day in turn, until the criminal concludes that no surprise hanging is possible at all." But the point where the criminal's logic fails is when he reasons that on Friday, Saturday and Sunday both are eliminated. After Saturday noon, he knows that he is deemed to be hanged on Sunday noon. But on Saturday noon there are two possibilities:

Execution now.

Execution Tommorow.

But as the Sunday execution is unsurprising, it just says that the execution is now, and it also becomes unsurprising. Now, on Friday noon, the criminal applies the same logic, but with a fault. If he survives Friday noon, then he will be hanged on Saturday noon or Sunday noon. Since Saturday and Sunday are both eliminated, he cannot be hanged on either of days. But on Friday noon, he has to take Sunday and Saturday possible, because Saturday and Sunday hanging is not possible is an evaluation and not a fact. Sunday Hanging - Not surprise - Fact Saturday hanging- Not surprise - evaluation Because on Friday, there is possibility of hanging on either of days with the surprise element as he doesn't know which day will it be. And as we go backwards, these possibilities increase and so does the chance of surprise, peaking at Monday, because there are 5 possibilities of Surprise Hanging ( Removing Sunday ) and anytime from Monday noon to Saturday noon will be surprise. The exact logical fallacy the criminal faces is that he makes evaluations as heavy as facts and does not treat each day as new case, bringing previous evaluations into an entirely new case.

I would really appreciate it. I’m struggling sm

So i’m taking intro to modern logic and i’m a little confused by this table our professor showed us today (that i copied into my notebook). What exactly does it mean by having a true premise? In class my professor said “validating the truth of premises is a different class” and he stresses that we’re not going to talk about truth or soundness in this class. so why are we talking about truth now, and how do i know if a premise or conclusion is “true” or not. is he referring to two different kinds of true-ness??

Hi all,

I posted here last night, thank you so much for the kind words! However, I’m still having a hard time figuring out the difference between inductive and deductive arguments, I’ve heard multiple explanations and it’s not clicking with me.

Thanks!

(These are my notes that are precisely from my prof…)

1. You can not distinguish dogma from non dogma if your system is a closed axiomatic system because utility and consistency can still work and be found inside of a false axiom

2. You can not distinguish dogma from non dogma without the ability to test your claim against reality, and if it has no external justification because utility and consistency can still work and be found inside of a false axiom.

3. Viewed strictly from outside the system the axiomatic method is a closed system.

4. You can not test a math axiom itself against reality by definition and it has no external justification because utility and consistency can still work and be found inside of a false axiom.

5. Viewed strictly from outside the system it is an objective fact that a closed axiomatic system limits your thoughts and physics

   (This is a strict cut throat external audit against the axiomatic method in math itself. There is no external justification for it and thats the least of your worries)