r/logic

I'm no mathematician so maybe I misunderstand but it seems to me like something in zfc might be arbitrary. I think I understand the concept of a set, where the quantity of 5 is a set of 5 thus numbers are sets. However, let's take the idea of an empty set.

Now my understanding of what an empty set is, is a box of chocolates w/o any chocolate. It's purely a mental overlay of reality when we say the box is an empty set. But the question is does nature deal in empty sets outside of the one's invented by our minds?

It seems to me that if mathematics may be said to exist in some capacity, such as if math is merely the laws or rules of existence, that it would not be meaningful to have an "empty set". As that's saying there is something ontologically more to a set than it being the collection of things in a set. In one instance your saying a set is a thing in and of itself, in the other "set" just refers to the things collectively considered such that an absence of the things leaves you with no set rather than something that's empty.

This "something" that is called a "set" such that it can even be empty seems like something that has no ontological reality and things that have no ontological reality can't be said to exist.

I guess the question is if mathematics exists mind independently can an empty set actually exist also or is it merely invention and if so how can the concept be said to be a "foundation" of math? Thoughts?

Hi everyone,

I’m looking for a logic check on a question from a Swedish entrance exam (MAFY) held a few days ago. The question presents a specific theorem and asks which conclusion follows strictly from it:

The Theorem: The angle opposite a longer side in a triangle is larger than the angle opposite a shorter side.

The Options:

(a) If a triangle's three angles are equal, then its three sides are equal in length.

(b) If a triangle's three sides are equal in length, then its three angles are equal.

(c) The base angles of an isosceles triangle are equal.

(d) None of the above follow from the theorem.

The official answer posted by the universities is option a, but to me that seems to be wrong. They have already adjusted some of thier official answers so it is not far fetched that they got this one wrong as well.

In previous years the logic question on the exam was always that you are given a theorem or a statement and the right answer have always been the contrapositive, but that approach does not seem to work this year.

I understand that a,b and c are all correct geometrically but don't beleive that any of them are a consequence of the provided tehorem.

what would you guys answer and why?

• A mathematical group exist no where in raw concrete reality (true)

• Logic says mapping an unanchored map is a major fallacy for modeling raw concrete reality (true)

• A mathematical group is unanchored to raw concrete reality, fully abstract. This is what logic says (true)

• It doesn’t matter if math claims to model reality or not, because we treat math as if it does model reality(physics, engineering) (true)

• Consistency and utility can still work and be found inside of a false axiom (true)

so here we have it, logic calling this a massive fallacy

I am trying to do an exercise in Elliot Mendelson's Intoduction to mathematical logic.

It is exercise 1.47 (d); the proof of ⊢ (¬(C) → ¬(B)) → (B → C).

It is the fourth in a series of exercises following a reflexivity lemma, which follows three axioms of a formal system. This is the formal system L:

1. The symbols of L are ¬, →, (, ), and the statement letters A_1,...,A_k.

2.  
    - All statement letters are wfs.
    - If B and C are wfs, then so are ¬B and B→C.

3. If B, C, and D are wfs of L, then the following are axioms of L: 
    - B→(C→B)
    - (B→(C→D))→((B→C)→(B→D))
    - ((¬C)→(¬B))→(((¬C)→B)→C)

4. The only rule of inference is modus ponens.

I have been fortunate to own a book whose previous owner jotted down the proofs of the first three exercises - in a different language too, as well as to have had help learning the syntax and methodology on math.sx. This has allowed me to complete the derivation of these first three proofs proofs.

However, having used this crutch, I have not developed the intuition which guides the justification synthesis. *How am I supposed to synthesize these types of derivations when I encounter future proofs on my own?

Even though I can follow the logic in (a) -- (c), I cannot synthesize my own pattern of it to solve (d).

This is what I have derived to far: (It is computer generated by a theorem proving algorithm I wrote). I don't know how to be creative/smart when combining previous theorems and axioms with modus ponens. My goal is not only to solve (d), but to be able to solve other theorems on my own without assistance.

Note: I use "IE" to mean modus ponens, because it also means implication elimination.

Note: Cut means that an assumption is removed when it is the proposition of a known assumptionless theorem.

Note: I have not labeled the steps as I reached them in the table itself, but they are line 8. (lemma, given), line 11. (a), line 17. (b), and line 23. (c).

Line.    Logic                                                  Reasoning
   1.    (B → (C → B))                                          axiom
   2.    ((B → (C → D)) → ((B → C) → (B → D)))                  axiom
   3.    ((¬(C) → ¬(B)) → ((¬(C) → B) → C))                     axiom
   4.    ⊢ ((B → ((B → B) → B)) → ((B → (B → B)) → (B → B)))    2.[C:=(B → B), D:=B]
   5.    ⊢ (B → ((B → B) → B))                                  1.[C:=(B → B)]
   6.    ⊢ ((B → (B → B)) → (B → B))                            IE[5, 4]
   7.    ⊢ (B → (B → B))                                        1.[C:=B]
   8.    ⊢ (B → B)                                              IE[7, 6]
   9.    ⊢ (¬(B) → ¬(B))                                        8.[B:=¬(B)]
  10.    ⊢ ((¬(B) → ¬(B)) → ((¬(B) → B) → B))                   3.[C:=B]
  11.    ⊢ ((¬(B) → B) → B)                                     IE[9, 10]
  12.    ⊢ ((C → D) → (B → (C → D)))                            1.[B:=(C → D), C:=B]
  13.    (C → D) ⊢ (C → D)                                      assumption
  14.    (C → D) ⊢ (B → (C → D))                                IE[13, 12]
  15.    (C → D) ⊢ ((B → C) → (B → D))                          IE[14, 2]
  16.    (B → C) ⊢ (B → C)                                      assumption
  17.    (B → C), (C → D) ⊢ (B → D)                             IE[16, 15]
  18.    (B → (C → D)) ⊢ (B → (C → D))                          assumption
  19.    (B → (C → D)) ⊢ ((B → C) → (B → D))                    IE[18, 2]
  20.    ⊢ (C → (B → C))                                        1.[B:=C, C:=B]
  21.    ((B → C) → (B → D)), (C → (B → C)) ⊢ (C → (B → D))     17.[B:=C, C:=(B → C), D:=(B → D)]
  22.    ((B → C) → (B → D)) ⊢ (C → (B → D))                    Cut[21, 20]
  23.    (B → (C → D)) ⊢ (C → (B → D))                          Cut[22, 19]

Is there a logical fallacy or something similar that describes arguments that simply dismiss issues present? For example:" your steak is perfectly fine, just ignore the burnt parts" dismissing that the food is in reality not "perfectly fine" and putting it on the eater to ignore the burnt parts

Or a situation where someone suggests that you sort of try to forget part of something that brings down an experience overall . In refrence to a video game : "just dont use that overpowered item if you think it cheapens the game" dismissing that the overpowered item is in the game, the player knows about it, and it is the best move for winning even though it isnt fun.

This might also apply to seperating art from the artist by asking someone to simply ignore or forget that terrible thing the artist did that makes you unable to enjoy their work anymore

Would this whole ballpark just be called "being dismissive"?

In our Logic with Ethics class, exams are straightforward: right = plus, wrong = minus. The rules are clear, consistent, and fair. Your grade reflects the accuracy of your reasoning.

But when I look at how procedures play out in politics—like in the Committee on Justice—it feels like a different kind of “logic.” Even if the process is flawed or the arguments don’t hold, the vote can still end up in favor of an abused congressman. It’s as if the outcome is predetermined by power dynamics rather than truth or fairness.

This contrast makes me wonder:

- Should logic always be tied to ethics, or can it be twisted to serve interests?

- Is political procedure just another “exam,” but one where wrong answers can still pass if enough people agree?

Curious to hear your thoughts. Do you think the way we practice logic in academics could ever be applied to politics, or are they fundamentally different games?

Hey, I’ve been wrestling this philosophical question for years. What must be true for anything to be true? Also, tell me in one sentence, what would the ontological bedrock have to do or what property would it have to possess for us to call it unconditional bedrock?

Hopefully this thread is the right place for my inquiry. I’ve read a few preprints on this topic and peer reviewed work. One stood out undeniably. But most fall short. To make sure I’m not missing anything I’d love to hear from the Reddit experts.

Hey all logicians. Logic is a great tool to manage ideas and make debate coherent. When learning about logic and argumentation, I realized a lot of the real world is stunted intellectually on the fundamentals of the discipline. When you find a formal contradiction in someone's beliefs, instead of admitting their inconsistency or incoherency they will straight up just deny logic. What do you have to say about this? Do you have any stories?

Edit: I'm talking mostly about debate setting. Like someone trying to argue that Socrates is immortal even though they affirmed that all men are mortal and Socrates is a man.

Premises

Tortilla is more than Singer

Jumper relates to Tortilla in the same way that Iron relates to Singer

Singer is more than Iron

Conclusion

Tortilla to Jumper

is unlike

Singer to Iron

My logic is that if singer is more than iron, than jumper is more than tortilla since singer relates to iron same way as jumper relates tortilla. So their relationship is the same, thus this should be FALSE? What am I missing here? Because the answer is TRUE

I’m studying frames in model logic and the case where R is a euclidean relation means that:
possible p —> necessary( possible p)

however when i’m looking at the worlds, my understanding js the definition of Euclidean is
if Rwu and Rwv, then Ruv.

as a consequence, also Rvu

so if in w: possible p, p is then true in at least one world accessible form w. I’m gonna to say p true at u and p false at v as my example.

then, for possible p —> necessary (possible p) to hold, I can see at v, possible p is true, but at u it seems possible p does not hold since p is not true at v and u doesn’t have any other accessible worlds? Since it doesn’t hold for both u,v then not necessary (possible p))

i’d greatly appreciate any help

Is the Compactness Theorem equivalent to Proposition 2? I ask because of the following:

1. For all X1, if X1 is a theory in FOL and for all X2, if X2 is a finite subset of X1 then X2 is consistent, then X1 is consistent.

2. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is not finite or X1 is a theory in FOL and X2 is not a subset of X1 or X1 is a theory in FOL and X2 is consistent, then X1 is consistent.

3. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is not finite then X1 is consistent.

4. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is not a subset of X1 then X1 is consistent.

5. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is consistent, then X1 is consistent.

Hey everyone, new to this sub. Most of my interests are specifically in theology but I would argue that proficiency in logic is a necessary prerequisite the deeper you dive into deep theological truth claims.

I’ve been exploring the ‘Logical Problem of the Trinity’ which I’ll quickly summarise here:

Consider the claims made by the following set S of natural language sentences:
(S1) The Father is God
(S2) The Son is God
(S3) The Holy Spirit is God
(S4) The Father is not the Son
(S5) The Father is not the Holy Spirit
(S6) The Son is not the Holy Spirit
(S7) There is exactly one God

If we follow the law of absolute identity we run into a self defeating syllogism, or a form of modalism which again contradicts the premises. Peter Geach comes along and proposes that his theory of relative identity can solve this:

objects can be the same "F" (a specific sortal concept) without being the same "G" (another concept), meaning identity is relative to a sortal term rather than absolute

My question is (forgetting all the theological context):

Within contemporary formal logic, is identity best treated as an absolute, or can a coherent logical system be formulated in which identity is relative to a sortal (in the sense of Peter Geach)?

Are there any papers or recourses that are somewhat simple to understand for an amateur in the study of logic?

I want to hand off the volumes of * for free, see

ISBN 1603861823

Pm