1. ### I Translate compound proposition p → q (implication) to p↓q question

I hope someone can help me or point me in the right direction. I am reading Discrete Mathematics with its Applications by Rosen. I am trying to self learn discrete math. I am actually able to do most questions but I have a question about a solution (not the question itself.) The question is...
2. ### A Range of values for ##2^{\aleph_0}##

Ok, so assume we have a model for ZFC where CH does not hold. What values may ##2^{\aleph_0}## assume over said models?
3. ### I Modus Tollens on {B|α} → R?

I can't figure out this: Say we have event B given α, denoted as {B|α}. If B happens, that implies that R happens: {B|α} → R. Now I want to apply modus Tollens. So if I do, do I get the result: ¬R → {¬B|α}? I mean, I hope I can keep the α unaffected. Is that the case? ¬X meaning X does not happen.

In Tao's Analysis 1, Lemma 5.3.6, he claims that "We know that ##(a_n)_{n=1}^{\infty}## is eventually ##\delta##-steady for everyvalue of ##\delta>0##. This implies that it is not only ##\epsilon##-steady, ##\forall\epsilon>0##, but also ##\epsilon/ 2##-steady." My question is, why do we need...
5. ### B ##A \subset B \iff \exists x \in B \land x \not\in A##?

##A \subset B## means that ##\exists x \in B ## such that ##x \not\in A##. Is this logically equivalent to ##\exists x \in B \land x \not\in A##? Formally, $$A \subset B \iff {(\exists x \in B \land x \not\in A)\land(\forall y \in A \land y\in B)}$$ I have tried consulting...

7. ### Simple Induction Direct Proofs regarding Induction

Summary:: . When asked to prove by Induction, i'm asked to prove a statement of the form: Prove that for all natural numbers ##n##, ## P(n) ## Which means to prove: ## \forall n ( P(n) ) ## ( suppose the universe of discourse is all the natural numbers ) Then, I see people translating...
8. ### I Question regarding quantifier statement

Suppose I have the following ( arbitrary ) statement: $$\forall x\in{S} \ ( P(x) )$$ Which means: For all x that belongs to S such that P(x). Can I write it as the following so that they are equivalent? ( although it is not conventional ): $$\forall x\in{S} \land ( P(x) )$$ Can I write...
9. ### I Who copied the assignment?

If someone is lying, who copied the assignment? Alex: Cate copied the assignment. Cate: David copied the assignment. David: Cate is lying. Keil: I didn't copy. I think Cate is lying. If Alex is true and there is only one person who is lying, Cate and David can't be true at the same time...
10. ### I Intro to Symbolic Logic: Replacement Rules

Basically the problem starts with these given premises: 1. ~ (A ∨ (B⊃T)) 2. (A ⋅ C) ∨ (W ⊃ ~D) 3. ~(P ∨ T) ⊃ D 4. ~P ≡ ~(T ⋅ S) And from these premises, I must prove ∴ ~W. This is what I have done so far: 5. (~P ⊃ ~(T ⋅ S)) ⋅ (~(T ⋅ S) ⊃ ~P) B.E. 4 6. ~P ⊃ ~(T ⋅ S)...
11. ### What did Hans Bethe think of von Neumann's quantum logic?

Nobel laureate Hans Bethe was a friend of mathematician-physicist John von Neumann, and he once said: "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man" and "[von Neumann's] brain indicated a new species, an evolution beyond man"...
12. ### I What did Omnès mean with this?

Summary: What did Omnès mean with this? I found an old article by Roland Omnès which analyzes the EPR paradox and offers a solution to it (https://www.sciencedirect.com/science/article/abs/pii/0375960189900182). At some point, the article says: "Some macroscopic systems do not satisfy the...
13. ### A Logical foundations of smooth manifolds

Hi I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This...
14. ### I Cardinality of a set of constant symbols (model theory)

First, I want to be pedantic here and underline the distinction between a set (in the model, or interpretation) and a sentence (in the theory) which is fulfilled by that set, and also constant symbols (in the theory) versus constants (in the universe of the model) Given that, I would like to...
15. ### Can random, unguided processes produce a rational brain?

I am fascinated by Einstein’s quote that the most unbelievable aspect of the universe was that it was intelligible. So my question is does anyone know whether it is so unlikely as to be absurd to suppose that random unguided processes could produce a rational brain in man in as little as 3...
16. ### Prime factors of odd composites

Homework Statement Let ##n## be odd and a composite number, prove that all of its prime is at most ##\frac{n}{3} ## Homework Equations Some theorems might help? Any ##n>1## must have a prime factor if n is composite then there is a prime ##p<√n## such that ##p|n## The Attempt at a Solution...
17. ### Logic as a dynamical system?

This thread is a shoot-off from this thread. Assuming some relation between human language and logical reasoning, how would this relate, let's say, to the arrival and evolution of logic in human and animal psychology? I would presume that some logic, for example classical logic, can be more or...
18. ### I "Laws of Form" by G. Spencer-Brown (1969)

I have received (unasked) a digital edition of "Laws of Form" (1969) by G. Spencer-Brown; I have glanced at it, and also at the Wikipedia article https://en.wikipedia.org/wiki/Laws_of_Form. OK, another logical system; logical journals (e.g. by ASL) are full of them, and I am not sure whether...
19. ### I Truth, lie and random confusion

So I am very, very new to logic based questions, and in the past have solved some with relative ease but whilest scrolling through the next to find some example stuff I came across a web site that gives a question and hints to the question if stuck, so I thought this would be good practice. But...
20. ### B Does retrocausality follow from logic?

I can't remember where this subject came forward in my topics, so I created a new topic. Suppose that: If X happens, we observe A, and: If Y happens, we observe B. Could we then say: If we observe A, Y did not happen, and: If we observe B, X did not happen, if we apply this to...
21. ### B Does anybody think that there is an underlying reality?

It seems like we keep chasing "reality", and by "reality" I think Physicists would mean the apparent rules of quantum physics which we hope would (if applied) lead to all the apparent known rules of macro-physics. However ... It seems like we have had to create a few things to do that: 1. Ideas...
22. ### I Implication problem

I just saw this proof.... And, I don't understand why this is true. How am I supposed to think about problems like this one? Edit: Here's another one: The only steps here I understand are 1 to 5. I don't know why 6 and 7 are true.
23. ### Basic Logic Problem P(x,y)

Homework Statement I refer to part G of this little problem: I don't see how to arrive at any conclusion, especially when I can't even see how ##z## comes into play. Assistance in interpreting the problem is appreciated! Homework Equations The Attempt at a Solution I know that the...
24. ### Proof by Induction of shortest suffix of concatenated string

Homework Statement Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
25. ### A Multiverse theory with impossible universes?

I found an article written by physicist George Ellis that confused me a little. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.498.4569&rep=rep1&type=pdf At some part, he says: 3.2 Non-uniqueness: Possibilities There is non-uniqueness at both steps. Stating “all that is possible...
26. ### I Why the emphasis on first-order logic?

It seems to me that, despite several systems developed for higher-order logics, almost all the attention in Logic is devoted to first-order. I understand that higher-order logics have some drawbacks, such as the compactness theorem and the Löwenheim-Skolem theorems and other such not holding in...
27. ### I Loophole in Godel's Incompleteness Theorem?

Gödel's incompleteness theorem only applies to logical languages with countable alphabets. So it does not rule out the possibility that one might be able to prove 'everything' in a language with an uncountable infinite alphabet. Is that a loophole in Godel's Incompleteness Theorem? Doesn't...
28. ### A Does the Frauchiger-Renner Theorem prove only MWI is correct

Hello all, I have only seen this paper brought up here once before based on the search function 2 years ago, and the thread devolved into something off topic within the first page. I am asking in reference to this paper: https://arxiv.org/pdf/1604.07422.pdf Which claims to show that single...
29. ### A About the “Axiom of Dependent Choice”

I learned something new today: the “Axiom of Dependent Choice”: The axiom can be stated as follows: For every nonempty set ##X## and every entire binary relation ##R## on ##X##, there exists a sequence ##(x_n)_{ n \in \mathbb{N} }## in ##X## such that ##x_nRx_{n+1}## for all ##n \in...
30. ### Logic Proposition Proof

Homework Statement Need to demonstrate this proposition: (P→Q)↔[(P ∨ Q)↔Q] . My textbook use truth tables, but I'd like to do without it. It asks me if it's always truth The Attempt at a Solution Im unable to demonstrate the Tautology and obtain (¬Q) as solution. I start by facing the right...