Logic Definition and 1000 Threads

Which part of Newton's second law is definition and which part is law content?It seem that there is a violation in logic, because we define the notion of force through the notion of mass, then we define the notion of mass through the notion of force when we consider the second law.

Physicist Craig Hogan has proposed that the universe is based on holographic principle. To prove that the universe is a "hologram" he (and other physicists) have designed an experiment named "The Holometer" to measure quantum fluctuations that would become fuzzy at Planck scale...

Goldrei's Propositional and Predicate Calculus states, in page 13: "The countable union of countable sets is countable (...) This result is needed to prove our major result, the completeness theorem in Chapter 5. It depends on a principle called the axiom of choice." In other words: the most...

Goldrei's Propositional and Predicate Calculus states (in my words; any mistake is mine) that first-order logic is complete, i.e. any logic deduction from a set of axioms (written in first-order logic) is equivalent to proving the theorem for all models satisfying the axioms. Completeness is...

I am stuck on these questions and don't really know how to start/solve them. prove the following sequent: 1. $(\exists x) Fx \to (\forall x) Gx \vdash (\exists x)(Fx \to (\forall x)Gx)$ 2. $(\forall x)(Fx \to (\forall y)\neg Fy) \vdash \neg(\exists x)Fx$ 3. $(\exists x)Fx, (\forall x)(Fx \; à...

I'm confused as to why this bubble sort I'm implementing is setting _all_ items to one of the items (I have no idea which one as array is too big) the data type is of ... [[1128 1026 1192 1023]] [[ 771 195 858 196]] [[ 953 1799 955 1738]]] when I have an array of int, this same algorithm...

Has anybody ever heard of this? I learned about it in a discrete math class in grad school, and I've never heard of it anywhere else !? For example, logical disjunction (OR) and set-theoretic UNION are isomorphic in this sense: 0 OR 0 = 0. {0} UNION {0} = {0}. Similarly, logical AND & set...

I like using the Euler–Lagrange equations to solve simple mechanical systems, but I'm not perfectly clear on the theory behind it. Is it derived by assuming that action is minimized/stationary? Or does one define a system's Lagrangian according to what makes the Euler–Lagrange equations...

I don't understand why I can't answer this question as a bernuli trial. There are 6 possible correct integers out of 40, and 34 incorrect integers out of 40. I'd assume it would look like this: (6c1)(6/40)(34/40)^5 I guess, it's because when you choose and incorrect or correct integer, the...

1. If we descended from an unknown common ancestor that was a step up from chimpanzees and bonobo's, then by biological definition, shouldn't chimpanzees and bonobos be much more wise and more intelligent than humans as they are the ones who came first through evolution before our unknown common...

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...

Hello, Given the domain as: D = {a,b}; ~Ba & Bb & Laa & ~Lab & Lba & ~Lbb Why is the interpretation false? (∀x)[Bx ⊃ (Lxx ⊃ Lxa)] I am having trouble understanding why that is the case because (Lxx ⊃ Lxa) evaluates to true in any case as long as Lxa is true in all cases, so the overall...

Usually the truth values of propositions of a logic are structured into a lattice, with 0 (False) on (say) the bottom and 1(True) on (say) the top, and the connecting lines being implication. In paraconsistent logics, there is at least one node which is not implied by 0. Can one safely say that...

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...

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...

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...

My school uses Rosens book which is an awful book, would not recommend. Can anyone please help me solidify if what I am saying is true: If you are presented an argument, to even take this argument into consideration, you must prove whether the argument is valid or not. After this is done, you...

A logic puzzle I just found online: $$ 5+3+2 = 151022\\ 9+2+4 = 183652\\ 8+6+3 = 482466\\ 5+4+5 = 202541\\ 7+2+5 = ? $$ [/SPOILER] Edit: The answer to the problem is not 14. Please post how long did the problem take you to solve:wink:

Hi, the question and Ke logic rules are attached. This is my attempt at the question. $1. P \land (R\implies Q) $ Premise $2. ( P \land Q ) \implies \lnot S) $ Premise $3. ( P \land S) \implies R) $ Premise $4. \lnot S $...

Homework Statement Homework Equations - The Attempt at a Solution Here's my work : When I added 0000 0001 to 0000 0000 , I didn't get the correct answer . Could someone check where is my mistake please ?

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...

I have a question about predicate logic and modal logic. Namely, do any of them overlap with one another? To give an example, does existential quantification apply to counterfactual statements? A counterfactual statement can be something like "A possible world where I won the lottery." I...

Homework Statement Design a circuit with four inputs and two outputs which realizes the functions : f1 = Σm(0,2,4,5,6,14) f2 = Σm(0,1,4,6,12,14) Homework Equations - The Attempt at a Solution Here's my work : 2 levels (NAND / OR ) 10 gates 17 gate inputs 2 levels ( NAND / OR ) 7 gates 11...

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...

Homework Statement In the logic circuit shown below , write the Boolean expression for the output function X . Which gate(s) could be removed without affecting the function X ? Homework Equations - The Attempt at a Solution [/B] So , we can get rid of ##A \overline{B} C ## . gate # 4. Right ?

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 website that gives a question and hints to the question if stuck, so I thought this would be good practice. But...

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...

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...

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.

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 EquationsThe Attempt at a Solution I know that the answer for...

Consider the following proof from Copi's "Symbolic Logic", p. 109: 1. (∃x) ¬Fx Assumption 2. ¬Fy Assumption 3. (∀x) Fx Assumption 4. Fy...

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...

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...

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...

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...

Is ohmic resistance a "necessary evil" in order for transistors to be able to function as logic gates? I mean, I have seen some circuits for the NAND gate in RTL and TTL and they both seem to involve ohmic resistances. Can we make a super conducting NAND gate that will have total ohmic...

Can you figure out what the answer of (x-a)(x-b)(x-c)...(x-z) is? This problem seems very tricky and you might think you need to expand one by one, but if you think carefully, you will find out that the answer is very simple! Solution:

For the folowing two problems determine the truth value of each statement: assume a and b are true and c and d are false. not(a V b) -> s not( T V T) -> s F -> s T r -> [(d -> w) <-> (a ^ c )] r -> [(F -> w) <-> ( T ^ F)] r -> [T <-> F] r -> F I am fairly certain I did the first one correct...

So i was thinking if i was a contestant and i got a question i had no clue what the answer was. and i wanted to use Ask the Audience and 50-50 lifelines, which one should I pick first? Is there a math logic to this? Lets pretend its a question very doubtful the majority of the audience would...

Hello, I don't know much about logic gates and how they work. I was wondering if this (Image) can work. (Is something like this possible?) The idea is that if we have logic 1, AND gate gives 1 while NOR gives 0. And vice versa.

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...

Please help me with this thing. I'm so frustrated I can't understand propositional logic Demonstrate this: (p ∧ q) ↓ q ≡ ¬q PLEASE.

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...

I'm working on a research project on logic and would be interested in real examples of the use of the theory of formal logic for any scientific or technological research. I'm well aware that logic, to begin with our intuitive sense of logic, is necessary to just about everything we do in life...

What exactly is the usefulness of formal logic theory? And how much useful is it? I believe that most of us have an intuitive sense of logic, which has to be very useful throughout our lives and just about for every situation we can think of. Given that, I came to wonder what more does formal...

Hello,Lately, I have been interested in philosophy and in particular the branches of philosophy where there is a tie with mathematics and physics. From this point, I think the best branch to start studying would be the most robust one for scientists. i.e Logic. But I am having a problem finding...

Homework Statement suppose I have a function defined as: G: ℚ--->ℚ f(x)= { 2/ 3x if x does not equal to 0, 0 if x=0} Homework Equations Injective:if for all x,y in ℚ, f(x)=f(y) then x=y. or if x does not equal to y then f(x) does not equal to f(y)The Attempt at a Solution I am confused as to...

Why do we sometimes neglect resistors and power sources when we draw circuits and other times not? Is there different terminology for diagrams that are only concerned about the logic involved and diagrams where the voltage and resistance is accounted for?

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 truthThe Attempt at a Solution Im unable to demonstrate the Tautology and obtain (¬Q) as solution. I start by facing the right side...

I want to do tricky physics and maths questions that require only A-level knowledge of maths and physics to solve. Questions that don't have a straightforward answer and will help me to develop my logic and problem-solving skills. I'm looking books, online resources and etc.