implication Definition and 76 Threads

Usually, transitivity of implication is welded into the axioms and/or rules of inference for a theory. However, I am trying to explain to someone that, except for explosive contradiction, there is nothing "sacred" in logic. I tried to get the general idea across with the example of...

I would ask for a clarification about the "link/connection" between syntactic vs semantic logical implication (entailment). As explained here logical consequence, syntactic logical implication (entitlement) ##\vdash## is about the existence of a formal proof within a formal system such that...

I'm still confused about the use of material implication (material conditional) ##\to## vs logical implication ##\implies##. From MSE the writing ##P \implies Q## makes the meta-logical assertion/statement that the logical statement ##Q## is logically implied by the logical statement ##P##...

A question about the Modus Ponens deduction rule used in theory of proof within an axiomatic system. $$[(p \to q) \land p ] \to q$$ One can easily check the above is a tautology (i.e. it evaluates always True for any truth value of propositions ##p## and ##q##). Now what is the logic argument...

I've seen a lot of people use implication and equivalence logic incorrectly. For example, when solving equations (i.e. ##x - 2 = 3 \implies x = 5##). Implication is not reversible, thus it only works in one way. By saying, ##x - 2 = 3 \implies x = 5##, you are essentially saying that it is...

Dear everybody, I am having some trouble proving the implication (or the forward direction.) Here is my work: Suppose that we have an arbitrary linear functional ##l## on a Banach Space ##B## is continuous. Since ##l## is continuous linear functional on B, in other words, we want show that...

Hi, Let's say we have the Gram-Schmidt Vectors ##b_i^*## and let's say ##d_n^*,...,d_1^*## is the Gram-Schmidt Version of the dual lattice Vectors of ##d_n,...,d_1##. Let further be ##b_1^* = b_1## and ##d_1^*## the projection of ##d_1## on the ##span(d_2,...,d_n)^{\bot} = span(b_1)##. We have...

We say that an implication p --> q is vaccuously true if p is false. Since now it's impossible to have p true and q false. That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent. So we take the implication as true. For eg...

There is something I don't understand that I want to ask quantum physics experts here: Suppose the happening of event X results logically speaking in the happening of event A. So we could for instance have the following logical implication ##X.happens \rightarrow A.happens##. If this is...

I am a graduated mathematician. Do you know that Kurt Godel proofed that it is not possible to proof that math is not noncontradiction? Physicist and a lot of other sciences are based od math. So, how can you proof that whole physics' is noncontradiction?You just believe in it as someone in God?

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...

If we have the statement: "If we prepare, we'll win the war", then according the rules of the truth table for this implication, this statement is only false if we prepared and still lost the war. This is what I'm having trouble with about implication. I understand that the only way to falsify...

I have a hypothesis of which I wonder if it's sound. Perhaps you guys can advise me: Suppose ##x_n\Rightarrow a_n## (logical implication) for some set X and set A. I think we have to assume a bijection. Then, if ##a_m = False##, ##x_m## should be ##False##, right? So, in case of a bijection...

Hello, I’m having difficulties understanding logical relations “A implies B” and “A if and only if” using Boolean expressions and Venn diagrams, there is something where I’m wrong, but I could not find it out. Please, be benevolent and tell me where I’m wrong. Thanks Note : Obviously I’m...

I had posted a similar question on another forum but didn't get much of a discussion. I'm interested to know what people here think. So consider a spaceship midway between stars A and B and initially at rest in the reference frame of the stars. The ship then accelerates away from A to some...

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.

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the proof of Proposition 2.2.9 ... ... Duistermaat and Kolk's Proposition 2.2.9 and its proof read as...

The proposition ¬(P→Q) is equivalent to ¬P^Q Does someone maybe have an idea how you can prove (directly) ¬P^Q from ¬(P→Q) by means of natural deduction? I do not manage it. Thanks in advance!

If an object type ◌ exists and within the set of these objects exist some which have both well defined properties A and B, and some which only have the property A. There is at least one ◌ which does not have both properties. Thus say we create the set of such objects such that A or B (inclusive...

Hello all, I am trying to find if the following two claims are true or false: 1) If \[\alpha \models \left ( \beta \rightarrow \gamma \right )\] then \[\alpha ,\beta \models \gamma\] 2) If \[\alpha \models \left ( \beta \rightarrow \gamma \right )\] then \[\alpha \vee \beta \models \gamma\]...

If the spectrum of a hermitian operator is continuous, the eigenfunctions are not normalizable. I have been told that these eigenfunctions do not represent possible physical states, but what exactly does that mean? Is there a good interpretation of the physicality of these eigenstates? If a...

Edited by moderator In the attached file it says that the formula for when n=2 implies the next one but I don't understand why it suddenly became (s-t0)^2 inside the integral instead of (s-t0) in the next one and keeps this pattern, it doesn't seem to be implied anywhere for me.

What does it mean that a Killing vector and a total differential of a certain theory are related to bilinears? In other words, why would bilinears (e.g. of the forms ##<\gamma_0\epsilon, \gamma_5\gamma_{\mu}\epsilon>## and ##<\gamma_0\epsilon, \gamma_{\mu}\epsilon>## tell us anything about...

OK, this is embarrassing, but I never looked carefully at this elementary point. We say that if p implies q P is the set of all things for which p is true Q is the set of all things for which q is true then Q ⊆ P. Also that the set of all things for which p&q is true equals P∩Q But p & q...

Hey guys I am having a bit of a difficult time with this question, if some one could help me out it would be appreciated, thanks. [FONT=Arial]Consider the following argument. "If the weather is fine, and the train is early, then the dog will sit on the tuckerbox. The train will be early, (but)...

In analysis of frames, we say that "The members of the frames are axially inextensible". In context of frames with vertical legs, what I understand is that the length of the chord joining the member ends in the displaced condition would be same in the axial direction. However, what does it imply...

Hi! (Smile) I am looking at the proof of the following sentence: For each natural number $n$ it holds that $n \notin n$. Proof : We define the set $X=\{ n \in \omega: n \notin n\}$. It suffices to show that $X$ is an inductive set, because then $X=\omega$. Obviously $\varnothing \in X$. We...

Given Hardy's "excess baggage" theorem showing that the size of the ontic state space must scale exponentially with the number of systems, does this necessarily pose any "threat" to the ontic approach to QM? Leifer writes: Is the quantum state real? A review of ψ-ontology theorems...

P -> Q P Q P->Q T T T T F F F T T F F T I understand the condition "P is sufficient for Q". But I'm not getting the meaning of why "Q is necessary for P". What does this signifies, Please explain ..

Hello! (Smile) I am looking at the proof of the follwing sentence: Let $a,b \neq 0$. The common multiples of $a,b$ are the same as the multiples of $[a,b]$, where $[a,b]$ is the least common multiple of $a \text{ and } b$. Let $a \mid m, b \mid m$ $$m=q \cdot [a,b]+r , \ 0 \leq r < [a,b]...

[SIZE="4"]HI,... This is my first post here on this forum ... I wonder if anyone here can help me to clarify some concepts to me in the paper of alan guth 2007 named "Eternal inflation and its implications" and this is my question here ... he says in the abstract the following...

Homework Statement I hope this does not violate copyright or anything but this problem originated from an assignment from Introduction to Mathematical Thinking in Coursera. I could not post there because the class ended and the discussion board there is dead. Let C be the set of all cars, let...

I'll be linking to this post when the question arises why an implication with a false premise is true.

Homework Statement A satellite is launched into a circular sun-synchronous orbit at a height of 900km above Earth's surface. What is the implication on the orbit's inclination (in deg) and on the change of the position of the right ascension of the ascending node per day.Homework Equations The...

Hello, I have some questions about the truth tables for impliocation and equivalence. for implication we have: p | q | p=> q T | T | T T | F | F F | T | T F | F | T Here I do not understand the last two lines, how can we say that p implies q when...

I have a non-zero measured subset X\subseteq\mathbb{R}^{n} on which \sum_{i=1}^{n}\psi_{i}x_{i}=0 for all x=(x_{1},\ldots,x_{n}) in X. How can I show that \psi_{i}=0 for i=1,\ldots,n?

A: (p => ~q) ^ (p v q) B: ~p v q does A => B (A implies B) ? does B => A ( B implies A) ? I did the truth tables for each: A => B: http://www4c.wolframalpha.com/input/?i=%28%28p+%3D%3E+NOT+q%29+AND+%28p+OR+q%29%29+%3D%3E+%28NOT+p+OR+q%29 B => A...

Homework Statement Is the argument below valid? If it's valid, write down the argument symbolically. If a movie is not worth seeing, then it was not made in England. A movie is worth seeing only if critic Ivor Smallbrain reviews it. The movie The Good, The Bad and The Mathematician was...

Unfortunately, my knowledge on this is limited to wiki which I trust is relatively correct but I would like to clear up some ambiguity. (I don't mind technical but please back it up with a general explanation since I am completely new to this. The general explanations I value more.) Wiki...

If P→Q, and P is false but Q is true, then why is P→Q true? To me, it seems as though we shouldn't be able to do proceed because there isn't enough information. Same goes when P and Q are both false, how does that suggest P→Q is true?

According to the truth tables in my computer architecture text, P → Q is false if P is true and Q false; and true otherwise. I cannot understand why it is "true" otherwise. For example, if P and Q are both true, P → Q is also true, but this makes no sense to me. Perhaps Q is true for...

First of all, I am fairly new to relativity, but not clueless. I am not saying that FTL is possible. I am not denying relativity principles. I am stating that FTL may be plausible. Relativity gives flexibility to how you can synchronize clocks and that does not affect outcomes of most...

Hello everyone! I was trying to prove the propositions that follow the addition axioms as a revision, I got a different proof for the following proposition: If $x+y=x+z$ then $y=z$ My proof was the following: $x+y=x+z$, $(-x)+x+y=(-x)+x+z$, $0+y=0+z$, $y=z$ Rudin however, in his book...

Homework Statement If lim (x->∞) [ln(x^(1/x))]=0 and lim (x->∞) x^(1/x)=1, then does this => lim (x->∞) [ln(x^(1/x))]= ln(lim(x->∞) [(x^(1/x))]) = ln(1)?? Homework Equations lim (x->∞) [ln(x^(1/x))]=0 and lim (x->∞) x^(1/x)=1 The Attempt at a Solution lim (x->∞) [ln(x^(1/x))]=...

Would it be possible for a truck, measuring 5ft tall when stationary, to pass under a 4ft barrier by accelerating towards the speed of light? If so what would a spectator see if standing next to the barrier, would the spectator see the truck shrink? What would you see of the barrier from the...

Suppose I know that [(P implies ~A) and (P implies B) and (P implies C)] is impossible. Does this means that the following statement is true: [(P implies A) and (P implies B) and (P implies C)]? Any help is greatly appreciated!

Hi guys. I'm rather new to number theory, and as part of an assignment I have been learning about various different primality tests. One of these tests is the Lucas primality test. As part of the reasoning behind the test, wikipedia states: "If [$a^{n-1} \equiv 1 \textrm{ (mod...

I understand that we just have to fill the last two raws in the truth table with any value, and that we randomly chose True, and that the value True makes matters easier sometimes (I don't know an example of that, but I read that somewhere). But the question is, since mathematics is tied to...

The truth table for implication looks like this p|q| p -> q ------------ T|T | T T|F | F F|T | T <----I'm trying to make sense of this one. My prof warned us that its strange. F|F | T I that implication means: "If p, then q" "q is necessary for p" "p is sufficient for q" "p, only...

Question about "if and then" statements. IE implication statements. Homework Statement When something is for example asking for: if |x-3|<δ, prove that |x+3| <δ + k (where k is a constant) are they supposing it's true? Like are they giving you a hypothesis? How do implication...