MHB Is (∀v Fv -> p) Equivalent to (∃u Fu -> p)?

  • Thread starter Thread starter agapito
  • Start date Start date
  • Tags Tags
    Equivalence Proof
Click For Summary
The discussion centers on the equivalence between the statements (∀v Fv -> p) and (∃u Fu -> p), with specific conditions regarding the free occurrences of variables. Participants seek clarification on the meaning of "occurring free at all" and the implications of variable u's occurrences in Fu. One suggested approach to prove the equivalence involves transforming the implications using logical identities, such as representing A -> B as ¬A ∨ B. The conversation highlights the need for a clearer understanding of the terms used in the equivalence. Overall, the equivalence remains a topic of interest, prompting requests for proof and further explanation.
agapito
Messages
46
Reaction score
0
Consider the equivalence:

(∀v Fv -> p) <=> (∃u Fu -> p)

Where variable v occurs free in Fv at all and only those places that u occurs free in Fu, and p is a proposition containing no free occurences of variable v.

Can someone please offer a proof of such equivalence. Many thanks. am
 
Physics news on Phys.org
I am not sure what it means to occur free "at all", and I don't understand the phrase "only those places that u occurs free in Fu". What is claimed about the places where u occurs free?

Perhaps your equivalence is $\forall v\,(Fv\to p)\iff (\exists u\,Fu)\to p$ or $(\forall v\,Fv)\to p\iff \exists u\,(Fu\to p)$. This is easy to show if you represent $A\to B$ as $\neg A\lor B$ and use de Morgan's law for quantifiers.
 

Thread 'A variant of the Monty Hall problem'

There is a nice little variation of the problem. The host says, after you have chosen the door, that you can change your guess, but to sweeten the deal, he says you can choose the two other doors, if you wish. This proposition is a no brainer, however before you are quick enough to accept it, the host opens one of the two doors and it is empty. In this version you really want to change your pick, but at the same time ask yourself is the host impartial and does that change anything. The host...
View full post »

Similar threads

MHB Logic Proof With Rules of Replacement
  • · Replies 1 ·
Replies
1
Views
1K
I Ways to put n balls in m boxes such that exactly k boxes are empty?
  • · Replies 29 ·
Replies
29
Views
4K
MHB How to give a proof of tautologies?
  • · Replies 4 ·
Replies
4
Views
4K
I Counterexample Required (Standard Notations)
  • · Replies 16 ·
Replies
16
Views
3K
I Homemorphism in quotient topology
  • · Replies 5 ·
Replies
5
Views
803
A Uncertainties in the proof of Proposition 4.4.2 in Hawking and Ellis
  • · Replies 2 ·
Replies
2
Views
710
Equivalence of (P -> R) V (Q -> R) and (P ∧ Q) -> R
  • · Replies 3 ·
Replies
3
Views
14K
MHB Solving a Logic Proof: Existential Hypothesis Rule
  • · Replies 4 ·
Replies
4
Views
3K
A Proof of Seifert-Van Kampen Theorem
  • · Replies 2 ·
Replies
2
Views
3K
Confusing Axiomatic Set Theory Proof
  • · Replies 1 ·
Replies
1
Views
2K