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

  • Context: MHB 
  • Thread starter Thread starter agapito
  • Start date Start date
  • Tags Tags
    Equivalence Proof
Click For Summary
SUMMARY

The discussion centers on the logical equivalence between the statements (∀v Fv -> p) and (∃u Fu -> p). It is established that the equivalence holds under the condition that variable v occurs free in Fv only where variable u occurs free in Fu, and p does not contain free occurrences of v. A proof approach involves representing implications as disjunctions and applying de Morgan's laws for quantifiers, leading to a clearer understanding of the equivalence.

PREREQUISITES
  • Understanding of first-order logic and quantifiers
  • Familiarity with logical implications and their transformations
  • Knowledge of de Morgan's laws in the context of logic
  • Basic comprehension of free and bound variables in logical expressions
NEXT STEPS
  • Study the transformation of implications using logical equivalences
  • Learn about de Morgan's laws and their application in quantifier logic
  • Explore the concepts of free and bound variables in formal logic
  • Investigate proofs of logical equivalences in first-order logic
USEFUL FOR

This discussion is beneficial for students of logic, mathematicians, and anyone interested in formal proofs and logical equivalences in first-order logic.

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.
 

Similar threads

MHB Logic Proof With Rules of Replacement
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad 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
Undergrad Counterexample Required (Standard Notations)
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Homemorphism in quotient topology
  • · Replies 5 ·
Replies
5
Views
1K
Graduate Uncertainties in the proof of Proposition 4.4.2 in Hawking and Ellis
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad 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
Graduate Proof of Seifert-Van Kampen Theorem
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Confusing Axiomatic Set Theory Proof
  • · Replies 1 ·
Replies
1
Views
2K