Testing Logical Equivalence

In summary, the conversation discusses determining whether two equations are logically equivalent and suggests using the biconditional to determine this. It also mentions using analogies and deMorgan's laws to understand the concept better.
  • #1
axellerate
4
0
Hello hello, I'm not looking just for an answer per say, but am also wondering the thought process in solving problems such as the following:

Hopefully this doesn't take up too much of someones time.

Determine whether the following equations are logically equivalent:

1) (∃x)( P(x) → Q(x) )

2) (∀x)P(x) → (∃x)Q(x)
 
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  • #2
Try to write the implication in terms of other connectives.
 
  • #3
After you have followed micromass's suggestion, the next step can make more sense to you if you contemplate an analogy (I stress that this is a way of thinking: it would not work as a formal proof)
all quantifier like a large "and",
existence quantifier like a large "or"
"and" like "intersection"
"or" like "union"
deMorgan Laws.
Formally, if you are not an intuitionist, you can try playing around with the equivalence between "[itex]\forall[/itex]x P" and "~[itex]\exists[/itex]x ~P", or between "[itex]\exists[/itex]x Q" and "~[itex]\forall[/itex]x ~Q"

(by the way, it's "per se")
 
  • #4
If you are familiar with how to determine a formula is logically true, then you can use the fact that formulas are logically equivalent just in case their biconditional is logically true. If there is an interpretation that makes the biconditional of (1) and (2) false, then they are not logically equivalent. If there is no such interpretation, then they are logically equivalent.
 
  • #5



As a scientist, my thought process in solving problems like this would involve breaking down each equation and analyzing its components. In this case, both equations involve quantifiers (∃ and ∀) and conditional statements (→). I would then consider the definitions and properties of these logical operators.

For the first equation, (∃x)(P(x) → Q(x)), I would first focus on the inner conditional statement, P(x) → Q(x). This statement can be read as "if P(x) is true, then Q(x) must also be true." This is known as the material implication. In order for this statement to be true, either P(x) is false or Q(x) is true (or both). Therefore, the statement (∃x)(P(x) → Q(x)) is true if there exists at least one value of x for which either P(x) is false or Q(x) is true.

For the second equation, (∀x)P(x) → (∃x)Q(x), I would again focus on the inner conditional statement. In this case, it can be read as "if P(x) is true for all values of x, then there exists at least one value of x for which Q(x) is true." This is known as the universal generalization. In order for this statement to be true, P(x) must be true for all values of x, and at least one value of x must make Q(x) true. Therefore, the statement (∀x)P(x) → (∃x)Q(x) is true if and only if P(x) is true for all values of x and Q(x) is true for at least one value of x.

Based on these analyses, it can be seen that the two equations are not logically equivalent. The first equation allows for the possibility that P(x) is false for some values of x, while the second equation requires P(x) to be true for all values of x. Therefore, the two equations have different truth values and are not logically equivalent.
 

What is logical equivalence?

Logical equivalence is a concept in mathematics and logic that refers to two statements or propositions that have the same truth value. This means that if one statement is true, then the other must also be true, and if one statement is false, then the other must also be false.

How is logical equivalence tested?

Logical equivalence is tested by using logical equivalences laws and rules. These include the commutative, associative, and distributive laws, as well as De Morgan's laws and the double negation law. By applying these rules and simplifying the statements, we can determine if they are logically equivalent.

Why is it important to test for logical equivalence?

Testing for logical equivalence is important because it helps us to identify whether two statements or propositions are essentially saying the same thing. It allows us to simplify complex statements and to better understand the relationship between different statements.

What is the difference between logical equivalence and material equivalence?

Logical equivalence and material equivalence are similar concepts, but they refer to different types of statements. Logical equivalence is used to compare two logical statements, while material equivalence is used to compare two statements that are based on facts or observations in the real world.

What are some common mistakes when testing for logical equivalence?

Some common mistakes when testing for logical equivalence include confusing logical equivalence with material equivalence, not following the correct rules and laws, and not simplifying the statements enough to determine their equivalence. It is important to carefully apply the rules and to double-check the results to avoid these mistakes.

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