# Error in Intuitive Understanding of First-Order Logic w/ 'And' and 'If-Then

• julypraise
This way, you will be using the given information and proving the desired conclusion, rather than assuming the conclusion and trying to prove the given information.
julypraise
One of the ordered field, F, property is the following (i):

(i) for every x, y, and z, if both x, y, z in F and y < z, then x + y < x + z.

(i') for every x, y, and z, if x, y, z in F implies y < z, then x + y < x + z.

I cannot prove that (i) and (ii) are equivalent simply using some simple logical rules. But when I inspect (i') in a intuitive sense, it seems (i') does not have a difference in terms of meaning with (i). Moreover, (i') is equivalent to (i'') which is the following:

(i'') for every x, y, z in F, if y < z, then x + y < x + z.

Especially when I think about this (i''), it seems really similar in terms of meaning to (i''). Yet, I cannot make (i) in the same form as (i'').

Thus, my question are these:

(1) Is (i) equivalent to (i')?
(2) If they are not, why does my intuitive understanding of those two sentences' meaning makes error? Any opinion?

My plausible guess for the question (2) is that it is because (i') implies (i). But anyway, I'm not sure whether it is a right anwer.

Further question:

What is wrong with the following proof, or is the following proof correct:

(->). Suppose (X and Y) implies Z. Suppose X. Suppose Y. Since X and Y, Z is true. Thus, Y implies Z. Thus X implies (Y -> Z).
(<-). Suppose X implies (Y implies Z). Suppose X and Y. Since X is true, Y implies Z. Since Y is true, Z is true too. Thus (X and Y) implies Z.
In conclusion, we proved that that (X and Y) implies Z is equivalnet to that X implies (Y -> Z).

Remark.
When proving the above in Fitch form style, I didn't find any contradiction. Please try that too and comment whatever you think.

The proof is incorrect. The problem lies in the first step, where you assume that "(X and Y) implies Z". This does not necessarily have to be true, so you cannot use it as a starting point for your proof. Instead, you should start by assuming that X implies (Y -> Z), and then try to prove that (X and Y) implies Z.

## 1. What is the difference between 'and' and 'if-then' in first-order logic?

In first-order logic, 'and' is a logical connective that represents the conjunction of two statements, meaning both statements must be true for the overall statement to be true. On the other hand, 'if-then' is a conditional statement that represents an implication, meaning if the first statement is true, then the second statement must also be true.

## 2. How do 'and' and 'if-then' statements work together in first-order logic?

In first-order logic, 'and' and 'if-then' statements can be combined to create complex statements. For example, an 'and' statement can be used to connect two 'if-then' statements, creating a compound conditional statement. Additionally, both 'and' and 'if-then' statements can be nested within each other to create even more complex statements.

## 3. Can 'and' and 'if-then' statements be used interchangeably in first-order logic?

No, 'and' and 'if-then' statements serve different purposes in first-order logic and cannot be used interchangeably. 'And' represents the conjunction of two statements, while 'if-then' represents an implication between two statements.

## 4. What is the importance of understanding the correct usage of 'and' and 'if-then' in first-order logic?

Understanding the correct usage of 'and' and 'if-then' in first-order logic is crucial for constructing accurate and valid logical arguments. Misusing or misunderstanding these logical connectives can result in flawed reasoning and incorrect conclusions.

## 5. How can one improve their intuitive understanding of 'and' and 'if-then' in first-order logic?

Improving one's intuitive understanding of 'and' and 'if-then' in first-order logic can be achieved through practice and familiarizing oneself with various logical statements and their connectives. It can also be helpful to study and understand the properties and rules associated with these connectives.

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