Logic Statements: Understanding F(x) & \forall y (F(y)

[nobahar]

Hello!

1) $$\forall x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))$$
2) $$\exists x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))$$
3) $$\forall x (F(x) \land \forall y (F(y) \rightarrow y=x))$$
4) $$\exists x (F(x) \land \forall y (F(y) \rightarrow y=x))$$

If 1) is true then 2) is true; if 1) is false then 2) may or my not be true
If 2) is true then 1) may or may not be true; if 2) is false then 1) is false

If $$\forall y (F(y) \rightarrow y=x))$$ is true then:
If 1) is true then 2) is true; if 1) is false then 2) may or my not be true
If 2) is true then 1) may or may not be true; if 2) is false then 1) is false

If $$\forall y (F(y) \rightarrow y=x))$$ is false then:
3) and 4) are always false.

I understand $$(F(x) \rightarrow \forall y (F(y) \rightarrow y=x))$$ to mean that if F(x) is true then $$\forall y (F(y) \rightarrow y=x))$$ is true. So the first two are determined by whether or not all x or there is some x that make F(x) true. I understand $$(F(x) \land \forall y (F(y) \rightarrow y=x))$$ to mean that they are independent, and F(x) and $$\forall y (F(y) \rightarrow y=x))$$ can be true or false separately.
Is this correct?

Thanks in advance.

 

[mmwave]

Hello! Thank you for your question and for sharing your understanding of these statements. Your understanding is mostly correct, but there are a few things that I would like to clarify.

First, let's look at the statements 1) and 2). As you mentioned, if 1) is true, then 2) is also true, but if 1) is false, then 2) may or may not be true. This is because the truth of 2) depends on the existence of an x that makes F(x) true. If such an x exists, then 2) is true. However, if there is no x that satisfies F(x), then 2) is false.

Similarly, if 2) is true, then 1) may or may not be true. This is because the truth of 1) depends on whether or not all y satisfy the condition \forall y (F(y) \rightarrow y=x)). If all y satisfy this condition, then 1) is true. However, if there is at least one y that does not satisfy this condition, then 1) is false.

To address your question about 3) and 4), I would like to clarify that they are not always false. In fact, they are both equivalent to the statement \forall x (F(x)), which means "for all x, F(x) is true." This is because if F(x) is true for all x, then it is also true for any y (since y is also included in the "for all" quantifier). Therefore, \forall y (F(y) \rightarrow y=x) is also true for all x, making 3) and 4) true. However, if there is at least one x for which F(x) is false, then \forall x (F(x)) is false, and therefore 3) and 4) are also false.

I hope this helps clarify your understanding of these statements. If you have any further questions, please don't hesitate to ask. it is important to have a clear understanding of logical statements and their implications in order to accurately interpret and communicate scientific findings. Best of luck in your studies!