4

As you read the question above what has a higher operation priority from these two logical operations $"\models" or "\Rightarrow"$ ?

I stumbled on a proof of a tautological entailemnt without parenthesis stating $\models A \Rightarrow B$. How do you read that ? The claim is:

If $\models A$ and $\models A \Rightarrow B$ then $\models B$.

The proof is that because $\models A$ , $\upsilon_\alpha(A) = T$ is valid. From $\models A \Rightarrow B$ comes $\upsilon_\alpha(A \Rightarrow B) = T$ ... and so on.

I am really confused because i remember that the $\Rightarrow and \Leftrightarrow$ would have the lowest operation priority

Marko Majstorovic
  • 43
  • 3

1 Answers1

1

$1.$ In this case, it means $\big((\models A)\wedge\big(\models (A\Rightarrow B)\big)\big)\Rightarrow(\models B)$

Since $$\upsilon_\alpha(A) = T\text{ and }\upsilon_\alpha(A \Rightarrow B) = T$$

That implies $\upsilon_\alpha(B)=T\tag*{$\square$}$

$2.$ Even $\big((\models A)\wedge\big((\models A)\Rightarrow B\big)\big)\Rightarrow(\models B)$ is also a valid statement

Assume it's the second case, the proof would looks like

Since $$\upsilon_\alpha(A)=T\text{ and }\upsilon_\alpha(A)\Rightarrow B$$

Therefore $\upsilon_\alpha(B)=T\tag*{$\square$}$

Compare to the given proof, clearly, it's $1.$

Ethan
  • 5,291
  • Thank you for your reply. Yeah, i thought it would be that way around just wanted some approval the book i am reading is written in a really strange way. – Marko Majstorovic Oct 26 '19 at 09:50