Symbolic Logic - Sentential Logic

Which of the following statements is part of the definition for sentential logic?

If and are formulas then $A\Leftrightarrow B$ is a formula as well.

Every lower case letter is a formula.

Anything and everything can be considered a formula.

If is a formula then so is $B\times A$ .

If is a formula then so is $\sim a$ .

Explanation

It is important to recall that sentential logic has a very specific definition that outlines and describes different formulas.

There are seven different statement criteria when discussing sentential logic and they are as follows.

I. If is a formula then $\sim A$ is a formula as well.

II. If and are formulas then $A\vee B$ is a formula as well.

III. If and are formulas then $A\rightarrow B$ is a formula as well.

IV. If and are formulas then $A\Leftrightarrow B$ is a formula as well.

V. If and are formulas then $A\wedge B$ is a formula as well.

VI. All upper case letters are formulas

VII. Nothing else is a formula.

Looking at the possible answer selections only IV is part of the sentential logic definition thus, "If and are formulas then $A\Leftrightarrow B$ is a formula as well." is in the definition.

Which of the following statements is NOT a definition of sentential logic?

Only , , , and are formulas.

If is a formula then $\sim A$ is a formula as well.

If and are formulas then $A\vee B$ is a formula as well.

If and are formulas then $A\rightarrow B$ is a formula as well.