In logic, a well-formed formula (wff) refers to a syntactically correct expression constructed according to the formal rules of the logical system. Let's analyze each of your expressions in terms of their well-formedness.

1. $A \vee A$:
   Yes, this is a wff.
   It follows the correct syntax for propositional logic where $\vee$ represents disjunction (OR). The same propositional variable appearing twice doesn't affect the formula's well-formedness.

2. $A + A$:
   No, this is not a wff.
   The symbol $+$ is not a standard logical operator in classical propositional logic. It is more commonly used in arithmetic or boolean algebra.

3. $A \& \sim A$:
   Yes, this is a wff.
   $\&$ represents conjunction (AND), and $\sim A$ is a valid negation of $A$. The formula $A \& \sim A$ is well-formed.

4. $\sim (\sim Z \supset (Z \vee Z))$:
   Yes, this is a wff.
   Negation $\sim$, implication $\supset$, and disjunction $\vee$ are all valid logical operators. This formula is properly constructed.

5. $(A \supset B) \& (C \equiv D) \vee ((E \vee F) G)$:
   No, this is not a wff.
   The subexpression $(E \vee F) G$ is not well-formed because the formula should contain logical operators between propositional variables. The juxtaposition of $(E \vee F)$ and $G$ is syntactically incorrect.

6. $A \sim \supset (E \vee D)$:
   No, this is not a wff.
   $A \sim \supset$ is not a valid combination. The negation operator $\sim$ should precede a proposition or formula, but here it is placed between two components in an incorrect manner.

7. $A \supset \sim A$:
   Yes, this is a wff.
   Implication $\supset$ and negation $\sim A$ are used correctly. This formula represents "if $A$ then not $A$."

8. $\sim A \supset (A \vee A)$:
   Yes, this is a wff.
   This formula is well-formed. It consists of a negation $\sim A$, implication $\supset$, and a valid disjunction $(A \vee A)$.

9. $A \sim Z \supset A \vee A$:
   No, this is not a wff.
   The subexpression $A \sim Z$ is not valid, as $\sim$ should apply to a single propositional variable or formula, but here it is placed between $A$ and $Z$ incorrectly.

10. $A \sim A \supset \sim B$:
    No, this is not a wff.
    The subexpression $A \sim A$ is not valid. As mentioned earlier, $\sim$ should apply to a single formula, not between two propositional variables.

In summary, the well-formed formulas (wffs) are:

• $A \vee A$
• $A \& \sim A$
• $\sim (\sim Z \supset (Z \vee Z))$
• $A \supset \sim A$
• $\sim A \supset (A \vee A)$

Would you like more details or have any other questions?

5 Related Questions:

1. What are the formal rules for constructing well-formed formulas in propositional logic?
2. Can you explain the difference between logical operators like $\supset$, $\vee$, and $\&$?
3. What are some examples of non-wffs in propositional logic and why they fail?
4. How does propositional logic differ from first-order logic in terms of wff construction?
5. What role does syntax play in ensuring a formula is a wff?

Tip:

Always ensure that logical operators are applied correctly and that every part of a formula follows the syntactic rules of the logic system in use.