The image you uploaded appears to be a logical proof exercise involving symbolic expressions and premises. It includes a table for demonstration of inference laws, where the premises are given and the reasoning for conclusions is outlined.

Let me break down the information based on the symbols provided in the image:

Given Symbolic Expression:

$\left( \left( p \land q \right) \land \left( (q \rightarrow r) \land (s \leftrightarrow r) \right) \land (s \vee t) \right) \rightarrow t$

Premises:

• P1: $p \land q$
• P2: $q \rightarrow r$
• P3: $s \leftrightarrow r$
• P4: $s \vee t$

Table:

The table at the bottom lists inference laws, showing premises applied and the corresponding justification for each.

Laws used:

• P5: Law of addition (from P1)
• MPP: Modus Ponens (from P2, P5)
• MTT: Modus Tollens (from P3, P6)
• MTP: Modus Tollens again (from P6)

Would you like a deeper analysis or clarification on how these logical steps are derived? If so, feel free to ask for further explanations!

Related Questions:

1. What is the Law of Addition in propositional logic?
2. Can you explain the Modus Ponens rule?
3. What is the difference between Modus Ponens and Modus Tollens?
4. How do you derive conclusions from logical premises in symbolic logic?
5. What is the role of the biconditional ($\leftrightarrow$) in logical proofs?

Tip: Always ensure you clearly understand each logical operator (like $\land$, $\vee$, $\rightarrow$, and $\leftrightarrow$) to follow along with logical proofs.