The first two rules of inference are the two valid conditional forms from Valid and Invalid Argument Forms: modus ponens and modus tollens. In a proof they are written MP and MT.
The letters stand for any formulas, simple or compound.
Rule
From
Infer
MP (modus ponens)
$p \to q$ and $p$
$q$
MT (modus tollens)
$p \to q$ and $\neg q$
$\neg p$
🔑 Key idea
MP: from a conditional and its antecedent, infer its consequent. MT: from a conditional and the negation of its consequent, infer the negation of its antecedent. Each cites two lines.
💡 Why it works
MP is valid because the only row that makes $p \to q$ true and $p$ true also makes $q$ true. MT is valid because if $q$ is false and $p \to q$ is true, $p$ cannot be true (that would make the conditional T → F). Because these are forms, $p$ and $q$ can be replaced by any formulas: from $(r \land s) \to \neg t$ and $r \land s$, MP gives $\neg t$.
Two uses of modus tollens in a row.
Worked examples
✏️ Worked example: Modus ponens with compound parts
From line 1, $(p \lor q) \to r$, and line 2, $p \lor q$, what does MP give?
The antecedent of line 1 is $p \lor q$, which is line 2.
MP gives the consequent.
Answer: $r$, justified "MP 1, 2".
✏️ Worked example: Modus tollens
From $s \to \neg t$ and $\neg \neg t$, what does MT give?
Applying MP with the consequent instead of the antecedent (from $p \to q$ and $q$, inferring $p$), or applying MT with $\neg p$ (from $p \to q$ and $\neg p$, inferring $\neg q$). Those are affirming the consequent and denying the antecedent, the two invalid look-alikes. A check: MP needs the part before the arrow; MT needs the negation of the part after it.
MT needs the exact negation of the consequent. If the consequent is $\neg t$, MT needs $\neg \neg t$. If you have $t$ instead, you will be able to turn it into $\neg \neg t$ later with a rule of replacement (DN).
The order of the cited lines does not matter: "MP 1, 3" and "MP 3, 1" say the same thing. What matters is that one cited line is a conditional and the other is exactly its antecedent (for MP) or exactly the negation of its consequent (for MT). "Exactly" means the very same formula, not one that merely says the same thing.
Most proofs you will meet use MP somewhere. When you see a conditional among the premises, ask at once: do I have, or can I get, its antecedent (for MP) or the negation of its consequent (for MT)?
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 1 XP.