Modus ponens · printable packet

A derivation is a numbered list of lines, each with a rule and the lines it cites; modus ponens, modus tollens and the two conjunction rules are enough to start.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

By the end of this lesson you will be able to write a derivation as numbered lines with rules and citations, apply modus ponens, modus tollens and the conjunction rules, say what each rule demands of the lines it cites, find the line of an attempted proof that its rule does not produce, and build a compound line that is not a premise.

2. What you already have

You can decide validity with a table and you know the named forms. A derivation is those forms used one at a time, written down in a way that can be checked line by line without any table at all.

3. Terms to use precisely

4. Deriving line by line

Term	What it means
Modus ponens	MP: from A -> B and A, infer B.
Conjunction introduction	andI: combine two available formulas as a conjunction.
Conjunction elimination	andE: extract either part of a conjunction.
Reference	The number of an earlier proof line supplying an input to a rule.

A derivation replaces the table with a list. Every line is a formula together with the rule that produced it and the numbers of the lines that rule was applied to; a line may cite only lines above it, and the last line is the conclusion. The basic rules are these. MP: from $\phi \to \psi$ and $\phi$, write $\psi$. MT: from $\phi \to \psi$ and $\neg \psi$, write $\neg \phi$. andI: from any two lines, write their conjunction. andE: from $\phi \wedge \psi$, write either half. Nothing is assumed about which lines are premises: a line built by a rule is a line like any other and may be cited afterwards. The gain over a table is not certainty — the table was already certain — but size: a derivation grows with the length of the argument, while a table doubles with every atom.

Another way: steps

Write every premise as its own line, copied exactly.
Look at the conclusion and ask which rule could produce it.
Ask what that rule would need, and whether those formulas are lines yet.
Work forwards from the premises until they are, then take the last step.

Another way: example

From $P \wedge Q$ and $Q \to R$, derive $R$. Line 3 is $Q$ by andE from 1; line 4 is $R$ by MP from 2 and 3. Two lines of work, and no table.

5. Match a whole input

The letters A and B in the schema can stand for compound formulas. From (P & Q) -> R, MP requires P & Q as one available line. P on its own is insufficient. If P and Q are separately available, first use andI to produce their conjunction, then cite that line and the conditional. Conversely, from P & Q you can use andE to obtain P when a conditional needs only P.

The consequent can also be compound. From P -> (Q | R) and P, MP gives Q | R, not Q and not R separately. An inference rule preserves exactly the structure it supplies. Copying just the part you wanted changes the conclusion.

Write premises exactly as provided, with rule premise and no references. Number lines from one. A derived line names its rule and cites the earlier lines that supply the inputs. This bookkeeping lets another person check a derivation without guessing which claim you intended to use. It also prevents a future conclusion from serving as its own support.

6. From a truth condition to an inference rule

The material conditional P -> Q has one false case: P is true and Q is false. Modus ponens supplies a second premise P. Because that premise is true in every assignment relevant to the argument, the rows with P false are no longer candidates. Of the remaining rows, the conditional eliminates the one with Q false. The only surviving possibility has Q true. This is a reason the inference preserves truth, rather than merely a name attached to a familiar pattern.

Notice the two different jobs the premises do. The conditional connects a condition to a result. The atomic premise establishes the condition within the argument. A conditional alone does not produce its consequent: P false and Q false would satisfy P -> Q. An antecedent alone does not produce an arbitrary result either: P true with Q false would satisfy P. The guarantee arises from the two premises together.

In a derivation, MP makes this reasoning available as a reusable rule. You do not have to draw a four-row table every time you use it. You do have to check that the two formulas you cite have the required relationship. A rule justified once by its truth conditions can be applied to new formulas, but it cannot be stretched to inputs that only resemble the required shape. The proof checker therefore examines the formulas on the cited lines, not your intention or the plausibility of the final sentence.

If the conditional is ~P -> Q, its antecedent is ~P. You need a line asserting that complete denial before MP yields Q. A line asserting P does not supply it. If the conditional is P -> ~Q and P is available, MP yields ~Q. The presence of a negation does not turn this into modus tollens. The rule is identified by how the inputs match, not by whether the output contains a negation sign.

The same point applies to larger inputs. From (P | Q) -> R and P | Q, MP yields R. From (P | Q) -> R and P, a preliminary orI step can construct P | Q before MP is used. From (P & Q) -> R and P, there is no corresponding shortcut that manufactures Q. Inclusive disjunction needs one true disjunct; conjunction needs both conjuncts. Which construction is possible depends on the connective's truth conditions.

7. Reference checking is part of the reasoning

Suppose a proof lists P & Q as its first line, P -> R as its second, P as its third by andE, and R as its fourth by MP. The last line must cite the conditional and the extracted P. Citing the conjunction instead of the extracted P leaves the written step unsupported, even though the missing extraction is easy to imagine. A reader should not have to supply an unrecorded inference to make a line correct.

Write the premise lines before derived lines. Copy their formulas rather than paraphrasing them. Parentheses are especially important when an antecedent or consequent is compound. The formula P -> (Q & R) has P as its complete antecedent and Q & R as its complete consequent. A proof that writes Q immediately after applying MP has skipped another rule: first derive Q & R, then use andE if Q is the actual target.

Rule names and references give different information. MP names the operation. The references identify its inputs. A proof can contain the right formula and the right rule name but the wrong references, just as a calculation can use the right arithmetic operation on the wrong numbers. When correcting an error, inspect the cited lines first. Then ask whether the problem is a missing intermediate formula, a reversed conditional, or merely a mistyped line number.

The target helps you decide which available conditional matters. If the goal is R, a conditional ending in R is a promising candidate. Its antecedent becomes a smaller local goal. If that antecedent is not yet a line, inspect the other premises for a way to build it or extract it. This backward planning does not change the direction of justification: the completed proof must still write each input before the line that uses it.

You may have more premises than a proof needs. An unused premise does not make a valid proof invalid. However, silently assuming a missing premise does. Distinguish these cases by writing the supplied list clearly. A long derivation with many available facts may still require only one of its conditionals and one matching antecedent. Adding irrelevant lines does not strengthen that specific inference.

8. What a successful derivation leaves to investigate

A completed proof tells you that its conclusion follows if its premises are true. It does not inspect a parcel, verify a record, or establish that a conditional policy accurately describes an organization. For an actual decision, separate the formal check from those factual checks. If a premise later turns out false, the derivation may still display valid reasoning from that premise, but it no longer supplies a sound argument about the actual situation.

This separation also helps diagnose disagreement. One person may accept the conditional but dispute whether its antecedent holds. Another may accept the reported event but reject the claimed guarantee. Writing both inputs makes those disagreements visible. Saying only 'therefore R' hides which step needs evidence. A careful explanation names the conditional, names the case that satisfies its antecedent, and then states the consequent the rule permits.

Finally, test a tempting reverse inference. P -> Q together with Q does not yield P. Choose P false and Q true: both given premises hold, yet the proposed conclusion fails. That countermodel explains why knowing a result is not enough to infer one particular sufficient condition. The safe habit is to read the whole antecedent before using MP, rather than treating every arrow as a two-way connection.

9. Following a dispatch authorization

A fictional workshop has two records: the parcel is packed and labeled; if the parcel is packed, pickup may be requested. Let P mean packed, L mean labeled, and R mean pickup may be requested. The premises are P & L and P -> R. The goal is R.

Write P & L as line 1 and P -> R as line 2. The antecedent needed by line 2 is P, while line 1 is a conjunction. At line 3, use andE on line 1 to extract P. At line 4, use MP on lines 2 and 3 to derive R. The label information is not used by this particular conditional; it remains part of the original record but is not an extra requirement of the stated rule.

Change the rule to (P & L) -> R and supply only P. Pickup permission no longer follows. An assignment with P true, L false, R false keeps both premises true and the conclusion false. The need to match the whole antecedent therefore has a practical consequence: a missing required condition must not disappear during the proof.

Neither proof establishes that the workshop's policy is sensible or that the parcel actually was packed. It checks the relation between the supplied records and the declared permission rule. The operation and its evidence are separate from the formal derivation. A careful record keeps both the premise evidence and the proof available for inspection.

10. Where this goes wrong

The first error is citing a formula that is inside a line rather than a line: $P$ occurring as half of $P \wedge Q$ is not the line $P$ until andE has written it down. The second is naming a rule that the citations do not fit, usually because the new line is obviously true; being true is not a rule. The third is stopping one line early, when the conclusion has been made available but not written.

11. Extract the required antecedent

12. Build a compound antecedent

13. Use a compound consequent in a second inference

14. A denial can be the antecedent

15. Guided practice

From the premises P & Q and Q -> R, derive R. Give one line at a time, with the rule and the lines it uses.

16. Guided practice

Complete the structural check for a derivation from P, Q and (P & Q) -> R. Count the steps after the premises, keeping one inference per line.

17. Guided practice

P means packed and R means pickup permitted. From P -> R and P, derive R, giving line numbers and rules.

18. Practice

19. Practice

20. Somewhere new

From the premises P, Q and (P & Q) -> R, derive R. Give one line at a time, with the rule and the lines it uses.

21. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

22. Test question

23. What you can do now

You can produce a short derivation and check somebody else's line by line. Say in your own words why a formula occurring inside a line is not yet available as a line.

Working for the steps left to you

The result is the whole consequent, not an arbitrarily chosen disjunct.

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P	Q	((P -> Q) & P) -> Q

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