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14 / 18 · Account for every step

Borrow an assumption, then return it

Use conditional proof without leaking temporary assumptions.

Builds on A proof is a chain of permissions

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A question to keep in mind

If you assume P to prove Q, have you proved Q unconditionally?

To prove P → Q, open a subproof by assuming P. If Q follows inside that subproof, close it and infer P → Q by conditional introduction (→I). The conclusion is conditional; you have not established P as a fact.

A line derived under an open assumption depends on it unless that dependency is discharged by an appropriate rule. After a subproof closes, its internal lines are not freely available as stand-alone facts. This is why scope matters in a proof as much as in a formula.

Conditionals can be proved even if the temporary assumption is not true. This is a method of reasoning about a possibility, not a report that the possibility occurred. Keep permanent premises separate from temporary assumptions in your notes.

Work through an example

  1. 1. P → Q (premise). 2. Q → R (premise). Open: 3. assume P.
  2. Inside: 4. Q (→E, 1,3). 5. R (→E, 2,4). Close: 6. P → R (→I, 3–5).
  3. The premises entail P → R, not R by itself. R could be false when P is false.

Your turn

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Practice 1Not checked

Justify the proof of P → R. The indented lines belong to the P subproof.

Your answer
  1. 1. P → Q · premise
  2. 2. Q → R · premise
Read solution · does not award completion
  1. 3. [inside] P → Temporary assumption
  2. 4. [inside] Q → →E, 1 + 3
  3. 5. [inside] R → →E, 2 + 4
  4. 6. [closed] P → R → →I, discharge 3–5

P is assumed only inside the subproof. The final →I exports the conditional, not its consequent alone.

Practice 2Not checked

Show why the permanent premises do not entail R alone.

Your answer
Premises · all must be trueP → QQ → R
Conclusion · must be falseR
Read solution · does not award completion

P = F · Q = F · R = F

P = Q = R = F is a counterexample. The temporary P assumption was essential to obtaining R.

Practice 3Not checked

Inside a subproof, assume P and derive Q. After closing it, what can →I export?

Your answer
Read solution · does not award completion

P → Q

Record both what was shown and under what assumption it was shown.

Bring it back to your own work

Find an estimate that begins 'assuming growth continues'. Separate its conditional conclusion from a claim that growth will continue.

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