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
Go to practice ↓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. P → Q (premise). 2. Q → R (premise). Open: 3. assume P.
- Inside: 4. Q (→E, 1,3). 5. R (→E, 2,4). Close: 6. P → R (→I, 3–5).
- The premises entail P → R, not R by itself. R could be false when P is false.
Your turn
0 / 3Justify the proof of P → R. The indented lines belong to the P subproof.
Read solution · does not award completion
- 3. [inside] P → Temporary assumption
- 4. [inside] Q → →E, 1 + 3
- 5. [inside] R → →E, 2 + 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.
Show why the permanent premises do not entail R alone.
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.
Inside a subproof, assume P and derive Q. After closing it, what can →I export?
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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Keep your own answer before consulting an AI candidate. Check sources, calculations, inference and conditions; explain adoption or rejection. Then solve a new-context task independently and apply the checks to this lesson.
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