Fitch Strategy Trainer Natural deduction in the forall x: Calgary system. Learn which assumption to make, and why.

Strategy card Why do these work? →

Three questions before any assumption. (1) What shape is my goal? Its main connective tells you the last rule, and that rule tells you what to assume. (2) Which premise is locked, and what key does it need? (3) Am I stuck? If nothing introduces the goal, assume its negation and hunt for ⊥.

Goal shape → what to assume

Goal	Assume	Aim for	Close
𝒜 → ℬ	𝒜	ℬ	→I
¬𝒜	𝒜 (exactly the sentence under the ¬)	⊥	¬I
an atom, or anything you can't build directly	¬goal	⊥	IP
𝒜 ∨ ℬ	nothing if one disjunct is in hand; otherwise ¬(𝒜 ∨ ℬ)	— / ⊥	∨I / IP
𝒜 ∧ ℬ	nothing; prove each half	𝒜, ℬ	∧I

Premise shape → how to unlock it

Premise	Gives	Needs first
𝒜 → ℬ	ℬ	𝒜. If you lack it but hold ℬ, manufacture it (trick 1).
𝒜 ∨ ℬ	your current goal	two cases, each ending in that goal (∨E)
¬𝒜	⊥	𝒜 itself: make deriving it the sub-goal
𝒜 ∧ ℬ	𝒜, ℬ	nothing

Two tricks

1 · Free conditional. Holding ℬ, you get 𝒜 → ℬ for any 𝒜: assume 𝒜, reiterate ℬ, →I.

2 · Explosion. A clash gives ⊥, and ⊥ gives any sentence (X). This is how a letter nobody mentions appears.

Two mistakes

¬I concludes the negation of what you assumed, nothing else. And an assumption you never cite inside its subproof is a sign the subproof is doing nothing.