MATH 457
Introduction to Mathematical Logic
Spring 2016
Dr. Jason Rute
Handout: Natural Deduction Proofs by Example
In proof theory, we would like to prove a wff ϕ from a set of hypotheses Σ. This will be
written Σ ` ϕ. Our proof trees represent a proof of Σ ` ϕ. Before, getting into more details
about the notation Σ ` ϕ, let us give some examples of wffs provable with our proof trees.
1
Examples
1.1
Minimal logic
Minimal logic is the logic which avoids both ⊥E (ex falso) and reductio ad absurdum. The
proofs are quite direct and intuitive.
Proposition 1. The following are all provable in minimal logic.
1. Commutativity of ∧: ϕ ∧ ψ ↔ ψ ∧ ϕ
2. Commutativity of ∨: ϕ ∨ ψ ↔ ψ ∨ ϕ
3. Associativity of ∧: (ϕ ∧ ψ) ∧ θ ↔ ϕ ∧ (ψ ∧ θ)
4. Associativity of ∨: (ϕ ∨ ψ) ∨ θ ↔ ϕ ∨ (ψ ∨ θ)
5. Distributivity of ∧ over ∨: ϕ ∧ (ψ ∨ θ) ↔ (ϕ ∧ ψ) ∨ (ϕ ∧ θ)
6. Distributivity of ∨ over ∧: ϕ ∨ (ψ ∧ θ) ↔ (ϕ ∨ ψ) ∧ (ϕ ∨ θ)
7. (ϕ → (ψ → θ)) ↔ (ϕ ∧ ψ → θ)
8. (ϕ → ψ) → ((ψ → θ) → (ϕ → θ))
9. ((ϕ ∨ ψ) → θ) ↔ (ϕ → θ) ∧ (ψ → θ)
10. ¬(ϕ ∨ ψ) ↔ ¬ϕ ∧ ¬ψ
11. ¬(ϕ ∧ ¬ϕ)
12. ¬(ϕ → ψ) → ¬ψ
The best way to get a feel for natural deduction is to work through as many proofs as
you can.
Exercise 2. Prove some results from the above list (avoiding ex falso and reductio ad
absurdum) until you get a feel for the proofs.
1
Example 3. This is a proof of (ϕ ∧ (ψ ∨ σ)) → ((ϕ ∧ ψ)∨(ϕ ∧ σ)).
y
ϕ ∧ (ψ ∨ σ)
ψ∨σ
y
y
ϕ ∧ (ψ ∨ σ)
ϕ ∧ (ψ ∨ σ)
x
ψ
ϕ
ϕ
σ x
ϕ∧ψ
ϕ∧σ
(ϕ ∧ ψ) ∨ (ϕ ∧ σ)
(ϕ ∧ ψ)∨(ϕ ∧ σ)
(ϕ ∧ ψ) ∨ (ϕ ∧ σ)
y
(ϕ ∧ (ψ ∨ σ)) → ((ϕ ∧ ψ) ∨ (ϕ ∧ σ))
Example 4. This is a proof of ¬(ϕ ∨ ψ) → ¬ϕ ∧ ¬ψ.
y
¬(ϕ ∨ ψ)
ϕ x
ϕ∨ψ
z
⊥
¬ϕ
¬(ϕ ∨ ψ)
z
⊥
¬ψ
x
ψ
ϕ∨ψ
y
¬ϕ ∧ ¬ψ
z
¬(ϕ ∨ ψ) → ¬ϕ ∧ ¬ψ
Remark 5. Remark, you may have noticed that ⊥ in minimal logic behaves the same as a
sentence symbol A. (This is assuming we interpret ¬ϕ as ϕ → ⊥.)
1.2
Intuitionistic logic
Intuitionistic logic is minimal logic plus ⊥E (ex falso) (but not and reductio ad absurdum).
Proposition 6. The following are all provable in intuitionistic logic.
1. ⊥ ↔ ϕ ∧ ¬ϕ
2. ¬ϕ → (ϕ → ψ)
3. (¬ϕ ∨ ψ) → (ϕ → ψ)
4. ϕ ∨ ψ → (¬ϕ → ψ)
5. ¬(ϕ → ψ) → ¬¬ϕ
6. ¬¬(ϕ → ψ) ↔ (¬¬ϕ → ¬¬ψ)
7. ϕ ∨ ⊥ ↔ ϕ
8. ϕ ∧ ⊥ → ⊥
Exercise 7. Prove some results from the above list (avoiding reductio ad absurdum) until
you get a feel for the proofs.
Example 8. This is a proof of ϕ ∨ ψ → (¬ϕ → ψ).
¬ϕ
ϕ∨ψ
z
y
⊥
ψ
ϕ
x
⊥E
ψ
y
¬ϕ → ψ
ϕ ∨ ψ → (¬ϕ → ψ)
2
ψ
z
x
x
1.3
Classical logic
Classical logic is intuisionistic logic plus reductio ad absurdum.
Proposition 9. The following are all provable in classical logic.
1. Law of excluded middle: ϕ ∨ ¬ϕ
2. Double negation elimination: ϕ ↔ ¬¬ϕ
3. ϕ ∨ ψ ↔ ¬(¬ϕ ∧ ¬ψ)
4. ϕ ∧ ψ ↔ ¬(¬ϕ ∧ ¬ψ)
5. (ϕ → ψ) ↔ (¬ϕ ∨ ψ)
6. deMorgan’s law: ¬(ϕ ∨ ψ) ↔ ¬ϕ ∧ ¬ψ
7. deMorgan’s law: ¬(ϕ ∧ ψ) ↔ ¬ϕ ∧ ¬ψ
8. ¬(ϕ → ψ) ↔ (ϕ ∧ ¬ψ)
9. (ϕ → ψ) ↔ (¬ψ → ¬ϕ)
10. (ϕ → θ ∨ η) → ((ϕ → θ) ∨ (ϕ → η))
11. (((ϕ → ψ) → ϕ) → ϕ)
Exercise 10. Prove some results from the above list until you get a feel for the proofs.
Example 11. This is a proof of ϕ ∨ ¬ϕ.
¬(ϕ ∨ ¬ϕ)
¬(ϕ ∨ ¬ϕ)
ϕ x
ϕ ∨ ¬ϕ
y
⊥
¬ϕ x
ϕ ∨ ¬ϕ
y
⊥
ϕ ∨ ¬ϕ
y RAA
Example 12. This is a proof of ¬¬ϕ → ϕ.
¬¬ϕ
y
¬ϕ
x
⊥ x RAA
ϕ
y
¬¬ϕ → ϕ
Example 13. This is a proof of ¬(ϕ ∧ ¬ψ) → (ϕ → ψ).
¬(ϕ ∧ ¬ψ)
ϕ
z
y
¬ψ
ϕ ∧ ¬ψ
⊥ x RAA
ψ
y
ϕ→ψ
¬(ϕ ∧ ¬ψ) → (ϕ → ψ)
3
z
x
2
Formal proofs
Informally, given a set Σ of wffs and a wff ϕ, say that ϕ is provable from Σ (or ϕ is deducible
from Σ or ϕ is a theorem of Σ), written Σ ` ϕ, iff there is a proof tree proving ϕ from
hypotheses (the uncovered leaves of the proof tree) in Σ.
Remark 14. It is ok if there are elements of Σ not used as hypotheses in the proof tree for
Σ ` ϕ.
Remark 15. Just as with , we have the following conventions.
• We write ` ϕ in place of ∅ ` ϕ.
• We write ψ, θ ` ϕ in place of {ϕ, θ} ` ϕ.
• We write Σ; θ ` ϕ or Σ, θ ` ϕ to mean Σ ∪ {θ} ` ϕ.
However, to clearly prove results about the provability relation `, it is necessary to give
an inductive definition of Σ ` ϕ. For example, if (we have a proof tree for) Σ; ϕ ` ψ, then by
the implication introduction rule, (we have a proof tree for) Σ ` ϕ → ψ (notice ϕ is removed
from the hypotheses, which corresponds to canceling ϕ in the proof tree). We can write this
succinctly as
Σ; ϕ ` ψ
→I.
Σ`ϕ→ψ
4