ronri.org

A naturalist's journal of logic,
found between pressed ferns
and propositional proofs.

March, in the year of the bracket fungus

Specimen No. 01

Modus Ponens

If the rain falls upon the forest floor, the mycelium awakens. The rain has fallen. Therefore, the mycelium awakens.

P → Q
P
∴ Q

Found scratched into birch bark near the eastern ridge. The philosopher's handwriting is unmistakable.

Formal Proof

1. P → Q    [premise]
2. P          [premise]
3. Q          [1,2 MP]

The simplest rule of inference, yet the mycelium network has known it for millennia.

Specimen No. 02

Sets & Forests

Let F be the set of all forest organisms. Let M be the set of all mycorrhizal fungi. Then M is a proper subset of F, yet M connects every element of F through underground networks no Venn diagram can contain.

F M

Pressed fern specimen attached. The spore pattern resembles a membership relation.

Set Notation

F = {x | x is a forest organism}
M = {x | x is mycorrhizal}
M ⊂ F
∀y ∈ F, ∃m ∈ M : connected(y, m)

The Wood Wide Web predates our formalisms by 450 million years.

Specimen No. 03

Truth Tables

The philosopher records truth values not as T and F, but as mushroom spore prints: dark for true, faint for false. The conjunction table becomes a field guide to certainty.

P Q P ∧ Q

The dark prints are Agaricus bisporus. The faint ones, barely visible, are Coprinus comatus at dawn.

Conjunction

P ∧ Q is true
iff both P and Q are true.

¬(P ∧ Q) ≡ (¬P) ∨ (¬Q)
— De Morgan's Law

Augustus De Morgan never walked these woods, but his laws govern every branching path.

Specimen No. 04

Contradiction

A thing cannot both be and not be, Aristotle declared. Yet the forest floor is thick with paradox: the dead log that teems with life, the darkness beneath the canopy that feeds the light above.

¬(P ∧ ¬P)

From contradiction, anything follows. Ex falso quodlibet. The philosopher's marginalia here grows frantic: "But what if the mushroom is the contradiction itself?"

Page stained with what appears to be chanterelle juice. The ink has bled into the proof.

Explosion Principle

P, ¬P ⊢ Q

From a contradiction,
any proposition can be derived.

The forest does not obey
classical logic.
Specimen No. 05

The Syllogism

All organisms that decompose matter are essential to the forest. All fungi decompose matter. Therefore, all fungi are essential to the forest.

∀x(D(x) → E(x))
∀x(F(x) → D(x))
∴ ∀x(F(x) → E(x))

Barbara, the first mood of the first figure. Named not for a person but for a pattern. Like the spiral of a snail shell.

Barbara (AAA-1)

Major: All M are P
Minor: All S are M
Concl: All S are P

Valid in all possible forests.

Specimen Wall

Click any specimen to turn it over

No. 06

Negation

¬P

The absence of a mushroom is itself a datum.

NOT Gate

What the forest floor lacks tells us as much as what it holds.

No. 07

Disjunction

P ∨ Q

The path forks. At least one branch leads somewhere.

OR Gate

Shaped like a split branch, because it is.

No. 08

Implication

P → Q

If the acorn falls, the oak follows. Given centuries.

Material Conditional

False only when the antecedent is true and the consequent is false. Nature rarely lies.

No. 09

Biconditional

P ↔ Q

The lichen and the tree. Neither exists alone.

If and Only If

Symbiosis is the biconditional of ecology.

No. 10

XOR

P ⊕ Q

Either the mushroom is edible, or it is not. Never both.

Exclusive Or

The most decisive of all connectives. Life or death in the identification guide.

No. 11

Universal

∀x P(x)

For all trees in this forest, roots reach downward.

For All

The universal quantifier dreams of completeness. The forest resists.

No. 12

Existential

∃x P(x)

There exists a clearing where logic and moss agree.

There Exists

The existential quantifier is satisfied by a single witness. One mushroom is enough.

No. 13

Tautology

P ∨ ¬P

Always true. Like gravity. Like decay.

Law of Excluded Middle

The intuitionists reject this. The mushrooms are silent on the matter.

No. 14

Modus Tollens

¬Q → ¬P

No fruiting body means no mycelium here. Or does it?

Denying the Consequent

The contrapositive is the shadow of the conditional, equally valid, equally true.

ronri.org

A journal of logic found in the forest.
The proofs are sound. The mushrooms, uncertain.

— ronri (論理) —