if and iff

Logical if (→) and iff (↔) are doing my head in.

→ is if, the "conditional connective". A → B means that A can be true only if B is true.

So → is false when A is true despite B being false. But otherwise it's true. Because we don't care about those scenarios. I don't know why. I suppose otherwise it would just be ∧ . I'm going to have to just accept this like the dot product, which also doesn't fit into my mathematical worldview, which is based on cakes.

↔ is iff, the "biconditional connective". A ↔ B is equivalent to [(A → B) ∧ (B → A)].

See? It's bi-conditional. So ↔ is false when A → B ∧ (B → A), or B → A ∧ (A → B) . Possibly I can come to terms with → as half of ↔ .

Next: the Axiom Schema Of Separation and how you can't model that using cakes either.

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