## Rules of Inference

Having a robust set of connectives allows us to express complex relationships, but the real power comes from the ability to draw new conclusions based on known relationships.

p,q := “given p and q.”

p∴q := “p, therefore q.”

**Modus Ponens**

The most basic rule of inference states that “given p and that p implies q, we can conclude q.”

p,p→q∴q

**New Symbols Redundant**

The statements “p, therefore q” and “p implies q” are synonymous in our natural languages. Classical predicate logic is too strong, requiring a new symbolism for the weakened version used in deduction.

With intuitionistic logic, the logic itself is weakened already. This allows us to express modus ponens as follows:

(p∧(p→q))→q

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