Logically Valid Inference

A logically valid inference preserves truth. If the premises are true,
the conclusion must be true too.

The classic example is modus ponens:

Other inference rules:
- Modus tollens: from $P \Rightarrow Q$ and $\neg Q$, conclude $\neg P$.
- Hypothetical syllogism: from $P \Rightarrow Q$ and $Q \Rightarrow R$, conclude $P \Rightarrow R$.

A syllogism is a logical argument with two premises and a conclusion. The
Greek tradition (Aristotle) formalized many of these — Barbara, Celarent, etc.

In modern logic, syllogisms are special cases of predicate logic with
universal statements.

Multiple representations are possible: unary $s(s(o))$, binary $10$, decimal $3$, Roman III — all the same number.

Before we had positional decimal (Hindu-Arabic) numbers, cultures represented quantities with one mark per unit — the unary representation. Tallies on bones, sticks, and cave walls going back ~8,000-10,000 years (the oldest known tally is the Ishango bone, ~20,000 BCE) used this idea. Even today we use unary in music ('four beats' = ♩ ♩ ♩ ♩), in protocol headers (e.g. a run-length encoding prefix), and in formal logic via the successor function $s(n)$. Unary is theoretically simplest because there is exactly one rule: 'count another unit'.