Propositional calculus: symbols, connectives, well-formed formulas

A formal language for true/false claims

The propositional calculus is the simplest logic in the course: a formal language for stating and combining claims that are wholly true or wholly false. It has three kinds of symbol:

• Propositional symbols — P, Q, R, S, … each names one atomic claim, e.g. P = "it rained on Tuesday".
• Truth symbols — the reserved constants true and false.
• Logical connectives — ¬ (not), ∧ (and), ∨ (or), → (implies), ≡ (equivalent).

The defining limitation, the thing that later forces us into predicate calculus, is that a propositional symbol is indivisible: P is an opaque token. There is no way to look inside it and talk about which day it rained or to say something general about all rainy days. One symbol, one fixed claim.

Building well-formed formulas

A well-formed formula (WFF), also called a legal sentence, is any expression you can build by these rules:

1. Every propositional symbol and every truth symbol is a WFF.
2. If $s$ is a WFF, so is its negation $\neg s$.
3. If $s_1$ and $s_2$ are WFFs, so are $s_1\wedge s_2$, $s_1\vee s_2$, $s_1\to s_2$, and $s_1\equiv s_2$.

That is the whole grammar, and it is inductive — every legal sentence is a symbol, or a connective applied to smaller legal sentences. So ((P \wedge Q) \to \neg R) is a WFF, while P \wedge \to Q is not (the \to has no left operand).

Well-formedness is purely syntactic: it says the expression was built correctly, not that it is true. P \wedge \neg P is a perfectly legal WFF even though it can never be true. Keep that separation — legal shape now, truth value in the next lesson.

Worked example: parts of an implication

In the implication $P\to Q$:

• $P$ is the antecedent (also called the premise).
• $Q$ is the consequent (also called the conclusion).

Take $P\to Q$ as raining → wet_ground. The antecedent is raining; the consequent is wet_ground. The crucial, exam-relevant fact about $\to$ is its truth definition: $P\to Q$ is false in exactly one case — when $P$ is true and $Q$ is false. In every other row it is true, including both rows where the antecedent $P$ is false (a false premise makes the whole implication vacuously true). We prove that with a truth table next lesson.

In $P\wedge Q$ the parts are the two conjuncts $P$ and $Q$; in $P\vee Q$ they are the two disjuncts. Those names matter the moment we apply inference rules: And-Elimination lets you pull either conjunct out of a true conjunction.