Logic
Propositional Logic
Propositions (truth variables): p, q, nice, etc. Operators: and (&, ×, .), or (∨, +), not (¬), => (implies, ⇒), <= (if, implied by, ⇐), <=> (iff, if and only if, ⇔)
 e.g., p and q => p.
Predicate Logic

Predicates (truth functions): p(), q(), odd(), parent(,), etc. Operators: and (&, ×, .), or (∨, +), not (¬), =>, <=, <=> Quantifiers: ∀ (forall), ∃ (there exists) Logical variables: X, Y, Child, Parent, etc. Constants: 123, charles, etc. Function names: f(), hcf(,), etc.
 e.g., ∀ C, P, GP, parent(C,P) and parent(P,GP) => grandParent(C,GP).
[Prolog]
 Prolog (Programming in Logic) is a programming language based on Predicate Logic; its restricted syntax is that of Horn Clauses:

facts, e.g., parent(charles, elizabeth).
odd(1).rules, e.g., grandParent(C,GP) <= parent(C,P) and parent(P,GP).
odd(s(s(N))) <= odd(N).queries, e.g., ?grandparent(henry, GP).  Each rule has exactly one predicate on its LHS, and a conjunction (and) of predicates and/or negated predicates on its RHS. A query is to be answered (yes or no) given a list of facts and rules which are to be taken as true.
 [Prolog interpreter]
 Example:
witch(X) <= burns(X) and female(X). burns(X) <= wooden(X). wooden(X) <= floats(X). wooden(woodBridge). stone(stoneBridge). floats(bread). floats(apple). floats(cherry). floats(X) <= sameweight(duck, X). female(girl). {by observation} sameweight(duck,girl). {by experiment } ? witch(girl). { After Monty Python (Sir Bedevere). }
 Commonly,
 ':' is used for '<=', and
 ',' is used for 'and' in queries and in the RHS of rules.
 All of the variables in a fact, or in a rule, are universally (∀) quantified, and all of the variables in a query are existentially (∃) quantified, so the quantifiers are taken as givens and are omitted.