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.