Nonstandard Logics and Extensions

Computability Logic - Wikipedia (free encyclopedia) article.
en.wikipedia.org/wiki/Computability_logic

Infinitary Logic - Article in the Stanford Encyclopaedia of Philosophy by John L. Bell. Infinitary Logic is a branch of formal logic where finitary formulae are replaced by potentially infinitary mathematical entities.
plato.stanford.edu/entries/logic-infinitary/

Logic System Interrelationships - Shows how a number of representative logics fit together. The interrelationships usually given as something of the form: System X is system Y plus the axiom Z. By John Halleck.
www.cc.utah.edu/~nahaj/logic/structures/

Non Standard Logics - A comprehensive listing of flavours of non-standard logic with brief descriptions and references, compiled by Peter Suber.
www.earlham.edu/~peters/courses/logsys/nonstbib.htm

What are Weak Arithmetics - Notes defining the subject. Available in HTML and PS formats.
www.univ-paris12.fr/lacl/jaf/html/wa.html