Logic by the Enlightened

[gusl]

(i.e. those who seem to share my philosophical convictions)

Paul Taylor - Practical Foundations of Mathematics

What is Mathematics: Goedel's Theorem and Around. Hyper-textbook for students, by Karlis Podnieks

Hintikka - Revolutions in Logic

The Principles of Mathematics Revisited, by Jaakko Hintikka (Author)

p.92 "Gödel's incompleteness result does not touch directly on the most important sense of completeness and incompleteness, namely, descriptive completeness and incompleteness", from here

p. 163
`In this chapter, it will be argued that [the] priviliged role of axiomatic first-order set theory becomes extremely dubious in the light of the insights that the game-theoretical approach has yielded ...'

p. 176
`... axiomatic set theory cannot be true to its own intended interpretation. When we try to construct the liar paradox for axiomatic set theory, the liar turns out to be axiomatic set theory itself.'
from here

Also, unrelated:
Ariel Rubinstein - Economics & Language (supposedly a very non-traditional text)