One can prove X1 is not regular using the pumping lemma. One can prove X2 is not regular either by using the pumping lemma (a version with bounds on the prefix) or by contradiction: if X2 is regular than X2 ∩ a∗b∗ = X1 is regular. One cannot prove X3 non-regular with the pumping theorem directly; however one can prove its regular by contradiction: if X3 is regular than X3 = X2 is regular. We wi...

The problems of the meaning and function of negation are disentangled from ontological issues with which they have been long entangled. TIle question of the function of negation is the crucial issue separating relevant and paraconsistent logics from classical theories. The function is illuminated by considering the inferential role of contradictions, contradiction being parasitic on negation. T...

After observing a contradiction between the 'official' ideology defined by open-source licenses and the actual behavior of hackers, we examine the actual customs which regulate the ownership and control of open-source software. We discover that they imply an underlying theory of property rights homologous to the Lockean theory of land tenure. We relate that to an analysis of the hacker culture ...