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		5-5 M is said to be conditionally consistent just in case for all , if , then . [TxtEx]cond-consistent denotes .

		Refute the following variant of Proposition 5.6: If conditionally consistent M identifies , then .

		5-6 Show that .

		5-7 is called total just in case for all there is such that Note that a total language need not represent a function (since it need not be single valued). M is called total-minded just in case for all , if then WM( s ) is total. [TxtEx]tot-minded denotes . Prove: There is such that (a) every is total, and (b) . Hint: Use Rogers [158], Theorem 5-XVI: the single-valuedness theorem.

		5-8 Prove that if contains only infinite languages, then TxtEx if and only if .

		5-9 Give a proof of Lemma 5.33.

		5-10 Show the following:

		(a) .

		(b) .

		5-11 M is said to be gradualist just in case for all , is finite. [TxtEx]gradualist denotes . Show that [TxtEx]gradualist = TxtEx.

		5-12 M is called cautious just in case for all  s , , is not a proper subset of WM( s ). Note that cautious scientists behave as if they never overgeneralize. Compare this strategy with conservative scientists that do not over generalize on languages they identify, but may over generalize on languages they do not identify. [TxtEx]cautious denotes . Show that .