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CONTENTS0. Preliminaries....................................................................... 71. Adding propositional connectives to $L_{\omega }{1}_{\omega }$............... 82. The propositional part of $L_{\omega }{1}_{\omega }$ (S)............................. 103. The operation S and the Boolean algebra $B_S$............... 114. General model-theoretic properties of $L_{\omega }{1}_{\omega }$(S)...... 175. Hanf number computations...................................................... 226. Negative results for $L_{\omega }{1}_{\omega }$(S)........................................ 277. Proposition...