We show that a coherent theory of partially ordered connectives can be developed along the same line as partially ordered quantification. We estimate the expressive power of various partially ordered connectives and use methods like Ehrenfeucht games and infinitary logic to get various undefinability results.

Shelah has shown that the number d, the smallest cardinality of a dominating family, is less than or equal to the number i, the smallest cardinality of a maximal independent family on ω. This was done using a downward Löwenheim-Skolem argument. Thus it is interesting to find a direct “elementary” proof. Here we show that this can be done.