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This study aims to introduce a modal extension M4CC of Arieli, Avron, and Zamansky's ideal paraconsistent four-valued logic 4CC as a Gentzen-type sequent calculus and prove the Kripke-completeness and cut-elimination theorems for M4CC. The logic M4CC is also shown to be decidable and embeddable into the normal modal logic S4. Furthermore, a subsystem of M4CC, which has some characteristic properties that do not hold for M4CC, is introduced and the Kripke-completeness and cut-elimination theorems for this subsystem are proved. This subsystem is also shown to be decidable and embeddable into S4.

Heterotrophically grown algae and fungal biomass and their residual materials from an industrial oil extraction process were used as components in marine larval and broodstock diets. Crypthecodinium sp. phospholipid extract and meal, used to enrich rotifers and Artemia nauplii, produced higher levels of DHA and higher DHA/EPA ratios in these zooplankters than Schizochytrium sp. algal whole cell preparation or fish oil-based emulsion. The improved enrichment resulted in enhanced growth of Atlantic halibut larvae, whereas several other marine larvae species (sea bream, European sea bass and striped bass) respond almost equally to all enrichment materials. In addition, a 60% replacement of menhaden oil with algal oil and meal in striped bass broodstock diets resulted in a similar growth increase to that obtained with standard commercial diets. Striped bass broodstock fish diets supplemented with an arachidonic acid (ArA)-rich oil obtained from heterotrophically grown fungi, Mortierella alpina , was shown to have significant benefits on the hatching rate of larvae. These findings demonstrate the potential of single cell heterotrophs as a partial substitute or replacement for fish-based ingredients in aquaculture diets.