Syllogistic Logic is a form of logic where conclusions are inferred from two premises. This is easier to understand with examples. I will have to note that this type of logic is rather narrow in scope and archaic. It can form logical arguments but other forms of more modern logic often supersede this. Still it is good to have a familiarity with it.
Premise: All dogs are carnivorous
Premise: Pugs are dogs
Conclusion: Therefore all pugs are carnivorous
Premise: Fish need water to breathe
Premise: Fish use gills for breathing
Conclusion: Gills use water for breathing