Final answer:
The Law of Syllogism is utilized to conclude that winning the division leads the Patriots to playoffs by relating the outcomes of winning the next game.
Step-by-step explanation:
The logical law used to draw the conclusion that If the Patriots win their division, then they will go the playoffs is the Law of Syllogism. This law allows us to draw a conclusion from two related conditional statements when the conclusion of the first is the hypothesis of the second. In the given situation, the first statement is 'If the Patriots win their next game, then they will win their division' (P > Q), and the second statement is 'If the Patriots win their next game, then they will go to the playoffs' (P > R). From these, we can infer the conclusion 'If the Patriots win their division, then they will go to the playoffs' (Q > R).