The following statements were used to illustrate how to write simple conditional syllogisms: Increasing the mass in a constant volume container increases the density inside of the container. Pumping up your tires increases the density inside of the tire. As stated the argument would be:
P1: If increasing the mass in a constant volume container increases the density inside of the container, then pumping up your tires increases the density inside of the tire.
P2: Increasing the mass in a constant volume container increases the density inside of the container.
C: Pumping up your tires increases the density inside of the tire.
How do we show that the argument is a sound argument that leads to the truthfulness of the conclusion? One way is to support each of the premises.
In order for the argument to be sound all of the premises must be true. If someone questions the truthfulness of any premise there would be an obligation to support that premise. In other words it may be necessary to write an argument for one or more of the premises. The premise then becomes the conclusion of a new argument and an intermediate conclusion (IC) or non-basic premise (NBP) for the original argument. This is one way of building what is known as a complex argument. When writing arguments it is generally best to start with the most basic argument and then only make it complex when necessary. A complex argument for the previous example where an argument is made for the second premise could be:
P1: If increasing the mass in a constant volume container increases the density inside of the container, then pumping up your tires increases the density inside of the tire.
P2a: If the density is directly proportional to the mass and inversely proportional to the volume, then increasing the mass in a constant volume container increases the density inside of the container.
P2b: The density is directly proportional to the mass and inversely proportional to the volume.
IC/NBP (P2): Increasing the mass in a constant volume container increases the density inside of the container.
C: Pumping up your tires increases the density inside of the tire.
It may also be necessary to argue for the first premise. An implicit premise is a premise that is assumed but not written down. In this example it is assumed in P1 (thinking of P1 as an intermediate conclusion) that pumping up your tire does not increase the volume inside of the tire. This would be an example of an implicit premise leading to the intermediate conclusion P1. That assumption may also need to be addressed if the goal is to create a sound argument. Think about one of the small racing tires on a bicycle. If the tire is completely flat the volume does change when more air is added. But if the tire has 75 psi and you add more air, the volume doesn't change enough to measure and can be assumed constant.
By-the-way, there is at least one other implicit premise assumed in P1 of this last argument. Can you identify it? Answer.
Notice that these implicit premises would be used in an argument that treats P1 as an intermediate conclusion. Sometimes premises are intentionally implicit because they are assumed to be common knowledge.
There are no implicit premises for the conclusion of the main argument. P1 and P2 are the only premises necessary to establish the conclusion in the main argument.