A logical argument is a claim that a set of premises support a conclusion. It is possible for a logical argument to have one or many premises, but there must be one conclusion. In this section we will look at types of arguments and how to determine the strength, validity and/or soundness of each type. There are two types of arguments we will explore in this section: inductive and deductive arguments.
When I went to the store last week, I forgot my wallet, and I forgot it again when I went back today. I always forget my wallet when I go to the store.
Before we analyze an argument, it is helpful to precisely state its premises and its conclusion. Most arguments you encounter in the real world won’t be stated in a precise “premise, premise, conclusion” form. Sometimes the conclusion will be stated before the premises, or the premises will be hidden within a bunch of rhetoric.
Notice that both premises make a claim about a specific instance – the specific instance last week when I forgot my wallet, and the specific instance today when I forgot my wallet. The conclusion, on the other hand, states what we can expect to happen more generally.
Unlike the first argument where the premises were specific and the conclusion was general, this argument’s first premise is a general statement and the conclusion is specific. We can determine whether an argument is inductive or deductive by looking at which part of the argument is general and which is specific. In the first example, the premises were specific and the conclusion was more general. This is an example of an inductive argument. In the second example, it was the premises that were more general and the conclusion that was specific. This is an example of a deductive argument.
In general, an inductive argument uses a collection of specific examples (i.e. data) as its premises and uses them to propose a general conclusion. This is shown in the image on the left below, where the argument moves from the narrow specific premises at the bottom up to the wide general conclusion at the top. A deductive argument uses a collection of general statements (i.e. definitions) as its premises and uses them to propose a specific conclusion. This is shown in the image to the right below, where the argument moves from the wide general premises at the bottom to the narrow specific conclusion at the top.
Juan’s dog Goober is having puppies. All three of Goober’s previous litters have had 5 puppies so Goober is bound to have 5 puppies in this litter as well.
Since the premises are general definitions and properties of numbers and the conclusion is a specific statement about the number 13, the argument is deductive.
This is an example of an inductive argument since it uses specific experiences/instances as its premises, and its conclusion is a general expectation based on those specific experiences.
A strong inductive argument is one that is well supported by its premises, while a weak inductive argument is one whose premises do a poor job of supporting the conclusion. The strength of an inductive argument is subjective, because where one person sees a strong argument, another may see a weak argument. Additionally, the strength and truth of an argument are not necessarily related; it is possible to have a weak argument that is true, and a strong argument that is false.
James Franco, Jodie Foster, Jennifer Lawrence, and Jack Nicholson have all won Academy Awards for acting. Actors whose names start with J are bound to win an Academy Award.
The inductive argument provides a number of specific cases as evidence for the conclusion. However, we would not be surprised if a J-named actor did not win an Academy Award, so the argument is weak.
Deductive arguments, on the other hand, can be proven and their validity and soundness can be evaluated. The validity of the argument is based on whether the conclusion follows logically from the premises, while the soundness of the argument is based on whether or not the premises are true. An argument cannot be sound if it is not valid, even if the premises seem reasonable.
Subsection1.8.4Evaluating Deductive Arguments Using Sets
One way to determine whether a deductive argument is valid is to illustrate the premises of the argument using sets and see if the conclusion logically follows if we assume the premises to be true.
First let’s write the argument in its “premise, premise, conclusion” form. For the problems we will be looking at, you will want to write the first premise as a qualified proposition (some, none, all) since this will form the basic structure of our diagram.
From the first premise we know that all cats lie inside the set of mammals (cats are a subset of mammals). From the second premise, we know that tigers lie inside the set of cats (marked with an X), and therefore also lie within the set of mammals.
This argument is valid because we were able to show that the conclusion follows logically from the premises. The argument is also sound since the premises “all cats are mammals” and “a tiger is a cat” are true.
From the first premise we know that all water bottles lie inside the set of plastic items (water bottles are a subset of plastic). From the second premise, we know that this particular water bottle must lie within the plastic items set.
This argument is valid because we were able to show that the conclusion follows logically from the premises. But the argument is not sound because the premise that all water bottles are plastic is not true. There are many versions of glass and metal bottles that are evidence that the first premise is not true. This argument is valid but not sound.
From the first premise we know that all firefighters lie inside the set of those who know CPR (firefighters are a subset of people who know CPR). From the second premise, we know that Jill is a member of the set of those who know CPR, but we do not have enough information to know whether she is also a member of set of firefighters.
Since we cannot determine which group Jill must be a part of, the argument is invalid. The statement that Jill is a firefighter does not follow logically from the premises that “all firefighters know CPR” and that “Jill knows CPR”. Since the argument is not valid, it cannot be sound.
Because it said “none” we draw disjoint sets – one set for my friends and a second set for people who like to dance. The second premise tells us that Kai doesn’t like to dance so they’re not in the set of people who like to dance. However, we can’t put Kai in the set of my friends either. They could be my friend, or someone I don’t know who happens to not like dancing. Therefore, the conclusion is not valid. And therefore, the argument is also not sound.
Because it said “some” we draw overlapping sets. The second premise tells us to put Tara in the set of young adults, but it doesn’t tell us if she makes minimum wage or not. So, like the previous example we cannot determine which region she is in. She could make minimum wage, or she could also make more. Therefore, the conclusion is not valid and therefore, not sound.
Rewrite each of the following arguments in their “premise, premise, conclusion” form, and determine whether the argument is inductive or deductive. If the argument is inductive, determine its strength. If the argument is deductive, use sets to illustrate and determine the validity of the argument, and state whether the argument is valid and whether it is sound.
Kiran collected data on the salaries of their friends. They found that female and nonbinary friends made less than male friends, so they concluded that women and nonbinary people make less than men.
Over the course of a year, data was collected on the number of students visiting the cafeteria. On average, there were 15-35 students present in the cafeteria during the peak hours of the data. We can expect there to be between 15 and 35 students in the cafeteria if we go during the peak hours of the day.
For each of the following, draw the appropriate illustration of sets (Subset, Disjoint or Overlapping). Then put an X to represent the subject of the conclusion or put two question marks to illustrate the subject could into two locations. Finally, state if the argument is valid and whether it is sound.
