9.2 Proving Validity for Complex Arguments
The purpose of this module is to give you practice taking arguments written in English, translating them, and them proving them valid using our existing rules of inference. Make sure you have with you notes with the valid propositional rules of inference and the valid categorical inference forms before you begin this module.
Table of Contents
- 9.2 Proving Validity for Complex Arguments
9.2.1 Propositional Rules: Proofs
A fragment of Euclid’s Elements, an ancient text used to teach proofs.
9.2.2 Propositional Rules: More Proofs
A depiction of Chrysippus, a Stoic logician and philosopher.
Note: Please disregard an error on some browsers in a few questions below, where the answer is marked incorrect, but is identical to the correct answer.
9.2.3 Categorical Rules: Proofs
A depiction of Aristotle teaching, from a medieval Arabic text.
Working with Categorical Syllogisms
We have already studied the rules for Categorical Syllogisms, including how to determine mood and form, and which moods and forms are valid. When adapting arguments in English into categorical syllogisms, there are a few steps to take in order to make sure the rules are easy to use:
- Simplify the language. Replace “every” before a singular term with “all” before a plural term: for instance, replace “every object” with “all objects”. Use only the words “All”, “Some”, “Are”, “no”, and “not”.
- When a predicate doesn’t begin with the word “are”, rephrase it by adding the words “things which” beforehand so that it could. For instance, “Some ducks have rabies” could be rephrased as “Some ducks are things which have rabies”, or “have a cold” would be rephrase “things which have a cold”, so that “Some kids have a cold” could be phrased as “Some kids are things which have a cold”.
- Change the order of the premises if necessary, so that P and M (or M and P) occur in the first premise, and S and M (or M and S) occur in the second premise, with S and P (in that order) in the conclusion line. Although this isn’t necessary to make the argument valid, it is necessary to determine the mood and form of the argument, which we use to figure out whether or not the argument is valid..
Activity: Categorical Syllogisms
First, rewrite each argument using S for the subject of the conclusion, P for the predicate of the conclusion, M for the middle term, and only the words “All” “Some” and “Are”, as well as “no” and “not”, as appropriate.
Then, identify the mood and figure of the argument.
Then, determine whether or not the argument is valid, based on its mood and figure.
9.2.4 Categorical Rules: More Proofs
Hybrid arguments switch back and forth between propositional and categorical rules.
Complex Arguments
Some arguments make use of both propositional rules and the rules which apply to categorical sentences. A more advanced system than we learn in this class, known as first-order logic, combines both into the same system in order to allow us to prove the validity of these arguments, but the rules of this system are more complicated to learn. So instead, for our class, we’ll learn some strategies for switching back and forth between propositional and categorical logic in a series of arguments.
Suppose we want to prove the validity of this argument:
1. If some of Cyla’s friends were asleep or out of town, then everybody at the party was having a good time.
2. Cyla’s friend, Lydia, was out of town and not having a good time.
C. Some of Cyla’s friends were not at the party.
Notice there are propositional operators: if…then, and, or; there are also quantifiers, some and everybody. Intuitively, the argument is valid. How do we prove it is valid?
Make it Explicit
The first step is to make the argument more explicit so that we reveal every bit of logical structure hidden in the argument. Whenever the word “and” or “or” is used, for instance, we need to show the whole atomic sentences on both sides of the “and” or “or”. When the word “and” is implied by a comma, we will add the word “and” back in. Whenever words that aren’t part of our vocabulary are used, we’ll replace them with ones in our logical vocabulary: we’ll replace “Everybody” with “All people” for instance. We’ll make sure to use a noun or noun phrase in the predicate, such as by adding “things that”, or “people.”
1. If some of Cyla’s friends were asleep or some of Cyla’s friends were out of town, then all people at the party were people having a good time.
2. Lydia is Cyla’s friend and Lydia was out of town and also Lydia was not having a good time.
C. Some of Cyla’s friends were not people at the party.
Now, we will need to switch back and forth between propositional and categorical logic to prove the argument valid. Typically it is best to begin with propositional logic first, then move to Categorical logic, and then move to propositional logic again if necessary. We will avoid symbolizing, so that it is easy to switch from one to the other; instead, we’ll continue to underline all of the logically significant words.
Apply Propositional Rules
The first rule we can easily apply is &E, since the word ‘and’ appears on line 2 twice:
1. If some of Cyla’s friends were asleep or some of Cyla’s friends were out of town, then all people at the party were people having a good time.
2. Lydia is Cyla’s friend and Lydia was out of town and also Lydia was not having a good time.
3. Lydia was not having a good time (2 &E)
4. Lydia is Cyla’s friend (2 &E)
5. Lydia was out of town (2 &E)
Now, notice that line 1 is a conditional. If we got the antecedent of that conditional, (“If some of Cyla’s friends were asleep or some of Cyla’s friends were out of town”), then we could get the consequent of the conditional (“All people at the party were people having a good time”) using modus ponens (MP). So, how might we get the antecedent, “Some of Cyla’s friends were asleep or some of Cyla’s friends were out of town”?
Remember Existential Generalization
Well, recall that our rule of Existential Generalization (EG) lets us infer from “x is S and x is P” that “Some S are P”. So, we could conclude from “Lydia is Cyla’s friend and Lydia was out of town” that “Some of Cyla’s friends were out of town”:
6. Lydia is Cyla’s friend and Lydia was out of town (4, 5 &I)
7. Some of Cyla’s friends were out of town (6, EG)
The rule of Disjunction introduction (vI) allows us to conclude that:
8. Some of Cyla’s friends were asleep or some of Cyla’s friends were out of town. (7 vI) So now, reiterating and then using modus ponens:
9. If some of Cyla’s friends were asleep or some of Cyla’s friends were out of town, then all people at the party were people having a good time. (1 Reit.)
10. All people at the party were people having a good time. (8, 9 MP)
Construct a Categorical Syllogism
Line 10 is the major premise of a syllogism. What we would need from line 10 in order to get the conclusion by categorical syllogism? We need to create the minor premise. Again, we’ll use EG:
11. Lydia is Cyla’s friend and Lydia was not having a good time. (3, 4 &I)
12. Some of Cyla’s friends were not people having a good time. (11 EG)
Let’s reiterate to see the form of the syllogism clearly:
13. All people at the party were people having a good time. (10 Reit.)
14. Some of Cyla’s friends were not people having a good time. (12 Reit.)
Now the conclusion follows from this AOO-2 Syllogism:
C. Some of Cyla’s friends were not people at the party. (13, 14 CS)
This is a complex argument, and it required a complex proof. It was possible, however, because we were patient, methodical, and make use of all of the logical structure we had available in propositional logic first before turning it into a Categorical Syllogism.
Submodule 9.2 Quiz
Licenses and Attributions
Key Sources:
- Watson, Jeffrey (2019). Introduction to Logic. Licensed under: (CC BY-SA).
Next Page: 9.3 Filling in the Missing Pieces