Disjunctive and Hypothetical Syllogisms

Up to now, we have only discussed categorical syllogisms. Syllogisms are called categorical syllogism when the propositions are categorical propositions: propositions that affirm or deny the inclusion of one category from another. But a syllogism may contain other sorts of propositions.

Categorical propositions can be considered as simple propositions: they have a single component which affirms or denies some class relation. In contrast, some propositions are compound statements, containing more than one component.

The first kind of compound proposition is the Disjunctive Proposition. An example would be: "Either you have a gun in your pocket, or you are happy to see me." The disjunctive holds that at least one of the two components are true, allowing for the possibility that both are true.

If we have a disjunction as one premise, and a denial of one of the disjuncts as a second premise, we can validly infer that the other disjunct component is true. Any argument of this form is a valid disjunctive syllogism:

Disjunctive Syllogisms
Recall an example used early on:

P1 Either that's a gun in your pocket, or you're happy to see me P2 You don't have a gun in your pocket(This is implied - implied premises are called "enthymemes") C: You must be happy to see me

This argument is valid, because it eliminates one of the disjuncts. Take a look at this argument:

P1 Either that's a gun in your pocket, or you're happy to see me P2 You have a gun in your pocket C: Therefore you're not happy to see me

This argument is invalid. It bears a superficial similiarty to the above argument, but rather than eliminate one of the disjuncts, it merely affirms one of them. Recall that both parts of a disjunct can be true at the same time. For this reason, affirming one of the disjuncts does not allow us to assume that the other is eliminated!

Some might put forth an argument like this, to rebut the above point:

P1: Either Smith is in New York or Smith is in Paris P2: Smith is in New York C: Therefore, Smith is not in Paris

Notice, however, that the disjunctive doesn't actually play a role in the argument. The argument, in fact, relies upon an unstated enthymeme: "Smith cannot be both in New York and in Paris", which can be stated in disjunctive form as : "Either Smith is not in New York or Smith is not in Paris" If we replace P1 with this statement, then the above disjunctive syllogism follows the proper form and affirms the validity of proper form!

Hypothetical Syllogism
The second kind of compound proposition is a conditional proposition: we can call these statements If/Then statements, where the "If" part is the antecedant and the part following after "Then" is the consequent. A conditional that contains conditional statements exclusively is called a pure hypothetical syllogism:

Example: P1: If you study (antecedent), then you will become a good student (consequent). P2: If you become a good student, then you will go to college Therefore, If you study, then you will go to college

Notice that the first premise and the conclusion have the same antecedent, and the second premise and the conclusion have the same consequent. It should be clear why hypothetical syllogisms provide the clearest example of why syllogisms preserve truth value - for this format also for a set of equivalencies.

It is also possible to mix up these two forms: the disjunctive and the hypothetical. There are two valid and two invalid forms of a mixed hypothetical syllogism. The first valid form is called modus ponens (From the Latin "ponere", "to affirm"), or "affirming the antecedent":

Modus Ponens If P is true, then Q is true P is true Therefore, Q is true

The next form, Affirming the consequent, is invalid:

Affirming the consequent If P then Q Q Therefore, P is true

Why is this form invalid? This argument differs from modus ponens in that its categorical premises affirms the consequent, not the antecedant. As we will see when we discuss Truth tables, there is no inconsistency in holding that P is false and Q is true: we can hold that the propositon "IF p, then Q" to be true, even if "P" is false, which would mean that we could have all true premises and a false conclusion: "If p, then Q" as a statement would be true, "q" would be true, and yet the conclusion, "P" all its own, would be false! - which, if we remember from earlier lessons, is not possible. Affirming the consequent can therefore be made valid, if the term "if" is replaced by the term "If and only If", so that P and Q can only be true when both are true.

The next valid form is called modus tollens (Latin: "To deny"), and it takes the following form:

Modus Tollens If P, then Q Not Q Therefore, not P

Here the syllogism denies the consequent of the conditional premise, and the conclusion denies the antecedant. Make sure not to confuse this form with the next form.

The next form, Denying the antecedent, is invalid:

Denying the antecedent If P, then Q Not P Therefore, not Q

This deductively invalid form differs from modus tollens in that it's categorical premise denies the antecedent rather than the consequent. This makes this form invalid because, while there is no case of all true premises and a false conclusion, the argument leads to a non sequitur. This can be made more clear with an example:

If it is raining, I will have an umbrella It's not raining Therefore, I can't have an umbrella

Such an argument confuses a correlative fact for a causal fact, where not causality has been established. For this reason, it can also be referred to as a vacuous implication. Denying the antecedent is valid if the first premise asserts that there is some necessary connection between the antecedent and the consequent, but using the term: "if and only if" rather than "if".

Those following the Course in Logic 101 should proceed to the next section: Propositional Logic

Those interested in testing their knowledge of syllogisms should try this test:

http://www.humboldt.edu/act/HTML/tests.html

Make sure to review the concepts of classical syllogisms, their components (i.e. major premise, minor premise, etc.) as well as enthymemes before attempting the test! There is also one question on inductive logic.