Conditional Propositions & Conditional Syllogisms

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Conditional Propositions & Conditional Syllogisms

(1) Conditional Propositions
— “if p, then q”
(2) Mixed Hypothetical Syllogisms
— modus ponens & modus tollens
— sufficient versus necessary conditions
(3) Invalid Mixed Hypothetical Syllogisms
— fallacy of denying the antecedent & fallacy of affirming the consequent
(4) Constructive Use of Conditionals
— making & using conditionals
— unless statements
— only if statements
(5) Negations & Double Negations
— “not q” versus “q”; “q” ≡ “not (not q)”
(6) Pure Hypothetical Syllogisms
— “if p, then q”; but “if q then r”; ergo, “if p, then r”
(7) Another Note on Mathematical Logic

A conditional proposition (sometimes referred to as a hypothetical proposition) such as “if P, then Q” differs from a categorical proposition by not asserting anything absolutely. What it does is identify a sufficient condition for Q being the case. That is, if P is the case, then Q is the case.

In other words, on the condition that P is the case, Q will also be the case. It does not assert, however, that P is in fact the case or that Q is in fact the case.

It is a hypothetical, after all!

For example: “If it is raining, then it is cloudy.”

This proposition tells us nothing about the weather presently. It simply tells us that on the condition that it is raining it will also be cloudy. Raining is sufficient for it to be cloudy.

All conditional propositions have an antecedent and a consequent. We can think of the conditional as composed of two categorical propositions. One appears as the antecedent and the other as a consequent. When we symbolize a categorical proposition, however, here we only use one letter (not three).

The antecedent refers to “it is raining.” Often the antecedent is symbolized as P.
The consequent refers to “it is cloudy.” Often the consequent is symbolized as Q.

Sufficient does not mean necessary! Thus, while raining is sufficient for it to be cloudy, a conditional proposition does not declare that raining is necessary for it to be cloudy. In general, it is entirely left open whether the consequent can be the case without the antecedent being the case. It can be cloudy without it being raining, for example.

Since the antecedent is not necessarily necessary for the consequent, in “if P, then Q”, other things in addition to P (such as R, S, or T) might bring about Q.

Understanding the distinction between sufficient versus necessary conditions additionally allows us to understand why “if it is cloudy, then it is raining,” is a bad proposition. It being cloudy is not sufficient for it to be raining. Just because it is cloudy outside doesn’t mean it is going to rain!

Therefore, notice that “if P, then Q” is not the same thing as “if Q, then P.”

Traditional Logic versus Mathematical Logic

It should be noted, even at this early stage, that Aristotelian-Scholastic logic treats conditional propositions differently than contemporary mathematical logic (e.g., propositional logic). That’s because, in Aristotelian-Scholastic logic, a good conditional proposition has some real causal connection or real relation of dependency between antecedent and consequent.

Mathematical logic, on the contrary, does not presuppose this. All that matters are “truth functional” relations between antecedent and consequent. Here a conditional is only false when there is a true antecedent but a false consequent. All other possibilities are defined as true. For example, “if squares are circles, then one plus one equals two” is said to be a “true” conditional.

This goes against the “common” way to use “if …, then.” First, there’s no direct relation between the antecedent and consequent in that example. In most argumentation, observe, people usually use “if …, then” propositions in a way that presupposes some more substantial connection between antecedent and consequent. Second, the antecedent is false in that conditional, yet the conditional is nevertheless said to be true. Paradoxes flow from this.

An Aristotelian-Scholastic approach fits better with ordinary English “if …, then” statements versus the mathematical construction. To be sure, that construction can be a useful construction; however, the meaning most people give to a “if …, then” statement in normal conversation and argumentation is not identical to that construction. It’s a major error to think it is!

**Sufficient *versus* Necessary Condition**

Mixed Hypothetical Syllogisms

In order to assert something absolutely, we require a categorical proposition. Mixed hypothetical syllogisms contain two premises, where one is a conditional proposition and the other is a categorical proposition.

The major premise contains the conditional proposition and the minor premise contains the categorical proposition.

A valid mixed hypothetical syllogism works on the distinction between sufficient versus necessary conditions.

This is fairly obvious with modus ponens. . .

Premise 1: “If it is raining, then it is cloudy.”
Premise 2: “It is raining.”
Conclusion: “Therefore, it’s cloudy.”

Here we have an example of a “modus ponens” argument. It’s of the general form. . .
If P, then Q. P is the case. Therefore, Q is the case.

As the antecedent is sufficient for the consequent (in premise one), once the antecedent is affirmed (as it is with the second premise), the consequent can then be affirmed (to reach the conclusion).

However, it’s not just that the antecedent is sufficient for the consequent, on closer inspection, the consequent is necessary for the antecedent! This becomes clear with an example.

Here we have modus tollens. . .

Premise 1: “If it is raining, then it is cloudy.”
Premise 2: “It is not cloudy.”
Conclusion: “Therefore, it’s not raining.”

This is an example of a “modus tollens” argument. It’s of the general form. . .
If P, then Q. Q is not the case. Therefore, P is not the case.

As the consequent is necessary for the antecedent (in premise one), once the consequent is denied (as it is with the second premise), the antecedent can then be denied (to reach the conclusion).

This makes perfect sense. After all, if the consequent is not the case, there is no way that the antecedent can be the case. Since, if the antecedent were the case, we would be back at a modus ponens argument.

Invalid Mixed Hypothetical Syllogisms

Four general forms of mixed hypothetical syllogisms are possible. Two of them are invalid. They are invalid precisely because they take what is truly a sufficient condition and erroneously treat it as a necessary condition.

Fallacy of Denying the Antecedent. . .

Premise 1: “If it is raining, then it is cloudy.”
Premise 2: “It is not raining.”
Conclusion: “Therefore, it’s not cloudy.”

Examples often help. It’s clear that something went wrong. Just because it’s not raining doesn’t mean it’s not cloudy. Yet the argument, in general, is formally invalid. It takes the antecedent as necessary for the consequent. That’s wrong! The antecedent is only sufficient; ergo, a false antecedent doesn’t imply a false consequent.

Fallacy of Affirming the Consequent. . .

Premise 1: “If it is raining, then it is cloudy.”
Premise 2: “It is cloudy.”
Conclusion: “Therefore, it’s raining.”

This is also helpful. The argument clearly reaches an invalid conclusion. A cloudy day doesn’t mean it must be raining. Formally the argument is invalid because it confuses sufficient and necessary conditions. A true consequent doesn’t imply a true antecedent.

Constructive Use of Conditionals

Logic is an “art” and a “science.” What good are conditionals if we cannot construct them and use them on our own? But context is everything. No one rule works for all cases.

In general, however, two things should be identified:

(1) a sufficient condition – which is connected to the antecedent
(2) a necessary condition – which is connected to the consequent

Once that’s figured out, the antecedent and consequent to a conditional can be formed. Moreover, a given conditional can be evaluated to determine if it is a good (or bad) conditional.

And there are different ways to express the “if …, then” relation. “If P, then Q” can be expressed as “If P, Q,” “Q, if P,” “P implies Q,” “P entails Q,” “P is a sufficient condition for Q,” “in the event that P, Q follows,” “provided that P, then Q,” “Q is necessary for P,” etc.

Sometimes “unless” statements can be translated as conditionals. For example, “You will be unhappy unless you have a meaningful life.” How should this be interpreted? Having a meaningful life will not lead to an unhappy life but a happy life. So, we can get: “If you do not have a meaningful life, you will be unhappy.” Generally speaking, the “unless” can be translated as “if not.” “Q unless P” is consequently “If not P, then Q.”

Sometimes “only if” statements can be translated as conditionals. It depends, as always, on the context. There are no exact rules. What’s confusing is the wording here. That’s because “only if” can imply something necessary, not sufficient. Thus, “P only if Q” would mean “If P, then Q.” (Q is necessary here; ergo, it appears in the consequent.)

“You are in Chicago only if you are in Illinois.” With this example, it is really clear that it wouldn’t make sense to turn this into “If you are in Illinois, then you are in Chicago.” It is necessary that you are in Illinois when in Chicago. Thus, the correct interpretation is “If you are in Chicago, then you are in Illinois.”

“Your parents will purchase a new car for you only if you pass the exam.” Here it may not be a sure thing that just passing the exam will automatically result in getting a new car. Passing the exam is necessary, however. There may be other conditions that must be met. So, we probably should translate this as “If your parents purchase a new car for you, then you must have passed the exam.”

**Example with Double Negation and Modus Tollens**

Negations and Double Negations

Entire categorical propositions can be negated. This means that an antecedent and/or consequent in a conditional proposition can be negated.

Our intuitions are usually sufficient when dealing with these situations. Still, let’s remember that “Q” and “not Q” are contradictory to each other. They thereby, respectively, deny each other. Knowing this comes in handy with some arguments (especially in the modus tollens form). Furthermore, “Q” and “not (not Q)” are equivalent to each other. To be as explicit as possible, for example, the denial of “not Q” is “not (not Q).” And the latter is equivalent to “Q.”

Pure Hypothetical Syllogisms

Another argument consists of only conditional propositions. The consequent of one conditional is the antecedent to another conditional; ergo, we can infer a new conditional based on this “link.” There’s a (conditional) chain reaction, so to speak.

Premise 1: “If today is Sunday, then Patrick spends time meditating.”
Premise 2: “If Patrick spends time meditating, Patrick reflects upon the Bible.”
Conclusion: “Therefore, if today is Sunday, Patrick reflects upon the Bible.”

Another Note on Mathematical Logic . . .

“P ⊃ Q” and “P → Q” are two common ways to symbolize “If P, then Q.”

(I prefer the arrow symbol.)

In mathematical logic, the atomic proposition P and the atomic proposition Q are related together through the conditional sentential connective (→) to form a molecular proposition P → Q.

As was stated above, it’s defined as being false only when there is a true antecedent and a false consequent. A “truth table” is a pedagogically useful way to document this. P can be true or false. Q can be true or false. Thus, there are two times two combinations possible with P → Q.

An ordinary English claim of “if …, then” would at least agree that a true antecedent but a false consequent must produce a false conditional. It would not agree, however, that that is the only requirement of a good “if …, then” statement. That’s because P and Q should have some connection between each other. Mathematical logic does not have that requirement at all.

From the mathematical construction, any conditional whatsoever with a false antecedent is true. Thus, we can construct pretty outlandish conditional propositions that are said to be “true” in mathematical logic. For example, “if elephants are humans, then Mars is a planet” and “if elephants are humans, then Mars is not a planet” are both said to be true.