Part II: Examples

I. Hypothetical Compound Propositions

Now, let’s consider a hypothetical compound proposition: “If frogs are amphibians, then they are animals”. 

The hypothetical is true because amphibians are a type of animal, so a frog cannot be an amphibian without being an animal. The truth of the antecedent implies the truth of the consequent. 

Now let’s consider the alternative example “If the animal is an amphibian, then it is a frog”. 

This hypothetical is false, as there are amphibians that are not frogs, so the antecedent may be true without the consequent having to be true. 

Finally, let’s take the case “If scorpions are insects, then frogs are amphibians”. In this case the hypothetical is true despite the fact that the antecedent is false (Scorpions are arachnids not insects). The truth of this proposition lies in the fact that frogs are in fact amphibians, so the hypothetical proposition would be true if scorpions turned out to be insects or not. Specifically, if scorpions were in actuality insects, then this hypothetical proposition could not be true without the proposition “Frogs are amphibians” being true, and we know the latter to be true.

II. Extra Challenge: Complex Compound Propositions

Finally, it should be understood that, as compound propositions are formed through the combination of two simpler propositions, they themselves can be used to form more complex compound propositions. Some examples from natural language could be: “If it is raining or snowing, then my car is covered in water” and “I have a dog and I have a cat, or I have a hamster.” The first sentence is a disjunctive proposition combined with a hypothetical proposition, and the second is a conjunctive proposition combined with a disjunctive proposition. While this will not be rigorously tested in this logic course, we can also think of these complex propositions in terms of more complex tables:

How ought we read this? This table is helpful, because one only needs to look at two columns at a time in order to fill it out. In order to fill out the blue column below, we only need to look at the green columns. That is, in order to evaluate whether ‘p or q’ is true, we only need to look at whether p is true and whether q is true.

Finally, in order to fill out the final column on the right, we only need to look at the green columns below. That is, if we wish to figure out if ‘p or q, and r’ is true, then we simply need to look at whether ‘p or q’ is true and whether r is true

We can make compound propositions infinitely large, and therefore we can (theoretically) make tables infinitely large. However, this course in logic will not deal with such large complex propositions.

Concrete Examples 

Some examples from natural language of these types of propositions could be: “If it is raining or snowing, then my car is covered in water” and “I have a dog and a cat, or I have a hamster”. Within both of these, as is commonly done in natural language, compound propositions are abbreviated into what appears to be a single simple proposition. From our examples above, “It is raining or snowing” really represents the disjunctive proposition “It is raining or it is snowing”.

 In order to understand the conditions for the truth of these types of statements, one must go to the simplest compound proposition, from which the complex proposition is composed, and determine its truth or falsity. Once this has been done, one should continue onto the next simplest compound proposition and determine the same; in continuing this process of stepping up to the next level of complexity eventually the truth or falsity of the whole proposition should be revealed. This process was exemplified above. From our second example in this section, “I have a dog and a cat, or I have a hamster”, let’s assume I have only a dog, then the compound proposition “I have a dog and I have a cat” will be false based on the truth conditions for conjunction discussed above. Next, let’s assume I do have a hamster, then the proposition “I have a dog and a cat, or I have a hamster” becomes a simple disjunction with one of its disjuncts being false, and the other being true. As we know from our discussion of disjunction, this will mean this complex compound proposition is true. (We take the disjunction to be inclusive here).