# Implication (Propositional Logic)

I am studying propositional logic and I am having real difficulty understanding implication (despite reading numerous examples). The book I am following has the following examples and answers, but no explanation:

Let JoelHappy stand for "Joel is happy" and AnnaHappy stand for "Anna is happy", determine whether JoelHappy => AnnaHappy or AnnaHappy => JoelHappy for the following statements:

1. "Joel is happy whenever Anna is happy."

2. "Joel is happy only if Anna is happy."

3. "Joel is happy unless Anna is not happy."

1. AnnaHappy => JoelHappy

2. JoelHappy => AnnaHappy

3. AnnaHappy => JoelHappy

I have tried using truth tables to determine these answers, but cannot get the correct answer. English is not my first language, so I am not sure if I am not understanding the sentences correctly.

If someone could explain a simple method for determining implication sentences using these examples, I would be very thankful.

• See S.C.Kleene, Mathematical logic, Dover (1967), page 63: $A \to B$ symbolize: "if A, then B", "A only if B", "B when A". Dec 31 '17 at 18:48
• And "A unless not B" is $A \lor \lnot B$, i.e. $B \to A$. Dec 31 '17 at 18:51

In essence, implication simply means that if one statement is true, then another must be true as well. For example take $A\Rightarrow B$. This simply means that if $A$ is true, then $B$ must also be true. An example with numbers would be, $$x<0 \Rightarrow x<10$$ Here if $x$ is a number that is less than $0$, then it is certainly less than any number greater than $0$ so we say $x<0$ implies $x<10$ or whenever $x<0$ is true, then $x<10$ must also be true. The best way to build an intuition for this is to do as many examples as you can, preferably simple mathematical ones such as the above.

For one of your examples, the statement that "Joel is happy whenever Anna is happy." is simply a different way to say that if Anna is happy, then Joel must also be happy. Notice how this sentence does not say what occurs when Anna is not happy. Then Joel can be either happy or sad. Implication in a sense was defined to only "care" about the case when the statement on the left of the $\Rightarrow$ is true. That is then, that the statement on the right side must also be true.

A different way to think about this is to consider some special circuit that contains two lights, one on the left and one on the right. Whenever the light on the left is turned on, then the light on the right is turned on as well. If you take the light as being turned on to mean "this statement is true", then it is similar to the logical meaning of implication.

Returning to the first example statement "Joel is happy whenever Anna is happy", you can consider Anna to be that leftmost light. Whenever it is true that Anna is happy i.e. the light is turned on, then the light on the right must be turned on as well i.e. it must also be true that Joel is happy.

Don't feel too intimidated by the wording of the propositions, the language that is used to phrase them is not as important as what happens to one statement when the other one is true. For your second example "Joel is happy only if Anna is happy", if you take "Joel is happy" as being the leftmost light, then in the way we created our little circuit, the left light can only be on at the same time as the right light is on, i.e. the leftmost light cannot be turned on and the right one turned off. I hope this intuition is helpful.

• To add to this, it helps to consider what happens to the other statement when each of the statements is true; i.e. if you find that the other statement is always true whenever you consider a statement to be true, then you can conclude that one implies the other. Dec 31 '17 at 19:42

Truth-tables won't help you; you need to get the right translation in the first place.

One important distinction that often helps with these translations is to distinguish between sufficient conditions and necessary conditions. Here is an example to illustrate the difference: to be a bachelor, you need to be unmarried. So: Bachelor => Unmarried. Being unmarried is thus a necessary condition for being a bachelor. But is it sufficient? Is anyone who is unmarried a bachelor? Clearly not: unmarried females are not bachelors. So, we don't have Unmarried => Bachelor. We can say, however, that being a bachelor is a sufficient condition for being unmarried. That is: once we know someone is a bachelor, then we know the person is unmarried. So, once again we have Bachelor => Unmarried.

Contrapositives can also often help you with making sense of these conditionals. For example, note that anyone who is married is not a bachelor. So, we have NOT Unmarried => NOT Bachelor. But the contrapositive of that is: Bachelor => Unmarried.

A final and simple trick is specific to the 'unless': read the 'unless' as 'if not'!

Now, let's apply these strategies to your statements:

The first statement, which states that Joel is happy whenever Anna is happy, is a good example of a sufficient condition. Apparently Anna being happy is sufficient for Joel to be happy. Or, in logic in terms: if we know that Anna is happy, then we immediately know that Joel is happy: nothing else needs to happen for Joel to be happy. So, this translates to AnnaHappy => JoelHappy

The second statement, which says that Joel is happy only if Anna is happy, is a good example of a necessary condition: Anna being happy is a necessary condition for Joel to be happy. Thus, Anna being happy may not be by itself sufficient for Joel to be happy. So, this time we can't say AnnaHappy => JoelHappy. We can, however, say JoelHappy => AnnaHappy: when we see Joel being happy, we can infer that Anna must be happy as well, because the only way for Joel to be happy is for Anna to be happy.

We can also use the contrapositive here: if Anna is not happy, then Joel will definitely not be happy either, and so we have NOT AnnaHappy => NOT JoelHappy, which by contraposition is the same as JoelHappy => AnnaHappy

Finally, for the third sentence 'Joel is happy unless Anna is not happy', we use the trick as indicated, and re[phrase this as 'Joel is happy if not Anna is not happy' ... which works out to: NOT NOT AnnaHappy > JoelHappy, and thus to AnnaHappy > JoelHappy

• For the reasons I give, "whenever" propositions are a textbook example of English sentences which it is perhaps tempting to treat as singular conditionals but which aren't. Dec 31 '17 at 22:51

I'll offer a different perspective. I like to think of this in terms of set theory. $S_A\subset S_B$ means
$A\Rightarrow B$. Here a set $S_A$ is the part of the universe where statement $A$ is true.

1. $A$ whenever $B$ means that $S_B\subset S_A$.
2. $A$ only if $B$ means that $S_A\subset S_B$.
3. $A$ unless $~B$ means that $S_A^c\subset S_B^c$ and thus $S_B\subset S_A$.

So for 1, the times and places that Anna is happy are a subset of the times and places that Joel is happy, therefore we know that AnnaHappy $\Rightarrow$ JoelHappy.

For 2, Joel is only going to be happy when Anna is happy but Joel may also be not happy when Anna is happy, therefore JoelHappy $\Rightarrow$ AnnaHappy.

For 3, the times and places where Joel is not happy are a subset of the times and places where Anna is not happy, therefore AnnaHappy $\Rightarrow$ JoelHappy.

• For 3: take AnnaHappy = False and JoelHappy = True. This is a true statement But Anna is clearly not Happy, but Joel is happy. This contradicts the given statement. Also take AnnaHappy = True and JoelHappy = false. This seems to sound true with the given statement. But Something True $\implies$ something false means that the whole statement is false. Oct 17 '19 at 12:50
• @JoshuaRMS: does that mean I have an error? Let me know and I'll try and fix it, or feel free to try and fix it yourself if you think my answer is salvageable. Oct 17 '19 at 18:05
• We have the sentence: "Joel is happy unless Anna is not happy." Let's restructure it to this sentence: "Anna is not happy implies that Joel is not happy." Now evaluate the 4 different combinations of the propositions Oct 17 '19 at 23:52

It does need to be said that (contrary to what is implied in some other answers) "Joel is happy whenever Anna is happy" can not be translated by something of the form $A \Rightarrow B$.

The reason is simple "Joel is happy whenever Anna is happy" is a quantified conditional -- it says that, for any time $t$, if Anna is happy at time $t$, then Joel is happy at time $t$. As the OP in effect says. So a logical regimentation could be something like $(\forall t)(Hat \to Hjt)$, where $Hxt$ says that $x$ is happy at time $t$.

The Joel proposition then belongs to the same family as e.g. "If a number is prime and greater than two, then it is odd". This is another generalisation. And this sort of quantified content cannot be represented by something from propositional logic of the form $A \Rightarrow B$. For that is a singular conditional relating complete propositions with determinate truth-values, telling us that if the proposition $A$ is true (e.g. Anna is happy here and now; 9 is prime and greater than two), then $B$ is true (e.g. Joel is happy here and now, 9 is odd). And we don't get the intended generality.

• Perhaps the downvoter would like to explain their dissent from what is a standard textbook point ... Jan 1 '18 at 17:50

Put your propositions that do not have a connective in the left colums and list each possible values so: F, F next row F,T and so on. Then move write the statement you want to calculate the value for as 1 column and divide it in such way that each column connects two propositions or the columns left and right of the connectives are propositions.

I write your propositions in the same order als you defined them. but written them in another order:

1. Can Joel be happy in this case when Anna is not happy? If so Anna being happy implies that Joe is happy (so it cant be that JoelHappy is false when AnnaHappy is true)

2. Again same question. It seems one side is true only if the other is also true. So we use a bi-implication $$\iff$$ between the propositions.

3. Can Joel be happy when Anna is not? Hmmm doesn't look like it. So when AnnaHappy is False or is false after a negation, we cant have Joel be happy. Again a implication

We then get the following truth table: You can split up the last colums to this order 1. $$\neg(AnnaHappy)$$ 2. $$\implies$$ 3. $$\neg(JoelHappy)$$

$$\begin{array}{ | m{5em} | m{1cm} | m{1cm}| } \hline JoelHappy & AnnaHappy & JoelHappy \to AnnaHappy & JoelHappy \iff AnnaHappy & \neg( AnnaHappy ) => \neg(JoelHappy) \\ \hline F & F & T & T & T\\ \hline F & T & T & F & F\\ \hline T & F & F & F & T\\ \hline T & T & T & T & T\\ \hline \end{array}$$