[SL03] Connectives

§1. What is a truth-table?

We start with truth-tables for the sentential connectives in SL. A truth-table shows how the truth-value of a complex WFF depends on the truth-values of its component WFFs. So what are truth-values?

In SL there are only two truth-values : T and F, which stands for truth and falsity. To say that a statement has truth-value T is just to say that it is true. To say that its truth-value is F is to say that it is false. Notice that in SL we assume the principle of bivalence : a WFF either has truth-value T or F. Some systems of logic, such as fuzzy logic, reject the principle of bivalence. Notice also that some logic or engineering textbooks use "1" and "0" in place of "T" and "F".

Let us now look at the truth-table of each of the sentential connectives.

§2. Negation

Consider the statement "whales are mammals." Let us use the sentence-letter "P" to translate this statement. (We can of course use any sentence letter we want.) This just means we are now taking the symbol "P" to have the same meaning as "whales are mammals." But suppose you disagree with the statement. Then you might express your disagreement by saying things like :

  • Whales are not mammals.
  • It is not true that whales are mammals.
  • It is not the case that whales are mammals.

In sentential logic, these three different sentences are all translated as "~P". Obviously, "P" and "~P" have opposite truth-values - if one is true then the other one must be false, and vice versa. The truth-table displayed here is the truth-table for the negation sign. It shows that when you have a WFF and you add the negation sign in front of it to make a new WFF, you end up with a WFF that has the opposite truth-value.

Notice that we use the Greek symbol "φ" in the table to stand for any WFF. So the table tells us that when "P" is true, "~P" is F, and when "(Q&~R)" is F, "~(Q&~R)" is T, etc.

§3. Conjunction

Now consider these two statements : "it is raining" and "it is hot". Each of them can be either true or false independently of the other, so in terms of their truth-values there are four different possibilities :

  1. It is raining. It is also hot.
  2. It is raining. But it is not hot.
  3. It is not raining. But it is hot.
  4. It is not raining. It is also not hot.

Now what about the truth-value of the complex statement "it is raining and it is hot"? What would be its truth-value in each of the four situations? This question is easy to answer because we know that this statement is true only in the first situation, and false in the other three.

In sentential logic we can translate this complex conjunction using "&" to conjoin the two conjuncts. The truth-table on the left shows how the truth-value of a conjucntion depends on the truth-values of the conjuncts, just as in the example we have looked at. The first row of truth-values tells us that when the first and second conjuncts are true, the whole WFF is true. The other three rows tell us that the conjunction is false in all other situations.

§4. Disjunction

The disjunction symbol "∨" is usually used to translate "or". The truth-table on the left tells us that a disjunction is false when both disjuncts are false. Otherwise it is always true.

Notice that the first row tells us that the disjunction is true when both disjuncts are true. In other words, if "(P∨Q)" is used to translate "Either Peter will leave, or Amie will leave.", and it turns out that they both leave, then the whole complex statement is still true.

There are two things to be said if you think this is counter-intuitive. First, one might say that a better translation of "(P∨Q)" is "Either P or Q (or both)." Second, it is arguable that there are certain uses in ordinary language where "either ___ or ___" is considered to be false even when the disjuncts are considered to be true. For example, when ordering from a set menu in a restaurant you might be told that you can either have the salad or you can have the soup, but presumably it is understood that you cannot have both! Here the statement is better understood as "either P or Q (but not both)." These two senses of "or" are called inclusive-or and exclusive-or respectively. The disjunctive sign "∨" in SL should always be understood in the inclusive sense. To express "P or Q" in the exclusive sense one might use the WFF "((P∨Q)&~(P&Q))" instead. This is an example of how formal logic can actually help us understand better the linguistic usages of natural languages.

In this web site, you should take "or" to mean inclusive-or, unless otherwise indicated.

§5. Biconditional

Suppose you are at a party, and you are wondering if your friend, Jane, is around. You asked another friend, and he replies, "Jane is at the party if and only if Matthew is at the party." If you accept this statement as true, what can you conclude from it? Of course this statement on its own does not tell you whether Jane is here or not. But it does tell you that if she is here, then Matthew is also here, and if one of them is not at the party, then the other person is absent as well. This sense of "if and only if" (or "iff") is captured in the truth-table on the left.

§6. The Material Conditional

The arrow sign is often translated as "If ... then ...". Its truth-table is probably the most difficult one to understand among the ones you have seen so far. Perhaps it will be easier to remember the truth-table the following way. Suppose you make a conditional statement such as "If I have lots of money then I shall be happy." Under what condition will this statement be false? Obviously, your statement is false if you have a lot of money but still you are not happy. In other words, when the antecedent is true and the consequent is false, the whole conditional statement is false. This is exactly what the second row of the truth-table says. Just remember that the conditional is true in all other situations.

This explanation is not quite the full and correct account of why the truth-table of "→" should look the way it does. We shall provide the full explanation in a separate tutorial in a different section.

Are these statements true or false?

  1. If "(P∨Q)" is true, either "P" is false or "Q" is false.
  2. If "(P&Q)" is false, then "P" is false and "Q" is also false.
  3. Whenever "(P∨Q)" is true, "(Q→P)" is also true.
  4. In order for "(P→Q)" to be true, "Q" must be true.

We can actually use circuit diagrams to represent truth-tables. Look at the following diagram. Try clicking the switches to see how you might turn the light on.

Suppose "P" means the switch on the left is down, and "Q" means the switch on the right is down. What WFF would you use to describe the switch settings that would turn on the light?

Assuming the same meaning for "P" and "Q", what WFF would you use to describe the switch settings that would turn on the light in this second circuit?

Now suppose "P" means the switch on the left is UP, and "Q" means the switch on the right is UP. Which is the right WFF to use?

Consider this diagram :

Now determine whether the following statements are true of the diagram, using the appropriate truth-tables to interpret the connectives :

  1. All squares are red if and only if all squares are green.
  2. If there is no red square, then there is a triangle.
  3. Either there is a green circle, or there are no orange squares.

Look at this animation for a few seconds (you should see them changing colors if your browser is working properly with Javascript enabled):

Suppose we use "P" to mean the circle is orange, and "Q" to mean the star is orange. Which of the following WFF is always true of the animation? Click the button next to your answer.

  1. (P&Q)
  2. (PvQ)
  3. (P→Q)

How about this one? Which WFF is always true?

  1. (P↔Q)
  2. ~P
  3. (PvQ)

How about this one? Which WFF is always true?

  1. (P↔Q)
  2. (P→Q)
  3. (P&Q)
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