In Sentential Logic, or SL, we use alphabetical letters to stand for whole statements, also called propositions. These can function as individual objects and simple categories such as those we used in the last chapter. For instance, A can stand for the statement, “That is Socrates”, and B can stand for “That is a man”, and then we can symbolize “If that is Socrates, then that is a man”, as “If A, then B”. Of course, we can symbolize any statement, no matter how complex, with a single letter, so if A stands for “I am alone and it is dark and scary”, and B stands for “I will sing in French and stand on my head or compose poetry in Arabic”, then “If A, then B”, would stand for “If I am alone and it is dark and scary, then I will sing in French and stand on my head or compose poetry in Arabic”.
Some logicians use lower case p, q and r to symbolize three basic statements. Others use a capital A, B and C, as we did in the last chapter. We can also use capital or lowercase letters that can be associated with the statements they symbolize, such as G standing for, “It is a gorilla”, and R standing for, “Rudolf is a reindeer”. We can also use the combination of letters and numbers to symbolize related statements, such as E1 standing for, “Eric is an elephant”, and E2 standing for, “Andrea is an elephant”. We will be using A, B and C for most of our examples and illustrations.
Modern logicians such as Ludwig Wittgenstein, who invented the truth table method we will use in this chapter, realized that you can use connectives to both modify statements and link several statements together. The connectives we use in SL are the same we use in QL, quantified logic, the second form of logic we will study. These five connectives are negation, conjunction, disjunction, conditional, and biconditional, also known as NOT, AND, OR, IF-THEN, and IF-AND-ONLY-IF. Each of these functions in a particular way that we can use truth tables to understand.
Negation, NOT, we will symbolize with the symbol ~, which you can produce by holding shift and pushing the key right below the escape key in the upper left corner of any keyboard. This is the same symbol used in Spanish above the letter n to give it a ‘nya’ sound. It is symbolized differently by various logic texts, but it is always a horizontal line and the only connective that modifies a single statement rather than linking two statements together. I am using the symbol ~ because that was the one I was taught, and it is easy to find on the keyboard.
If we say that A stands for, “I have an apple”, then not-A, or ~A, means, “It is not the case that I have an apple”, which is the same thing as saying, “I do not have an apple”. The simplest truth table is the one that defines negation, the only one that features a single statement with two possible truth values.
For any truth table, the cases of all possible truth values go on the left, and the truth values for the expression being evaluated go on the right. Notice that with one statement symbolized by a single letter, there are two possible truth values, the first case in which A is true (T), the second in which A is false (F). In the first case, when A is true, ~A is false, and in the second case, when A is false, ~A is true. There is only one other truth table we will study with only two cases and one symbol standing for a single statement, which is not-not-A:
Notice that first we flip and negate the original truth values for each case within the parentheses, and then flip and negate them again, placing them under the NOT sign outside the parentheses, showing us that in case one, in which A is true, ~(~A) is also true, and in the second case, when A is false, then ~(~A) is also false. Negating a truth value twice flips it twice, bring it back to its original truth value.
With conjunction, AND, we will need two symbols, A and B. We will symbolize the expression, “A and B” as A ^ B, using the symbol one gets by typing the number 6 while holding the shift key down. Some use the symbol & to symbolize AND, which is also acceptable, but I am using ^ because that is how I was taught.
With two symbols, we will need four cases of possible truth values. We will use this set of eight truth values in four cases many times, as we will be doing many truth tables with two symbols. With four cases for two symbols, the first case is both as true (TT), the second case is the first as true and the second as false (TF), the third case is the first as false and the second as true (FT) and the fourth case is both as false (FF). When we have a truth table with two symbols (A and B), to the left of the vertical line we will always see these four in this order.
Let us say that A stands for, “I have an apple”, and B stands for, “I have a banana”. This means that A ^ B means, “I have an apple and I have a banana”, or “I have an apple and a banana”. In the first case, when it is true that I have an apple and I have a banana, the statement, “I have an apple and I have a banana” is true.
In the second case, when it is true that I have an apple but I do not have a banana, as B is false, the statement, “I have an apple and I have a banana” is false. Notice that in everyday speak, we might say that I was half right, and my statement is a half-truth, but in formal logic we can only give absolute truth values to statements and sets of statements, so we must call my entire statement false.
In the third case, when I do not have an apple, as A is false, but I do have a banana, as B is true, the statement, “I have an apple and a banana” is false, for the same reason that it is false in the second case. It is only a half-truth, not fully true.
In the fourth and final case, when I am lying about having both an apple and a banana, as A and B are both false, the statement, “I have an apple and a banana” is entirely false.
Notice that the four truth values beneath the AND symbol (^) are T, F, F and F, meaning that the expression A ^ B is only true in the first case, when both A and B are true, and it is false in the second and third case, when one side is true but the other is false, and false in the fourth case, when both sides are false. This set of T, F, F and F down the middle is the final answer for the truth table, the truth values for each of the four cases for the expression being evaluated (A ^ B).
In the last chapter, we defined a contradiction as a statement that must be false or set of statements that must be false and inconsistent. As an example, we used the example, “It is both raining and not raining”. We can demonstrate contradiction with a truth table.
Notice that beneath the AND symbol there are two Fs, as “A and not A” is false when A is true and false when A is false, false in both possible cases.
Disjunction, OR, connects two statements, like AND and the rest of the connectives other than NOT. OR is symbolized with a lowercase V (v), the inverse of our symbol for AND (^). If we say that A stands for the statement, “I have an apple”, and B stands for the statement, “I have a banana”, then the expression A v B means, “I have an apple or I have a banana”, which can be shortened to, “I have an apple or a banana”.
Using the same four cases in the same order that we did for AND (TT, TF, FT and FF), we can show how OR functions. The first case (TT) presents us with an interesting case. If I say, “I have an apple or a banana”, and I have both an apple and a banana, we could say that what I say is false, as I have both, not simply one OR the other, OR we could say that what I say is true, as I have both, one AND the other. In everyday speech and judgement, we use “or” in both of these ways, exclusively and inclusively.
Suppose we are at a buffet with many different types of food. I tell you that you can pick whatever food you like, including eggs, or salad, or steak, or sausage, or broccoli, and you choose eggs and sausage. You would not expect me to look at you with horror, and say, “Hey, I told you you could have eggs OR sausage, not both!”, because at a buffet we naturally expect that when I say, “You can have eggs or sausage”, I am using OR inclusively. I could have said, “You can have eggs AND sausage”, and meant the same thing.
Now suppose we are at a car dealership, and I tell you that I will buy you a truck or a convertible or a van. If you picked out both a truck and a van, I would look at you with horror, and say, “Hey, you can have one OR the other, but not both!”. Unlike at a buffet, when someone is buying a car there is an expectation that only one thing with be exclusively chosen, so when I say, “You can have a truck or a van”, I am using OR exclusively.
INCLUSIVE OR: You can choose more than one.
EXCLUSIVE OR: You can only choose one.
The interesting thing is that we use the word “or” without indicating whether or not we mean it inclusively or exclusively. We can imagine that in English, or any other language, we could use two different words for exclusive and inclusive OR, but we do not. It is the context of the situation in which “or” is used that give it its meaning and use. Before this is explicitly brought to our attention, we do not even notice that we automatically use and hear “or” appropriately without problems. This is because we are good at sharing frames of context, although there can be problems.
In order to construct a consistent system of logic, we have to choose which way we will consistently use OR. This is the first ambiguity in logic that shows us formal logic is one possible logic, not the simple bedrock of human reasoning. If formal logic was nothing more than the complete way we use reasoning, we would not have to make a choice as to how a particular element should be used. We will encounter another similar ambiguity, another place where we must make a choice as to how a thing is to be used with two disjunctive possibilities, with the next connective we will study, the conditional (IF-THEN). Before that, however, we must make a choice and continue to see how OR functions.
In formal logic we use OR inclusively. This means that if I say, “I have an apple or a banana”, and I have both, we would choose to say what what I say is true, as if I had said, “I have an apple AND a banana”. Thus, in the first of our four cases for the OR truth table (TT), the expression is true. In the second case (TF), if I say, “I have an apple or a banana”, and I have an apple, what I say is true, which is the same in the third case (FT), when I do not have an apple but I have a banana. It is only in the fourth and final case (FF), when I have neither an apple nor a banana, that what I say is false.
Notice that in the first and fourth final case, OR functions the same way as AND. In the first case, when both A and B are true, A ^ B and A v B are both true. In the fourth case, when both A and B are false, A ^ B and A v B are both false. It is in the second and third cases that AND and OR are different. When one side is true but the other is false, AND is false but OR is true. AND is only true in the first case, when both sides are true, and OR is only false in the fourth case, when both sides are false. When we begin to combine connectives and work with complex truth tables, we will see that AND and OR, in accord with DeMorgan’s theorem, are inverses of each other.
A conditional, IF-THEN, is trickier than NOT, AND and OR. We will use the symbol > to symbolize a conditional. As with AND, IF-THEN is sometimes symbolized differently, some using an arrow, and others using a capital U on its left side like a finger pointing. We use > because, like ~, it is easy to find on any keyboard.
It is useful to think of IF-THEN as a promise, such that if I say, “If A then B”, I am promising that whenever A is true, B will also be true. Let us say that A stands for the statement, “You give me an apple”, and B stands for the statement, “I give you a banana”, such that A > B stands for “If you give me an apple, I will give you a banana”. What happens in each of our four cases?
In the first case (TT), if you give me an apple and I give you a banana, what I say is true, and my promise is unbroken. Just like for AND and OR, IF-THEN is true when both sides are true. In the second case (TF), if you give me an apple and I do not give you a banana, what I say is false and my promise is broken. So far, no problems.
Things get tricky in the third and fourth cases (FT and FF), when you do not give me an apple. Clearly, if you do not give me an apple, I do not owe you a banana. I can give you a banana if I feel like it, but I am not obligated to give you a banana unless you give me an apple. The question is, is my promise true? We could say we do not know whether my promise is true or not unless you give me an apple and we see whether or not I give you a banana. However, according to the Principle of Bivalence, in logic things must be either true or false, not neither true nor false, so we cannot leave the third and fourth cases open without assigning a truth value, so we must decide what to do.
We could decide that a promise is false unless it is proven true. This would make A > B false in the third and fourth case, giving it the same function as AND, only true in the first case when both A and B are true. We could also decide that a promise is true unless it is broken and proven false. This would make A > B true in the third and fourth cases. This is another ambiguity, another choice we must make, like the one we encounter with OR, which we choose to use inclusively. In formal logic, we make the choice to say A > B is true in both the third and fourth cases. This means that if you do not give me an apple, my promise is considered unbroken and still true whether or not I give you a banana.
Many have problems with the fourth and final case, thinking that it should be false when both sides are false as it is with both AND and OR, but this is not the way IF-THEN functions. It is useful to remember that A > B is only false in the second case (TF), when you give me an apple but I do not give you a banana, when my promise is explicitly broken.
A biconditional, IF-AND-ONLY-IF, is a conditional that works both ways. We will symbolize it with the symbol we use for conditionals along with its inverse, <>. Some symbolize it with an arrow with points on both ends, others with an equals sign with three instead of two bars, but again we will use < > because it is easy to find on any keyboard.
Let us again say that A stands for the statement, “You give me an apple”, and B stands for the statement, “I give you a banana”, as we did for the conditional. Consider that saying, “You give me an apple ONLY IF I give you a banana” (A > B) means the same thing as saying, “IF I give you a banana THEN you give me an apple” (B > A). This would mean that A <> B stands for, “You give me an apple if and only if I give you a banana”, making the condition work both ways.
In the first of our four cases (TT), if you give me an apple and I give you a banana, the biconditional statement holds true. In the second and third cases (TF and FT), when one of us gives the other something but does not receive what they were promised in return, the biconditional statement is false as our arrangement is broken. In the fourth and final case (FF), our arrangement is unbroken if neither gives the other anything, and the statement is considered true because it has not been shown to be false, like in the third and fourth case for conditionals. This is the truth table which shows how biconditionals function.
Exercise 3: On a separate piece of paper, construct the five basic truth tables, one for each of the five connectives. You should practice until you can construct each without looking at the truth tables above.