Read this tutorial, which will introduce you to truth tables. Truth tables are an objective way of determining the validity of an argument as a whole when the argument is expressed symbolically.
Complete the exercises for this tutorial and check your answers.
§1. What is a Truth-Table?
We start with truth-tables for the sentential connectives in sentential logic (SL). A truth-table shows how the truth-value of a complex well-formed formula (WFF) depends on the truth-values of its component well-formed formulas (WFFs).
So what are truth-values?
In sentential logic (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 it is true. To say that its truth-value is F is to say that it is false. Notice that in sentential logic (SL) we assume the principle of bivalence: a well-formed formula (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's now look at the truth-table of each of the sentential connectives.
Consider the statement "whales are mammals." Let's use the sentence-letter "P" to translate this statement. (We can 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 well-formed formula (WFF) and you add the negation sign in front of it to make a new well-formed formula (WFF), you end up with a well-formed formula (WFF) that has the opposite truth-value.
Notice that we use the Greek symbol "φ" in the table to stand for any well-formed formula (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.
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:
- It is raining. It is also hot.
- It is raining. But it is not hot.
- It is not raining. But it is hot.
- 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 conjunction 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 well-formed formula (WFF) is true. The other three rows tell us that the conjunction is false in all other situations.
|φ||ψ||( φ ∨ ψ )|
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 we consider "either ___ or ___" to be false even when the disjuncts are considered to be true.
For example, when ordering from a set menu in a restaurant you may be told that you can either have the salad or you can have the soup, but you cannot have both! We could say, "either P or Q (but not both)."
These two senses of "or" are called inclusive-or and exclusive-or respectively. We should always understand the disjunctive sign "∨" in sentential logic (SL) in the inclusive sense. To express "P or Q" in the exclusive sense you might use the well-formed formula (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.
|φ||ψ||( φ ↔ ψ )|
Suppose you are at a party and wonder whether your friend Jane is around. You ask 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? 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, the other person is also absent.
This sense of "if and only if" (or "iff") is captured in the truth-table on the left.
§6. The Material Conditional
|φ||ψ||( φ → ψ )|
The arrow sign is often translated as "If... then..." Its truth-table is probably the most difficult one to understand among the ones we have studied 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.
Consider this diagram:
Source: Joe Lau and Jonathan Chan, https://philosophy.hku.hk/think/sl/connectives.php
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