The Rules of Truth

 We said that simple sentences are either true or false.  We now will look at how the truth value of compound sentences are determined by the truth value of their component sentences.  Compound sentences are constructed from simple sentences and operators.  Consider the sentence:
 
 

Suppose it is raining. So the sentence "It is raining" is true.  What is the truth value of sentence 1?  Clearly sentence 1 is false.  So if a sentence p is true, ~p is false.  And by the same token, if p is false, then ~p is true.  That's our first "rule of truth", and we can express it in what we'll call the basic truth table for negation:

Notice that the truth value of a negation is completely determined by the truth-value of negated wff, together with the rule, which we can informally call the rule of flip and flop.

Let's turn to the rule for conjunctions.  Let's look at the following conjunction:
 

When is this sentence true?  What happens if one of the conjuncts is false?  Clearly, both conjuncts must be true for the whole sentence to be true. If Ignat isn't twelve, or if Fritz isn't ten, or if both of their ages are not as stated, then the conjunction is false. When is the conjunction true - only when both conjuncts are true.  The truth table is:
  

Informally we can remember the rule for conjunction as the bad apple principle. One bad apple spoils the barrel.  One false conjunct and the whole conjunction is false.

The word "or" in English can be understood in two ways, and so our presentation of the rule of truth for disjunction requires that we consider these two meanings.  See if you can distinguish the two meanings in the following two sentences:
 

Do you see the difference?  In sentence 3, Mathilda may take Algebra, or Shakespeare, or she may take both!  But in sentence 4, Frieda may be a junior or a senior, but she's not both!  We call the "or" of sentence 3 inclusive or and we call the "or" of 4 exclusive or.  We have only one symbol for "or" in PL, (the wedge), so which "or" do we use?

We will use inclusive or in PL, in which a disjunction is true when both disjuncts are true. Of course a disjunction differs from a conjunction in that it is also true when just one disjunct is true, and false only when both disjuncts are false.  The basic truth table:

Question:  Suppose we introduced a new symbol, "*", and used it to represent exclusive disjunction. What would the basic truth table for (p*q) look like? (This is for the reader to answer!)

The rule of truth for conditionals also requires some explanation.  The words "if...then" in English can be used in a number of senses.   Some of those senses are problematic, from a logicians point of view, because they are not senses in which the truth value of the whole sentences built out of them depend on the truth value of the component sentences.  Such uses are called non-truth functional.  The uses presented above, for negation, conjunction, and disjunction, in contrast, are truth functional.  Our rule of truth for conditionals will restrict itself to the truth-functional meaning of "if...then".  This isn't the only meaning, but it's the only one for which we can give a rule of truth.  More will be said about the distinction between truth-functional and non truth functional operators in a subsequent section.

So what is the truth-functional meaning of "if ... then ..."?   Consider the two sentences:

Sentence 6 has the same meaning as sentence 5.  Using the rules of truth for negation and disjunction, we can write the truth-table for sentence 6 as follows: (where "A" is "The weatherperson is right" and B is "there will be a cooling trend tomorrow")
 

Since (~A v B) means the same thing as (A ⊃ B), the truth table for (A ⊃ B) will be identical to this truth table, namely:
 

So a conditional is false when it's antecendent is true and its consequent false. Otherwise it's true.  Again, this doesn't conform to all uses of the conditional. For example, consider a definition of the fragility of an object:
 

Suppose I have a rock which has never been struck.  The antecedent of the conditional is false - the object has not been struck. But then the whole conditional is true, because conditionals are only false when the antecendent is true and the consequent false!  This demonstrates that the meaning of the conditional in logic is different from the meaning in a sentence like 7.

The rules of truth for conditionals have other odd consequences. Consider the following sentence:

Now suppose you have eggs for breakfast. Then the conditional is true, on that particular assignment of truth to the consequent. It doesn't matter what truth value we assign to the antecedent. Interpreting sentence 8 as a conditional in our sense (often referred to as the "material conditional") is odd. It's likely that something more complex is expressed in this sentence, namely "If I were to drown this morning, you would have eggs for breakfast." That's a subjective conditional, and capturing the meaning of such conditionals is outside the scope of our study.

Our final rule is the rule of truth for biconditionals.  Here again, we can explain this rule by pointing out how a biconditional has the same meaning as a wff for which we can already compute truth-values.  The triple-bar abbreviates the English expression "if and only if."  What does "if and only if" mean?  As we'll discuss in more detail later, "if" indicates the antecedent of a conditional and "only if" indicates the consequent of a conditional.  So "p if and only if q" means "p if q and q if p", in ordinary conditional form, "if p then q and if q then p".  Let's look at the truth table for this:
 

Since the wff in the final column means the same as "p ≡ q", it has the same truth table. So the rule of truth for the triple bar is that a biconditional is true when the parts of the biconditional have the same truth value, otherwise it's false. In truth-table form:

This completes the the presentation of the rules of truth for PL.  In motivating these rules, we've made use of some important semantical notions which will be discussed in more detail in what follows. In particular, we made use of the idea that two wffs can have the same meaning, e.g. "(~p v q)" means the same thing as "(p ⊃ q)", and that when two wffs have the same meaning, they have the same truth-table.

In the next section we'll briefly review the basic truth tables.