3. If  and  are well-formed sentences, then ( ) is a well-formed sentence.

As we see right in this rule, every conjunction will have two parts.  So our semantic rule for conjunctions will need to start with the truth values of these two parts, and from them figure out the value for the whole conjunction sentence.


For example, let's extend the little translation table we used in the earlier examples, to add a second sentence.

P: It's raining
Q: It's cold

We can take these two little English-language subject-matter sentences to build an English-language conjunction:

It's raining and it's cold.

And we know that this conjunction will (following the little translation table) get translated like this:

(PQ)

This conjunction, like all conjunctions, is built up out of its two parts.  So, we said, the truth value of the whole conjunction is going to depend on the truth-value of its two
parts.


Now, since we're starting with two basic sentences ("P" and "Q"), we know already (from Bivalence) that we'll need 4 valuations to go through all the possibilities.


All we need to do now is think about how a conjunction phrase – like "and" – really works in English (because our "" symbol is meant to be a faithful symbolic counterpart to the English word "and," and its various relatives).

So consider the first valuation.  This is a situation where the sentences "It's raining" and "It's cold" are both true.


In a situation where it's true that it's raining, and it's also true that it's cold, what will the sentence "It's raining and it's cold" be?  Pretty clearly, the whole conjunction will be true in that sort of situation.


Now let's consider the second valuation.


This is the sort of situation where it's false to say that it's raining, but true to say that it's cold – for example, a clear, cloudless, day when it's 30 degrees out.  In that sort of situation, the sentence "It's raining and it's cold" would be false.


Now the third valuation.


This is the kind of situation where it's true that it's raining, but it's false that it's cold – say, a situation where it's raining and 90 degrees out.  Here, the sentence "It's raining and it's cold" would again be false.


And finally we consider the fourth valuation.


In this type of situation, it's false to say that it's raining, and likewise false to say that it's cold – for instance, a clear, cloudless day when it's 90 degrees out.  In this kind of situation, the sentence "It's raining and it's cold" is obviously false.


That's really all there is to the semantic rule for conjunctions – the Conjunction Rule.

Once again, we illustrated the rule by building the truth table for a particular WFS –
"(P  Q)" – but the same semantic rule holds equally for all conjunctions, big or small.  So we can restate it more generally, without mentioning "P" or "Q", or any other particular WFS:


(Here again, " " and " " are not sentences at all, just little blanks where sentences would go.)

---

Let's look at some examples of our new rule – the Conjunction Rule – in action. 

Let's compare the truth tables for two WFS's, and see if they're truth-functionally equivalent or not.  Here are the two sentences:

(i) ~ (PQ)
(ii) (~P~Q)

At first glance, they might look like they're saying the same thing.  People just starting out in Logic have a natural expectation: they expect to be able to take sentence (ii) and "factor out" the tilde, like a minus sign in mathematics, and get sentence (i); and likewise, they expect to be able to take sentence (i), and "distribute in" the tilde to the parts, and get sentence (ii).   So they might expect that, yes, the two sentences are saying the same thing.

I said people might have this natural expectation; but is it correct?  We don't need to guess: let's build truth tables for the two WFS's, and see if they're truth-functionally equivalent.

Since both of these sentences use just "P" and "Q" as sentence letters, we can build both truth tables up out of "P" and "Q".


And since we're using two sentence letters, we know we'll need 4 valuations to go through all the possibilities.


Next we can build up "(PQ)" out of "P" and "Q".


And we know from the Conjunction Rule what the truth value of "(PQ)" will be in each of these valuations.  The Conjunction Rule says that a conjunction is only true in one kind of valuation: where both of the parts(both of the "conjuncts") are true.


As we can see, in the truth table we're building, both the parts of the conjunction are true only in the first valuation.


So, according to the Conjunction Rule, the conjunction "(PQ)" will only be true in this first valuation.


So it will be false in the other three valuations.


(No big surprise there: this looks just like our statement of the Conjunction Rule; and besides, we already built the truth table for "(PQ)" above.)
 
Next we take "(PQ)", and build up "~(PQ)".


Now, since "~(PQ)" is just the negation of "(PQ)", we can use the Negation Rule to build up the value of "~(PQ)" out of the value of "(PQ)," for each valuation.

The Negation Rule, as we know, is very simple: if the original sentence is true, its negation is false; and if the original sentence is false, its negation is true.  In other words: the tilde has the effect of turning a "T" into an "F", and an "F" into a "T".


In the first valuation of our truth table, "(PQ)" is true; so the Negation Rule says
"~(PQ)" will be false in this valuation.


And since "(PQ)" is false in the other three valuations, the Negation Rule says
"~(PQ)" will be true in those valuations.


So there we have the truth table for our first sentence.

(i) ~(P Q)
(ii) (~P ~Q)

Next, we build the truth table for our second sentence, "(~P~Q)".

This sentence is a conjunction, with two parts, "~P" and "~Q".  So we need to build up the whole sentence out of these two parts.


Building the truth table for "~P" is easy: we know from the Negation Rule that "~P" will have the opposite value of "P" in every valuation.  Since "P" is true in the first valuation, "~P" will be false there.


And likewise, since "P" is false in the second valuation, the Negation Rule says "~P" will be true there.


Of course, in the third and fourth valuations “P”  and “~P” will have the same values that they did in the first two.


The values of "~Q" are just easy to figure out, thanks to the Negation Rule.  The values of Q are:


So "~Q" will have just the opposite value in each valuation.


Now we can use the Conjunction Rule to figure out the values of the whole conjunction,
"(~PQ)".  In the first valuation, "~P" and "~Q" are both false; so the conjunction "(~P~Q)" will also be false.


In the second valuation, "~P" is true, and "~Q" is false; but the Conjunction Rule says both parts have to be true if we're going to have the whole conjunction true.  So "(~P~Q)" is still false here.


In the third valuation, "~P" is false, but "~Q" is now true.  Again, the Conjunction Rule says this isn't enough to make the whole conjunction true here – "(~P~Q)" is false in the third valuation.


Finally, in the fourth valuation, both "~P" and "~Q" are true.  Here, at last, are the ingredients we need to make "(~P~Q)" true.


Now we can settle our original question: are sentences (i) and (ii) truth-functionally equivalent?

(i) ~(PQ)
(ii) (~P~Q)

We can tell from the truth tables we've just built that sentences (i) are (ii) are definitely not truth-functionally equivalent.


In particular, these sentences have opposite values in the second and third valuations.

This isn't really so surprising, if we think about what sentences (i) and (ii) would look like in English.  Suppose we use the following translation table:

P: We'll have ice cream
Q: We'll have cake

OK, then sentence (i) is saying "It's not the case that we'll have both ice cream and cake" – or, more naturally put, "We won't have both ice cream and cake".

P: We'll have ice cream
Q: We'll have cake
(i) ~(PQ)  ("We won't have both ice cream and cake)
(ii) (~P~Q)

So sentence (i) leaves open the possibility that we could still have one or the other – just not both.

But sentence (ii), according to our translation table, says "We won't have ice cream, and we won't have cake".

P: We'll have ice cream
Q: We'll have cake
(i) ~(PQ)  ("We won't have both ice cream and cake")
(ii) (~P~Q) ("We won't have ice cream, and we won't have cake")

Sentence (ii) definitely does not leave open the possibility of our having one or the other.  So we can see, in natural English, that the two sentences do not mean the same thing.  And if truth-functional equivalence really is a good measure of when two sentences mean the same thing logically, then English speakers would predict that sentences (i) and (ii) should not be truth-functionally equivalent.  Lo and behold, our semantic rules yield exactly the correct results.  Good old semantic rules.