And just like in the case of conjunction, so here too we want our formal symbol for disjunction – the ‘vel’
(‘ ’) – to faithfully model the behavior of its English-language counterpart, the word ‘or’ (and its various relatives.  So to figure out the truth-and-falsehood behavior of disjunctions, we can just think about when an ‘or’ sentence in English would be true, and when it would be false.

We can start with two little subject-matter sentences.

P: We’ll have ice cream
Q: We’ll have cake.

And then we can build up a simple ‘or’-sentence in English out of these two parts.

We’ll have ice cream or we’ll have cake.

This will be translated into symbolic notation, by way of this little translation table, like so:

(PQ).

Since ‘(PQ)’ is built up grammatically out of its two parts, ‘P’ and ‘Q’, its semantic profile (its ‘truth table’) will do the same.


Here again we’re starting with two sentence letters; so we know we’ll need 4 valuations, to go through all the possibilities.


Now we just need to go through these four different types of situation, and decide when the ‘or’ sentence would be true, and when it would be false.

First valuation: suppose it’s true that we’ll have ice cream, and it’s true that we’ll have cake.  In that case, the sentence ‘we’ll have ice cream or we’ll have cake’ would be true.


(In case that seems wrong, keep in mind: the kind of ‘or’ we mean by the ‘vel’ is an inclusive ‘or’.  An inclusive ‘or’ says: ‘either one, or the other, possibly both’.  Since the ‘or’ here includes the possibility of both, the case where we have both ice cream and cake still makes the ‘or’ sentence true.)

 
Second valuation: suppose it’s false to say that we’ll have ice cream, but it’s true that we’ll have cake.


In a situation like that – with ice cream but no cake – the sentence ‘We’ll have ice cream or we’ll have cake’ turns out to be true.


If I’d said ‘We’ll have ice cream or we’ll have cake’ as a promise, then in an ice-cream-enhanced but cake-free situation like Valuation 2, I would have kept my promise.

Third valuation: suppose it turns out to be true that we’ll have ice cream, but it’s false that we’ll have cake.


In this sort of situation, the whole disjunction would again be true.


Once again, if I’d made a promise, ‘We’ll have ice cream or we’ll have cake,’ then in an ice-cream-less but en-caked situation like Valuation 3, I would have kept that promise.

Finally, the fourth valuation: here, it’s false to say we’ll have ice cream, and it’s also false to say we’ll cake.  Here we’re having neither.


Here, it’s clearly false to say that we’ll have either.  If I told you ‘we’ll have ice cream or we’ll have cake,’ but we ended up having no ice cream and no cake, you would rightly accuse me of having said something false.


And while we used the particular sentence letters ‘P’ and ‘Q’ as examples here, any inclusive-‘or’ sentence will follow the same pattern: it will be true as long as at least one of its parts is true.  So we can state the general semantic rule for disjunctions.

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As an illustration of our disjunction rule, we see how disjunctions act when they’re mixed with negations.  Once again, if only for a little more practice, let's compare the truth tables for two WFS's, and see if they're truth-functionally equivalent or not.

(iii) ~ (PQ)
(iv) (~P~Q)

Just like before (with sentences (i) and (ii)), where we had conjunctions and negations mixed together, we’re left wondering here if these two sentences are saying the same thing or not.  Truth tables will settle that question in a hurry.   

Sentences (iii) and (iv) both use just "P" and "Q" as sentence letters.  So we know these truth tables will start with "P" and "Q".


And we know the rule for how many valuations we’ll need: since we're using two sentence letters, we know we'll need 4 valuations to go through all the possibilities


Then we’re ready to build up "(PQ)" out of "P" and "Q".


Since “(PQ)” is a disjunction, it will (like all disjunctions) follow the Disjunction Rul,e from our semantic theory.  The Disjunction Rule says that a disjunction is only false when both the parts are false.


In our truth table, the only case where both parts of the disjunction are false is in the fourth valuation.


So, following the Disjunction Rule, the disjunction "(PQ)" will only be false in this last valuation.


So it will be true in the other three valuations.


Now we’re ready to build up "~(PQ)" out of “(PQ)”.


"~(PQ)" is just the negation of "(PQ)"; so we can use the Negation Rule to figure out the value of
"~(PQ)" in each valuation, based on the value of "(PQ)" in that valuation.

And by now we’ve got the Negation Rule pretty much memorized: if the original sentence is true, its negation is false; and if the original sentence is false, its negation is true.


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


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


We now have the truth table for sentence (iii).

(iii) ~(P Q)
(iv) (~P ~Q)

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

This sentence is a disjunction, with two parts, "~P" and "~Q".  If we want to build up a disjunction, we’ve first got to have its two parts.  So we’re not going to get to the truth table for “(~P~Q)” until we first build the truth tables for “~P” and “~Q”.


Start with “~P”: 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.


In the second valuation, "P" is false; so the Negation Rule says "~P" will be true there.


The third and fourth valuations will just be a repeat of the first two.


The values of "~Q" are likewise built up from the values of “Q”, following the Negation Rule.  The values of Q are:


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


And now we’re in a position to figure out the values of the whole disjunction, "(~P~Q)".  In the first valuation, "~P" and "~Q" are both false; and as we know from the Disjunction Rule, that’s the one thing that makes a disjunction false.  So the disjunction "(~P~Q)" is false in the first valuation.


But in the remaining three valuations, at least one of the parts (“~P” or “~Q”) is true; and the Disjunction Rule tells us that’s all it takes to make the whole disjunction true.  So the whole disjunction, “(~P~Q),” will be true in the remaining valuations.


Now we have all the information we need to answer our question: are sentences (iii) and (iv) truth-functionally equivalent?

(iii) ~(PQ)
(iv) (~P~Q)

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


Specifically: these sentences have opposite values in the second and third valuations.

Maybe you didn’t predict that outcome; but if we think about English versions of these two sentences, we can see that in fact this is exactly the right result.  Letting “P” stand for the subject-matter sentence “We’ll have ice cream,” and letting “Q” stand for “We’ll have cake, sentence (iii) in English would be “We’ll have neither ice cream nor cake”.

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

(iii) ~(PQ): We’ll have neither ice cream nor cake
(iv) (~P~Q)

Using this same translation dictionary for “P” and “Q,” the English version of sentence (iv) –
“(~P~Q)” – would be “Either we won’t have ice cream or we won’t have cake”.

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

(iii) ~(PQ): We’ll have neither ice cream nor cake
(iv) (~P~Q): Either we won’t have ice cream, or we won’t have cake

Now, as English speakers, we would conclude that these two sentence are saying different things: saying “We’ll have neither ice cream nor cake” is saying that we’ll go without both; whereas saying “Either we won’t have ice cream or we won’t have cake” is only saying that we’ll go without at least one of them.  So once again, the semantic rules seem to be giving us just the right predictions – predictions that agree perfectly with our natural understanding of “or” and “not” sentences.

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Finally, we can notice one more set of nice predictions that the semantic rules make, by looking at sentences (i) through (iv), all at once.


We already saw that sentences (i) and (ii) aren’t truth functionally equivalent, and that sentences (iii) and (iv) are likewise not truth functionally equivalent.  But having all four sentences lined up like this illustrates two further points.  First, sentences (ii) and (iii) are truth functionally equivalent.


And if we think about English examples, this makes sense.

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

(ii) (~P~Q): We won’t have ice cream and we won’t have cake
(iii) ~(PQ): We’ll have neither ice cream nor cake

These two sentences do mean the same thing; so the semantic rules get it just right when it says they’re truth functionally equivalent.

Moreover, we see that sentences (i) and (iv) have the same truth table – they’re truth functionally equivalent.


Once again, this makes sense, if we think about the sentences in English.

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

(i) ~(PQ): We won’t have both ice cream and cake
(iv) (~P~Q): Either we won’t have ice cream or we won’t have cake

A situation where sentence (i) is true – where we’re not having both –  is indeed a situation where we’re going without one or the other (possibly both).  The semantic rules are making all the right predictions.

Now, if we wanted to sit around all day and test pairs of sentences for semantic equivalence, truth tables can get the job done.  But this is just a passing amusement; we’re after bigger game.  What we’re really interested in – our whole reason for building formal semantics in the first place – is testing arguments for validity.  So we’ll do one final brush-up on our truth table technique, and then (at last) build our formal test of validity.