To make our logical language more easy and intuitive to use, we can now add to it elements that make it able to express the equivalents of other sentences from a natural language like English.  Our translations will not be exact, but they will be close enough that: first, we will have a way to more quickly understand the language we are constructing; and, second, we will have a way to speak English more precisely when that is required of us.

Consider the following expressions.  How would we translate them into our logical language?

Tom will go to Berlin and Paris.

The number a is evenly divisible by 2 and 3.

Steve is from Texas but not from Dallas.

We could translate each of these using an atomic sentence.  But then we would have lost—or rather we would have hidden—information that is clearly there in the English sentences.  We can capture this information by introducing a new connective; one that corresponds to our “and”.

To see this, consider whether you will agree that these sentences above are equivalent to the following sentences.

Tom will go to Berlin and Tom will go to Paris.

The number a is evenly divisible by 2 and the number a is evenly divisible by 3.

Steve is from Texas and it is not the case that Steve is from Dallas.

Once we grant that these sentences are equivalent to those above, we see that we can treat the “and” in each sentence as a truth functional connective.

Suppose we assume the following key.

P:  Tom will go to Berlin.

Q:  Tom will go to Paris.

R:  a is evenly divisible by 2.

S:  a is evenly divisible by 3.

T:  Steve is from Texas

U:  Steve is from Dallas.

A partial translation of these sentences would then be:

P and Q

R and S

T and~U

Our third sentence above might generate some controversy.  How should we understand “but”?  Consider that in terms of the truth value of the connected sentences, “but” is the same as “and”.  That is, if you say “P but Q” you are asserting that both P and Q are true.  However, in English there is extra meaning; the English “but” seems to indicate that the additional sentence is unexpected or counter-intuitive.  “P but Q” seems to say, “P is true, and you will find it surprising or unexpected that Q is true also.”  That extra meaning is lost in our logic.  We will not be representing surprise or expectations.  So, we can treat “but” as being the same as “and”.  This captures the truth value of the sentence formed using “but”, which is all that we require of our logic.

Following our method up until now, we want a symbol to stand for “and”.  In recent years the most commonly used symbol has been “&”.

The syntax for “&” is simple.  If Φ and Ψ are sentences, then

(Φ&Ψ)

is a sentence.  Our translations of our three example sentences should thus look like this:

(P&Q)

(R&S)

(T&~U)

Each of these is called a “conjunction”.  The two parts of a conjunction are called “conjuncts”.

The semantics of the conjunction are given by its truth table.  Most people find the conjunction’s semantics obvious.  If I claim that both Φ and Ψ are true, normal usage requires that if Φ is false or Ψ is false, or both are false, then I spoke falsely also.

Consider an example.  Suppose your employer says, “After one year of employment you will get a raise and two weeks vacation”.  A year passes.  Suppose now that this employer gives you a raise but no vacation, or a vacation but no raise, or neither a raise nor a vacation.  In each case, the employer has broken his promise.  The sentence forming the promise turned out to be false.

Thus, the semantics for the conjunction are given with the following truth table.  For any sentences Φ and Ψ:

Φ	Ψ	(Φ&Ψ)
T	T	T
T	F	F
F	T	F
F	F	F

5.2  Alternative phrasings, and a different “and”

We have noted that in English, “but” is an alternative to “and”, and can be translated the same way in our propositional logic.  There are other phrases that have a similar meaning: they are best translated by conjunctions, but they convey (in English) a sense of surprise or failure of expectations.  For example, consider the following sentence.

Even though they lost the battle, they won the war.

Here “even though” seems to do the same work as “but”.  The implication is that it is surprising—that one might expect that if they lost the battle then they lost the war.  But, as we already noted, we will not capture expectations with our logic.  So, we would take this sentence to be sufficiently equivalent to:

They lost the battle and they won the war.

With the exception of “but”, it seems in English there is no other single word that is an alternative to “and” that means the same thing.  However, there are many ways that one can imply a conjunction.  To see this, consider the following sentences.

Tom, who won the race, also won the championship.

The star Phosphorous, that we see in the morning, is the Evening Star.

The Evening Star, which is called “Hesperus”, is also the Morning Star.

While Steve is tall, Tom is not.

Dogs are vertebrate terrestrial mammals.

Depending on what elements we take as basic in our language, these sentences all include implied conjunctions.  They are equivalent to the following sentences, for example:

Tom won the race and Tom won the championship.

Phosphorous is the star that we see in the morning and Phosphorous is the Evening Star.

The Evening Star is called “Hesperus” and the Evening Star is the Morning Star.

Steve is tall and it is not the case that Tom is tall.

Dogs are vertebrates and dogs are terrestrial and dogs are mammals.

Thus, we need to be sensitive to complex sentences that are conjunctions but that do not use “and” or “but” or phrases like “even though”.

Unfortunately, in English there are some uses of “and” that are not conjunctions. The same is true for equivalent terms in some other natural languages.  Here is an example.

Rochester is between Buffalo and Albany.

The “and” in this sentence is not a conjunction.  To see this, note that this sentence is not equivalent to the following:

Rochester is between Buffalo and Rochester is between Albany.

That sentence is not even semantically correct.  What is happening in the original sentence?

The issue here is that “is between” is what we call a “predicate”.  We will learn about predicates in chapter 11, but what we can say here is that some predicates take several names in order to form a sentence.  In English, if a predicate takes more than two names, then we typically use the “and” to combine names that are being described by that predicate.  In contrast, the conjunction in our propositional logic only combines sentences.  So, we must say that there are some uses of the English “and” that are not equivalent to our conjunction.

This could be confusing because sometimes in English we put “and” between names and there is an implied conjunction.  Consider:

Steve is older than Joe and Karen.

Superficially, this looks to have the same structure as “Rochester is between Buffalo and Albany”.  But this sentence really is equivalent to:

Steve is older than Joe and Steve is older than Karen.

The difference, however, is that there must be three things in order for one to be between the other two.  There need only be two things for one to be older than the other.  So, in the sentence “Rochester is between Buffalo and Albany”, we need all three names (“Rochester”, “Buffalo”, and “Albany) to make a single proper atomic sentence with “between”.  This tells us that the “and” is just being used to combine these names, and not to combine implied sentences (since there can be no implied sentence about what is “between”, using just two or just one of these names).

That sounds complex.  Do not despair, however.  The use of “and” to identify names being used by predicates is less common than “and” being used for a conjunction.  Also, after we discuss predicates in chapter 11, and after you have practiced translating different kinds of sentences, the distinction between these uses of “and” will become easy to identify in almost all cases.  In the meantime, we shall pick examples that do not invite this confusion.

5.3  Inference rules for conjunctions

Looking at the truth table for the conjunction should tell us two things very clearly.  First, if a conjunction is true, what else must be true?  The obvious answer is that both of the parts, the conjuncts, must be true.  We can introduce a rule to capture this insight.  In fact, we can introduce two rules and call them by the same name, since the order of conjuncts does not affect their truth value.  These rules are often called “simplification”.

(Φ&Ψ)

_____

Φ

And:

(Φ&Ψ)

_____

Ψ

In other words, if (Φ&Ψ) is true, then Φ must be true; and if (Φ&Ψ) is true, then Ψ must be true.

We can also introduce a rule to show a conjunction, based on what we see from the truth table.  That is, it is clear that there is only one kind of condition in which (Φ&Ψ) is true, and that is when Φ is true and when Ψ is true.  This suggests the following rule:

Φ

Ψ

_____

(Φ&Ψ)

We might call this rule “conjunction”, but to avoid confusion with the name of the sentences, we will call this rule “adjunction”.

5.4  Reasoning with conjunctions

It would be helpful to consider some examples of reasoning with conjunctions.  Let’s begin with an argument in a natural language.

Tom and Steve will go to London.  If Steve goes to London, then he will ride the Eye.  Tom will ride the Eye too, provided that he goes to London. So, both Steve and Tom will ride the Eye.

We need a translation key.

T:  Tom will go to London.

S:  Steve will go to London.

U:  Tom will ride the Eye.

V:  Steve will ride the Eye.

Thus our argument is:

(T&S)

(S→U)

(T→V)

_____

(V&U)

Our direct proof will look like this.

 

Now an example using just our logical language.  Consider the following argument.

(Q→ ~S)

(P →(Q&R))

(T→ ~R)

P

_____

( ~S&~T)

Here is one possible proof.

 

5.5  Alternative symbolizations for the conjunction

Alternative notations for the conjunction include the symbols “^”  and the symbol “∙”.  Thus, the expression (P&Q) would be written in these different styles, as:

(P^Q)

(P∙Q)

5.6 Complex sentences

Now that we have three different connectives, this is a convenient time to consider complex sentences.  The example that we just considered required us to symbolize complex sentences, which use several different kinds of connectives.  We want to avoid confusion by being clear about the nature of these sentences.  We also want to be able to understand when such sentences are true and when they are false.  These two goals are closely related.

Consider the following sentences.

~ (P →Q)

( ~ P →Q)

( ~ P → ~ Q)

We want to understand what kinds of sentences these are, and also when they are true and when they are false.  (Sometimes people wrongly assume that there is some simple distribution law for negation and conditionals, so there is some additional value to reviewing these particular examples.)  The first task is to determine what kinds of sentences these are.  If the first symbol of your expression is a negation, then you know the sentence is a negation.  The first sentence above is a negation.  If the first symbol of your expression is a parenthesis, then for our logical language we know that we are dealing with a connective that combines two sentences. Notice that a connective can combine two atomic sentences, or two dependent sentences.

The way to proceed is to match parentheses.  Generally people are able to do this by eye, but if you are not, you can use the following rule.  Moving left to right, the last “(” that you encounter always matches the first “)” that you encounter.  These form a sentence that must have two parts combined with a connective.  You can identify the two parts because each will be an atomic sentence, a negation sentence, or a more complex sentence bound with parentheses on each side of the connective.

In our propositional logic, each set of paired parentheses forms a sentence of its own.  So, when we encounter a sentence that begins with a parenthesis, we find that if we match the other parentheses, we will ultimately end up with two sentences as constituents, one on each side of a single connective.  The connective that combines these two parts is called the “main connective”, and it tells us what kind of sentence this is.  Thus, above we have examples of a negation, a conditional, and a conditional.

How should we understand the meaning of these sentences?  Here we can use truth tables in a new, third way (along with defining a connective and checking arguments).  Our method will be this.

We make a truth table for ~(P→Q). This sentence is a conditional. Its two parts are atomic sentences. The definition of “~” tell us that ~(P→Q) is true when (P→Q) is false, and ~(P→Q) is false when (P→Q) is true.

P	Q	~	(P	→	Q)
T	T	F	T	T	T
T	F	T	T	F	F
F	T	F	F	T	T
F	F	F	F	T	F

This truth table tells us what ~(P→Q) means in our propositional logic.  Namely, if we assert ~(P→Q) we are asserting that P is true and Q is false.

We can make similar truth tables for the other sentences.

P	Q	(~	P	→	Q)
T	T	F	T	T	T
T	F	F	T	T	F
F	T	T	F	T	T
F	F	T	F	F	F

How did we make this table?  The sentence (~P→Q) is a conditional with two parts, ~P and Q.  Because Q is atomic, it will be on the left side.  We distribute the truth value assignment, then determine the truth value of the target sentence, using our definitions of the connectives, and keeping in mind that we begin with the inner-most parentheses, doing negations before any other connective. In so doing, we determine the truth value of ~P, which is the antecedent of our conditional. We are now in a position to determine the truth value of our target sentence, (~P→Q), which is only false when ~P is true and Q is false.

And:

P	Q	(~	P	→	~	Q
T	T	F	T	T	F	T
T	F	F	T	T	T	F
F	T	T	F	F	F	T
F	F	T	F	T	T	F

Such a truth table is very helpful in determining when sentences are, and are not, equivalent.  We have used the concept of equivalence repeatedly, but have not yet defined it.  We can offer a semantic, and a syntactic, explanation of equivalence.  The semantic notion is relevant here:  we say two sentences Φ and Ψ are “equivalent” or “logically equivalent” when they must have the same truth value.  (For the syntactic concept of equivalence, see section 9.2).  These truth tables show that these three sentences are not equivalent, because it is not the case that they must have the same truth value.  For example, if P and Q are both true, then ~(P→Q) is false but (~P→Q) is true and (~P→~Q) is true.  If P is false and Q is true, then (~P→Q) is true but (~P→~Q) is false.  Thus, each of these sentences is true in some situation where one of the others is false.  No two of them are equivalent.

We should consider an example that uses conjunction, and which can help in some translations.  How should we translate “Not both Steve and Tom will go to Berlin”?  This sentence tells us that it is not the case that both Steve will go to Berlin and Tom will go to Berlin.  The sentence does allow, however, that one of them will go to Berlin.  Thus, let U mean Steve will go to Berlin and V mean Tom will go to Berlin.  Then we should translate this sentence, ~(U&V). We should not translate the sentence (~U&~V).  To see why, consider their truth tables.

U	V	~	(U	&	V)	(~	U	&	~	V)
T	T	F	T	T	T	F	T	F	F	T
T	F	T	T	F	F	F	T	F	T	F
F	T	T	F	F	T	T	F	F	F	T
F	F	T	F	F	F	T	F	T	T	F

We can see that ~(U&V) and (~U&~V) are not equivalent.  Also, note the following.  Both ~(U&V) and (~U&~V) are true if Steve does not go to Berlin and Tom does not go to Berlin.  This is captured in the last row of this truth table, and this is consistent with the meaning of the English sentence.  But, now note:  it is true that not both Steve and Tom will go to Berlin, if Steve goes and Tom does not.  This is captured in the second row of this truth table.  It is true that not both Steve and Tom will go to Berlin, if Steve does not go but Tom does.  This is captured in the third row of this truth table.  In both kinds of cases (in both rows of the truth table), ~(U&V) is true but (~U&~V) is false.  Thus, we can see that ~(U&V) is the correct translation of “Not both Steve and Tom will go to Berlin”.

Let’s consider a more complex sentence that uses all of our connectives so far:  ((P&~Q)→~(P→Q)).  This sentence is a conditional.  The antecedent is a conjunction.  The consequent is a negation.  Here is the truth table, completed.

P	Q	((P	&	~	Q)	→	~	(P	→	Q))
T	T	T	F	F	T	T	F	T	T	T
T	F	T	T	T	F	T	T	T	F	F
F	T	F	F	F	T	T	F	F	T	T
F	F	F	F	T	F	T	F	F	T	F

This sentence has an interesting property:  it cannot be false.  That is not surprising, once we think about what it says.  In English, the sentence says: If P is true and Q is false, then it is not the case that P implies Q.  That must be true:  if it were the case that P implies Q, then if P is true then Q is true.  But the antecedent says P is true and Q is false.

Sentences of the propositional logic that must be true are called “tautologies”.  We will discuss them at length in later chapters.

Let’s consider another conjunction: (P&~P).

Here is the truth table, completed.

P	P	&	~	P
T	T	F	F	T
F	F	F	T	F

This sentence has an interesting property: it cannot be true. That is not surprising, once we think about what it says. In English, the sentence says: P is true and P is false. That must be false.

Sentences of the propositional logic that must be false are called “contradictions”. We will also discuss them at length in later chapters.

Finally, note that we can combine this method for finding the truth conditions for a complex sentence with our method for determining whether an argument is valid using a truth table.  We will need to do this if any of our premises or the conclusion are complex.  Here is an example.  We’ll start with an argument in English:

If whales are mammals, then they have vestigial limbs.  If whales are mammals, then they have a quadrupedal ancestor.  Therefore, if whales are mammals then they have a quadrupedal ancestor and they have vestigial limbs.

We need a translation key.

P:  Whales are mammals.

Q: Whales have have vestigial limbs.

R: Whales have a quadrupedal ancestor.

The argument will then be symbolized as:

Here is a semantic check of the argument.

premise

premise

conclusion

P

Q

R

(P

→

Q)

(P

→

R)

(P

→

(R

&

Q))

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

F

T

T

T

T

F

F

T

F

F

F

T

T

F

T

T

F

F

T

T

T

T

F

T

F

F

T

F

F

T

F

F

T

F

F

T

F

F

F

F

F

T

T

F

T

T

F

T

T

F

T

T

T

T

F

T

F

F

T

T

F

T

F

F

T

F

F

T

F

F

T

F

T

F

F

T

T

F

T

T

F

F

F

F

F

F

T

F

F

T

F

F

T

F

F

F

We have highlighted the rows where the premises are all true.  Note that for these, the conclusion is true.  Thus, in any kind of situation in which all the premises are true, the conclusion is true.  This is equivalent, we have noted, to our definition of valid:  necessarily, if all the premises are true, the conclusion is true.  So this is a valid argument.  The third column of the analyzed sentences (the column for (R&Q)) is there so that we can identify when the conclusion is true.  The conclusion is a conditional, and we needed to know, for each kind of situation, if its antecedent P, and if its consequent (R&Q), are true.  The third column tells us the situations in which the consequent is true.  The stipulations on the left tell us in what kind of situation the antecedent P is true.

5.7 Key Concepts

 

5.8 Exercises

Within this section, you will find two types of problems for the chapter material. Firstly there are interactive exercises that randomly test your knowledge. Secondly, there is a comprehensive list of exercise questions with all answers at the back of the text.