Boolean Connectives

In any language, there are words or phrases that are used to construct complex statements out of simpler ones. For instance, consider the following two statements:

1. It is cold.
2. College Park is in Maryland.

The words in boldface transform these two statements into new, more complex statements:

• It is not cold.
• It is cold or College Park is in Maryland.
• It is cold and College Park is in Maryland.

The terms "and", "or", and "not" are called Boolean connectives. We call "and" and "or" binary connectives since they connect two statements and "not" a unary connective since it applies to a single statement. Examples of other terms, called connectives, that connect two statements to form a new, more complex statement are:

Term	Example sentence
but	It is raining, but I have an umbrella.
either...or	Either Ann will have chocolate or vanilla ice cream.
if	Ann will get an A in PHIL 171 if she asks questions in class.
only if	Ann will ask questions during class only if the material discussed in the class is interesting.
if...then	If it is raining, then I will bring an umbrella.
unless	I will pick up Lauren from the airport unless Angelie picks her up.
because	I got wet because it is raining.

info

In natural language, "and" and "but" are used in different contexts. However, in propositional logic "but" is is synonymous with "and". For instance, the following two sentences express the same propositions:

1. It is raining, but I have an umbrella.
2. It is raining, and I have an umbrella.

In propositional logic, the crucial assumption is that the connectives are truth-functional:

Truth-Functional Connective

A connective is truth functional when the truth or falsity of a complex statement constructed using the connective is completely determined by the truth or falsity of the statements to which the connective is applied.

So, a connective is truth-functional if the truth or falsity of a complex sentence involving that connective is a function of the truth or falsity of the statements connected by the connective. For example, the truth or falsity of "$X$ and $Y$" is completely determined by the truth or falsity of $X$ and the truth or falsity of $Y$, regardless of what statements are substituted for $X$ and $Y$. For instance, if $X$ is true and $Y$ is false, then "$X$ and $Y$" is false (no matter what statements are substituted for $X$ and $Y$).

An example of a connective that is not truth-functional is "because". The truth of "$X$ because $Y$" depends on more than just the truth or falsity of $X$ and $Y$. First of all, note that "$X$ because $Y$" is only true when both $X$ and $Y$ are true. For example, "I am wet because it is raining" is not true if either I am not wet or it is not raining. Consider the following two complex sentence constructed from these statements and the "because" connective:

1. I am wet because it is raining.
2. It is raining because I am wet.

Suppose that it is raining and I am standing outside without an umbrella. Then, both "I am wet" and "It is raining" is true. In this situation, statement 1 is true, but statement 2 is false. The crucial point is that "because" expresses something about a causal or explanatory connection between the statements that it connects.

Material Conditional

The binary connective "if...then" is called a conditional. Conditionals play an important role in many arguments, especially in mathematical and scientific reasoning. There are many equivalent ways to express the connective "if...then" in English. For instance, the following two sentences express the same proposition:

• If it is raining, then I will carry an umbrella.
• I will carry an umbrella if it is raining.

Another way to express "if...then" is to use the phrase "...only if". For instance, the following two sentences express the same proposition:

• If I am carrying an umbrella, then it is raining.
• I am carrying an umbrella only if it is raining.

It is important not to confuse "...if" with "...only if". For example, the following two sentences mean different things:

1. I will wear a jacket if it is cold outside.
2. I will wear a jacket only if it is cold outside.

Consult this tutorial at the Kahn academy for a very good explanation of the difference between "...if" and "...only if".

When studying propositional logic, it is assumed that the conditional is truth-functional. It is an interesting question whether or not this is a good assumption. Some uses of the conditional are clearly truth-functional, especially in mathematical writing. For instance,

• If $x=5$, then $x+5=10$.
• If the lines are parallel, then the lines do not have a point in common.
  

Other uses of the conditional seem not to be truth-functional. For instance,

• If the match is struck, then it would light.
• If Shakespeare didn't write Hamlet, then someone else would have.

When "if...then" is assumed to be truth-functional we call it the material conditional.

Symbolization

We will use the following symbols to represent the Boolean connectives:

Name	English expression	Symbol
conjunction	and	$\wedge$
disjunction	or	$\vee$
negation	not/it's not the case that...	$\neg$
material conditional	if...then	$\rightarrow$

Using these symbols, we have the following:

• For all statements $X$ and $Y$, we write "$X\wedge Y$" instead of "$X\text{ and } Y$".
• For all statements $X$ and $Y$, we write "$X\vee Y$" instead of "$X\text{ or } Y$".
• For all statements $X$, we write "$\neg X$" instead of "$\text{ not }X$".
• For all statements $X$ and $Y$, we write "$X\rightarrow Y$" instead of "$\text{if }X\text{ then } Y$".

So, the abstract argument discussed in the argument form chapter:

$$X \text{ or } Y, \text{ not }X \Rightarrow Y$$

will be written as:

$$X \vee Y, \neg X \Rightarrow Y.$$

Some logic texts use different symbols for the Boolean connectives. (You may be interested in reading about the history of logical notation.)

Name	Symbol	Alternative symbol
conjunction	$\wedge$	$\&$
negation	$\neg$	$\sim$
material conditional	$\rightarrow$	$\supset$

Be careful not to confuse the symbols '$\rightarrow$' and '$\Rightarrow$'.

In this course, the symbol '$\Rightarrow$' is used to identify the conclusion in an argument, but '$\rightarrow$' is the symbol we are using to represent the material conditional. To illustrate the difference, consider the following:

• $X, Y\rightarrow Z$: This is a list of two statements $X$ and a conditional $Y\rightarrow Z$

• $X, Y \Rightarrow Z$: This is an argument with premises $X$ and $Y$ and conclusion $Z$.

• $X\rightarrow Y\Rightarrow Z$: This is an argument with one premise $X\rightarrow Y$ and conclusion $Z$.

• $X\Rightarrow Y\Rightarrow Z$: This is not well-formed since the conclusion of an argument must be a statement not another argument (similarly for premises: a premise must be a statement not another argument).

Paraphrases

Connectives are used to form complex statements from simpler statements. Often, you will need to paraphrase a sentence to discover the simpler statements that are used to form the complex sentence. For example, consider the sentence:

Sheba and Lauren went to the party.

This sentence can be rewritten as:

Sheba went to the party and Lauren went to the party.

The second sentence makes it clear that the sentence is a conjunction formed from the two statements:

1. Sheba went to the party.
2. Lauren went to the party.

Practice Questions

1. Paraphrase the following sentences to identify the basic statements that form the complex sentence.

   1. The winners are Lily and Asha.
   2. You will get an A or B in PHIL 171.
   3. I will eat steak, fish, or pasta.
   4. Ann didn't order steak.
   5. Ann studied a lot, but enjoyed college.
   6. Ann either ordered steak or she didn't.
   7. At least one of Ann or Bob went to the party.
   8. Ann and Bob played chess.
2. Challenge

   Rephrase the following sentence without unless: I will carry an umbrella unless it is not raining.
3. Challenge

   Rephrase the following sentence without unless: Ann will come to the party unless Bob does.