Logic and Probability

Boolean Connectives

In any language, there are words 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 propositions into new, more complex propositions:

  • 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 the Boolean connectives. We call "and" and "or" binary connectives since they connect two propositions and "not" a unary connective since it applies to a single proposition. There are other terms that can be used to connect two statements to form a new, more complex statement. What are some examples?


Some connectives are synonymous with "and", "or", or "not". For instance, "It is raining, but I have an umbrella" means the same thing as "It is raining, and I have an umbrella".

Challenge

In natural language, "and" and "but" are used in different contexts. Why do you think that in propositional logic "but" is interpreted as a conjunction (so it is synonymous with "and")?

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 proposition constructed using the connective is completely determined by the truth or falsity of the statements to which the connective is applied.

For example, the truth or falsity of "φ\varphi and ψ\psi" is completely determined by the truth or falsity of φ\varphi and the truth or falsity of ψ\psi (this holds regardless of what statements are substituted for φ\varphi and ψ\psi). For instance, if φ\varphi is true and ψ\psi is false, then "φ\varphi and ψ\psi" is false (no matter what statements are substituted for φ\varphi and ψ\psi).

An example of a connective that is not truth-functional is "because". The truth of "φ\varphi because ψ\psi" depends on more than just the truth or falsity of φ\varphi and ψ\psi. First of all, note that "φ\varphi because ψ\psi" is only true when both φ\varphi and ψ\psi 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. Suppose that it is raining and I am standing outside without an umbrella. Then, both "I am wet" and "It is raining" is true. 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.

In the situation described above, 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".

Challenge

A statement with unless" is also considered as conditional. Try rephrasing the following sentence using unless: If it is raining, then I will carry an umbrella. (Hint: you need to use negation.)

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=5x=5, then x+5=10x+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

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

Name

Symbol

Other symbols

conjunction

\wedge

&\&

negation

¬\neg

\sim

material conditional

\rightarrow

\supset, \Rightarrow

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

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

  1. The winners are Lily and Asha.
  1. You will get an A or B in PHIL 171.
  1. I will eat steak, fish, or pasta.
  1. Ann and Bob played chess.