## Translating English into Propositional Logic

# Goals

- Be able to identify simple propositions in complex sentences.
- Be able to translate English sentences into propositional logic.
- Complete the short quiz below.
- Complete the extra practice exercises in the PDF below.

### Identifying Simple Propositions

Simple propositions can be found by looking for facts or opinions in sentences. Most facts consist of a subject and a predicate. Facts are who or what we're talking about, and predicates are actions or states of the subject.

- Mary is here.
- Four is greater than two.
- All positive integers can be written as a multiple of 1.

When we identify a subject and a predicate, we can assign the pair a proposition such as P, Q, R, etc.

A simple proposition is always a positive statement. If we see words such as "not", or "don't", or "isn't", it's a complex proposition and needs a connective.

### Creating Complex Propositions

We can create complex well-formed formula (wffs), or complex propositions, by connecting two simple propositions together using an operator, or attaching a negation to a proposition. The following are operators that we have at our disposal:

**Negation**:*not, it is not the case that***Conjunction**:*and, but, although, yet, even though***Disjunction**:*or, unless***Conditional**:*if... then..., when... then...,***Biconditional***: if and only if, just in case that*

For example, the following sentence has three different propositions (underlined) joined by connectives.

- If
__x is an even number__, then__x can be written as 2a for any a__and__x is divisible by two__.

If we assign a letter to each proposition, we get the form such as:

- If E, then A and T.

We use different symbols to stand for different connectives. Each connective will behave differently.

**Negation**: ¬**Conjunction**: ∧**Disjunction**: ∨**Conditional**:**Biconditional***:***↔**

So, translating our sentence from before, we get:

- E → (A ∧ T)

### Understanding the Conditional

The conditional has a pattern to what goes before the arrow and what goes after. What comes before the arrow is called the *antecedent*, and what comes after is called the *consequent*.

- "if P then Q" translates to P → Q.

Even if we remove the word "then" or switch the order, the translation remains the same.

- "Q, if P" translates to P → Q.

We have terms for each type of conditional. Suppose that we have our original conditional P → Q.

**Inverse**: ¬P → ¬Q**Converse**: Q → P**Contrapositive**: ¬Q → ¬P

The contrapositive is equivalent to the original conditional, and the inverse and converse are equivalent to each other.

## Try Some Extra Practice!

If you can solve these practice problems, you've got it! Solutions with explanations are included.

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