Translating English into Propositional Logic
- 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.