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DA5. Detectives and Logic

Statement

Information overload is a common problem these days and understanding the basic principles of logical reasoning is the only way out. This is especially important in fields like law enforcement, where detectives and investigators must use logic and reasoning to solve complex cases.

Imagine you are a detective trying to solve a case. You have gathered various pieces of evidence, including witness statements, CCTV footage, and physical evidence. Now your job is to put them together to form a cohesive argument that will lead you to the truth.

In this assignment, you will apply the concepts of logical connectives and propositional logic to help you reason through evidence and solve the case.

To start, let’s explore the five logical connectives or operators you have learned: \(\lor\), \(\land\), \(\rightarrow\), \(\leftrightarrow\), and \(\neg\). These connectives are essential in constructing complex arguments and reasoning through evidence. For example, you might use the connective (and) to link two pieces of evidence together, indicating that both must be true for your argument to be valid.

Tasks:

  1. Explain each of the logical connectives in words and provide self-made examples related to a detective case. Represent them symbolically and discuss in detail the truth or false of the compound propositions. Additionally, explain the hierarchy of these operators if they are included in parentheses.
  2. Consider any conditional proposition of your choice related to a detective case and write the Converse, Contrapositive, and Inverse to the proposition. Represent all of them symbolically, and determine whether the statements or propositions obtained are true or false.

Solution

Task 1

Let’s differentiate between two types of logical connectives, namely, Unary and Binary. The unary connectives only take one argument, whereas the binary connectives take two arguments. The unary connective is the negation (\(\neg\)), and the binary connectives are the conjunction (\(\land\)), disjunction (\(\lor\)), conditional (\(\rightarrow\)), and biconditional (\(\leftrightarrow\)).

  • The logical Negation (not, \(\neg\)):
    • It is the only unary connective in the list; it takes an argument and flip its truth value.
    • The negation is true, only if the original argument is false, and vice versa.
    • Example: let \(p\) is the statement: “The suspect was at the mall at 5:00 PM.” Then, \(\neg p\) is the statement: “The suspect was not at the mall at 5:00 PM.”
    • Note, the statement \(s\): “The suspect was at gaz station at 5:00 PM.” is not the negation of the statement \(p\), however, it may be equivalent to \(\neg p\).
  • The logical Conjunction (and, \(\land\)):
    • It is a binary connective that takes two arguments and returns true if both arguments are true, and false otherwise.
    • Example: let \(p\) is the statement: “The suspect was at the mall at 5:00 PM.” and \(q\) is the statement: “The mall was open at 5:00 PM.” Then, \(p \land q\) is the statement: “The suspect was at the mall and the mall was open at 5:00 PM.”
  • The logical Disjunction (or, \(\lor\)):
    • It is a binary connective that takes two arguments and returns true if at least one of the arguments is true, and false otherwise.
    • Example: let \(p\) is the statement: “The suspect was at the mall at 5:00 PM.” and \(q\) is the statement: “The suspect was at the gaz station at 5:00 PM.” Then, \(p \lor q\) is the statement: “The suspect was either at the mall or at the gaz station at 5:00 PM.”
  • The logical Conditional (if-then, \(\rightarrow\)):
    • It is a binary connective that takes two arguments and returns false if the first argument is true and the second argument is false, and true otherwise.
    • Example: let \(p\) is the statement: “The suspect was at the mall at 5:00 PM.” and \(q\) is the statement: “The suspect was not at the gaz station at 5:00 PM” Then, \(p \rightarrow q\) is the statement: “If the suspect was not at the gaz station at 5:00 PM, then he must be in the mall”.
  • The logical Biconditional (if and only if, \(\leftrightarrow\)):
    • It is a binary connective that takes two arguments and returns true if both arguments have the same truth value, and false otherwise.
    • Example: let \(p\) is the statement: “The suspect was at the mall at 5:00 PM.” and \(q\) is the statement: “The mall was open at 5:00 PM” Then, \(p \leftrightarrow q\) is the statement: “The suspect was at the mall at 5:00 PM if and only if the mall was open.”

Task 2

Let’s assume

  • \(p\) is the statement: “The suspect was not at the gaz station at 5:00 PM”
  • \(q\) is the statement: “The suspect was at the mall at 5:00 PM.”
  • The conditional statement is \(p \rightarrow q\) which is: “If the suspect was not at the gaz station at 5:00 PM, then he must be in the mall”.

For the example above we need more data to determine the truth value of statements; but for simplicity we will assume that there are only two places the suspect could be at 5:00 PM, the mall or the gaz station (1).

  • The Converse is \(q \rightarrow p\) which is: “If the suspect was at the mall, then he was not at the gaz station”. Considering the assumption (1), the converse is true.
  • The Contrapositive is \(\neg q \rightarrow \neg p\) which is: “If the suspect was not at the mall, then he must not be at the gaz station”. Considering assumption (1), the contrapositive is false.
  • The Inverse is \(\neg p \rightarrow \neg q\) which is: “If the suspect was at the gaz station at 5:00 PM, then he must not be in the mall”. Considering assumption (1), the inverse is true.