Question

Show that the connective \(\downarrow\), with valuation function \([\varphi \downarrow \psi]=1\) iff \([\varphi]=[\psi \rrbracket=0\), forms a functionally complete set (neither \(\varphi\), nor \(\psi)\).

Short answer

The connective \(\downarrow\) is functionally complete, as it can express NOT, AND, and OR.

Step-by-step solution

Understand the Connective's Truth Table

The connective \(\downarrow\) is also known as the NOR operation. It returns true (1) only when both operands are false (0). A truth table for \(\downarrow\) shows four combinations: if both operands are 0, the result is 1; otherwise, it is 0. This resembles logical negation (NOT) and conjunction (AND) capabilities.

Show Equivalence to Basic Operations

We need to express basic logical operations (NOT, AND, OR) using only \(\downarrow\). - NOT \(\varphi\) = \(\varphi \downarrow \varphi\). - Use De Morgan's law for OR: \(\varphi \lor \psi = eg (eg \varphi \land eg \psi)\). Expressed with \(\downarrow\) as: \((\varphi \downarrow \varphi) \downarrow (\psi \downarrow \psi)\).

Verify Composition for Functionality Completeness

Proving functionality completeness requires expressing any logical statement using \(\downarrow\). Construct these compositions: - For AND \(\varphi \land \psi\), use: \(((\varphi \downarrow \varphi) \downarrow (\psi \downarrow \psi)) \downarrow ((\varphi \downarrow \varphi) \downarrow (\psi \downarrow \psi))\).- Check by composing more complex formulas using these primitive replacements.

Key concepts

Logical Connectives

Logical connectives are building blocks in logic, which help form complex statements by connecting simpler ones. These connectives include operations like AND, OR, and NOT.
They follow simple rules and help us derive truth values in logical statements. When we talk about logic, every statement in logic often uses these connectives to combine propositions into more complex expressions.
For example, with logical connectives, you can say, "If it rains, then I will take an umbrella," forming a single logical statement using connectives. The main logical connectives that people use are typically:

• AND (Conjunction): Both statements must be true for the result to be true.
• OR (Disjunction): If at least one statement is true, the result is true.
• NOT (Negation): It inverts the truth value of a statement.

Understanding these connectives is critical for evaluating logical expressions and establishing the basis for various logical computations.

NOR Operation

The NOR operation, denoted as \(\downarrow\), is a unique logical connective. It functions by returning true only when both of its inputs are false.
Mathematically, the expression \( \varphi \downarrow \psi \) is true if and only if both \( \varphi \) and \( \psi \) are false.This operation is distinctive because it combines both the NOT and OR operations into a single operation.
By nature, it says "not both are true," meaning if any of the inputs is true, the result is false. A few examples include:

• If both statements are false: \((0 \downarrow 0) = 1\)
• If one statement is true and the other is false: \((0 \downarrow 1) = 0\)
• If both statements are true: \((1 \downarrow 1) = 0\)

Understanding the NOR operation is crucial because it can represent all other logical operations alone, showcasing its functional completeness.

Truth Table

A truth table is an essential tool in logic that displays the output of a logical operation for all possible inputs. It helps us understand how a logical connective works.
Think of it like a map or chart that records what happens for each possible input combination.For two variables, \( \varphi \) and \( \psi \), their truth table using the operation \( \downarrow \) (NOR) looks like this:

• \( \varphi = 0, \psi = 0 \rightarrow (\varphi \downarrow \psi) = 1 \)
• \( \varphi = 0, \psi = 1 \rightarrow (\varphi \downarrow \psi) = 0 \)
• \( \varphi = 1, \psi = 0 \rightarrow (\varphi \downarrow \psi) = 0 \)
• \( \varphi = 1, \psi = 1 \rightarrow (\varphi \downarrow \psi) = 0 \)

The truth table demonstrates how powerful the NOR operation is, by being able to replicate other logical operations. This makes it a vital component of functional completeness.

De Morgan's Law

De Morgan's Law is a fascinating concept in logic, providing a bridge between different logical operations. It allows us to transform conjunctions (AND) into disjunctions (OR), and vice versa, using negations.
These laws help simplify logical expressions and can be very powerful when dealing with complex logical constructs. There are two main De Morgan's Laws:

• The negation of a conjunction is equal to the disjunction of the negations: \( eg(\varphi \land \psi) = eg\varphi \lor eg\psi \)
• The negation of a disjunction is equal to the conjunction of the negations: \( eg(\varphi \lor \psi) = eg\varphi \land eg\psi \)

Using De Morgan's Law with NOR operations, we can express OR operations in terms of NOR by knowing that OR is the negation of an AND, highlighting the functional completeness of the NOR operation. Understanding these laws can significantly deepen one’s grasp of logic and computational theory.