Chapter 1: Introduction to Truth Tables

In this Python tutorial, we will explore how to construct a truth table generator specifically for Propositional Logic (PL). If you're not well-versed in PL, it might be beneficial to first familiarize yourself with the basics by reading "How To Do Propositional Logic". Additionally, a fundamental understanding of Python programming is required to follow along with this guide.

Our primary objective is to develop a Python program that takes a compound proposition, such as "P ∧ Q", as input and produces the corresponding truth table.

Chapter 2: Writing Functions for Logical Connectives

To generate truth tables, our program must accurately determine the truth value of a given compound proposition. For example, the truth value of "P ∧ Q" is contingent upon the individual truth values of P and Q.

The initial implementation of a function to ascertain the truth value of a conjunction might look like this:

def conjunction(p, q, truthValues):

return truthValues[p] and truthValues[q]

Here, truthValues is a dictionary that contains the truth values for P and Q. When we call conjunction("P", "Q", truthValues), it will return False under certain conditions. However, since a conjunct can be more than just a variable (e.g., "(P ∨ R) ∧ Q"), we need a more robust function that can handle complex expressions.

An improved function for parsing conjunctions would look like this:

def parseConjunction(expr, truthValues):

# Logic to parse conjunctions

For disjunctions, the approach is nearly identical, simply substituting "and" with "or".

Next, let’s examine how to handle negations:

def parseNegation(expr, truthValues):

return not truthValues[expr[1:]]

Python lacks a direct keyword for conditionals, but note that "P → Q" is logically equivalent to "¬P ∨ Q". Thus, we can implement parseConditional like so:

def parseConditional(p, q, truthValues):

return not truthValues[p] or truthValues[q]

For biconditionals, which check if two propositions share the same truth value, we can define parseBiconditional as follows:

def parseBiconditional(p, q, truthValues):

return truthValues[p] == truthValues[q]

Chapter 3: Parsing Propositions

Now, let's delve into the primary function that evaluates the truth of a proposition, parseProposition().

This function accepts a proposition as a string (e.g., "(P ∨ (Q ∧ ¬R)) ↔ P") and a dictionary of truth values for the involved variables (P, Q, R) and returns the truth value of the proposition.

To achieve this, the function examines the proposition character by character to identify its components—whether they are negations, disjunctions, conjunctions, and so on. For instance, in the expression "(P ∨ (Q ∧ ¬R)) ↔ P", the biconditional connective "↔" is the outermost connective.

Here’s a high-level view of parseProposition():

def parseProposition(prop, truthValues):

# Logic to parse and evaluate the proposition

To ensure well-formed propositions, we utilize a helper function called isWellFormed().

def isWellFormed(prop):

# Logic to validate the structure of the proposition

This function checks for unbalanced parentheses and verifies that each opening parenthesis has a corresponding closing one.

...

Now, let's explore how to create the truth table itself with the writeTruthTable() function.

Chapter 4: Implementing the Truth Table Function

The writeTruthTable() function generates a truth table for any compound proposition. It initializes truth values for all variables and systematically alters them to produce every possible combination.

The full code for the truth table generator is as follows:

def writeTruthTable(prop):

# Logic to create and print the truth table

For example, invoking writeTruthTable("P ∨ (Q → (R ∧ T))") will yield:

P | Q | R | T | P ∨ (Q → (R ∧ T))

T | T | T | T | T

T | T | T | F | T

T | T | F | T | T

...

Thank you for following along! If you have any questions about this tutorial, feel free to leave them in the comments.