Propositional proof system
A propositional proof system is like a special set of instructions that helps us understand and prove different things about certain statements. It's kind of like a game, where we have certain rules and we follow those rules to show that a statement is true or false.
In this game, the statements we want to prove are called propositions. These propositions can be simple or complex. Simple propositions are like building blocks, while complex propositions are made up of smaller propositions combined together. It's like playing with Legos - you can make bigger things by putting smaller pieces together.
Now, in this game, we have a few basic rules that we follow. These rules are a bit like the rules of a puzzle game. We need to take a look at the propositions we have and see if we can use these rules to prove if they are true or false.
One rule we have is called the "And Introduction" rule. This rule says that if we know that two propositions are true, then we can say that the proposition formed by combining them with an "and" is also true. It's like saying that if I have an apple and a banana, then I can say that I have an apple and a banana together.
Another rule we have is called the "And Elimination" rule. This rule says that if we know that a proposition made up of an "and" is true, then we can say that the individual propositions that make it up are also true. It's like saying that if I have an apple and a banana together, then I can say that I have an apple and I have a banana separately.
We also have rules for "Or Introduction" and "Or Elimination". These rules are like the "And" rules, but they work with the word "or" instead. So if we know that either proposition is true, we can say that the whole proposition with the "or" is also true. And if we know that a proposition with "or" is true, we can say that at least one of the individual propositions is also true.
There are also rules for "Not" and "Implication". These are a bit trickier, but they help us understand how negatives and relationships between propositions work.
In this game, the goal is to take our initial propositions and use these rules to build a step-by-step proof. It's like playing with building blocks and putting them together in a specific way to create something bigger and more complex.
And if we can use these rules correctly and follow all the steps, then we can say that our initial proposition is true or false based on what we have shown. It's like solving a puzzle and finding the right pieces that fit together to make a complete picture.
So a propositional proof system is like a special set of instructions or a game that helps us prove if certain statements are true or false by following specific rules and building a step-by-step proof using those rules.
In this game, the statements we want to prove are called propositions. These propositions can be simple or complex. Simple propositions are like building blocks, while complex propositions are made up of smaller propositions combined together. It's like playing with Legos - you can make bigger things by putting smaller pieces together.
Now, in this game, we have a few basic rules that we follow. These rules are a bit like the rules of a puzzle game. We need to take a look at the propositions we have and see if we can use these rules to prove if they are true or false.
One rule we have is called the "And Introduction" rule. This rule says that if we know that two propositions are true, then we can say that the proposition formed by combining them with an "and" is also true. It's like saying that if I have an apple and a banana, then I can say that I have an apple and a banana together.
Another rule we have is called the "And Elimination" rule. This rule says that if we know that a proposition made up of an "and" is true, then we can say that the individual propositions that make it up are also true. It's like saying that if I have an apple and a banana together, then I can say that I have an apple and I have a banana separately.
We also have rules for "Or Introduction" and "Or Elimination". These rules are like the "And" rules, but they work with the word "or" instead. So if we know that either proposition is true, we can say that the whole proposition with the "or" is also true. And if we know that a proposition with "or" is true, we can say that at least one of the individual propositions is also true.
There are also rules for "Not" and "Implication". These are a bit trickier, but they help us understand how negatives and relationships between propositions work.
In this game, the goal is to take our initial propositions and use these rules to build a step-by-step proof. It's like playing with building blocks and putting them together in a specific way to create something bigger and more complex.
And if we can use these rules correctly and follow all the steps, then we can say that our initial proposition is true or false based on what we have shown. It's like solving a puzzle and finding the right pieces that fit together to make a complete picture.
So a propositional proof system is like a special set of instructions or a game that helps us prove if certain statements are true or false by following specific rules and building a step-by-step proof using those rules.