A propositional logic question

In summary: Therefore, using modus ponens, we can prove that P implies P, using the axiom schemas A1 and A2. In summary, axiom schemas in predicate calculus are chosen to represent logically valid formulas and are used with modus ponens to prove theorems and make inferences.
  • #1
Cinitiator
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Actually, I have several questions:
1) Why are axiom schemas the way they are? What do they represent? I know that infinitely many axioms can be written using the axiom schema form. However, what's the formal definition of axioms in predicate calculus? I've heard that the formal definition of axioms is any wff which has the axiom schema form. If that's the case, what's so special about some wffs which can have infinitely many forms? Do they have any distinctive properties at all?

2) Why and how are they used for proving theorems / making other inferences?

3) How is modus ponens used with such axiom schemas to prove theorems?
 
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  • #2
Cinitiator said:
Actually, I have several questions:
1) Why are axiom schemas the way they are? What do they represent? I know that infinitely many axioms can be written using the axiom schema form. However, what's the formal definition of axioms in predicate calculus? I've heard that the formal definition of axioms is any wff which has the axiom schema form. If that's the case, what's so special about some wffs which can have infinitely many forms? Do they have any distinctive properties at all?

2) Why and how are they used for proving theorems / making other inferences?

3) How is modus ponens used with such axiom schemas to prove theorems?

1)-2) Axioms can be different in different formal theories. But in all theories, the axioms of predicate calculus must be chosen so that the set of theorems (which can be derived from the axioms by the rules) is the same as the set of logically valid formulas. It we restrict ourselves to propositional calculus, logically valid formula is the same as tautology, so the axioms of propositional calculus are chosen so that the theorems are exactly the tautologies.

3) For example, suppose we have the following axiom schemas (among others):

A1. P->(Q->P).
A2. (P->(Q->R))->((P->Q)->(P->R)).

Then, let us derive the theorem P->P:

1. (P->((P->P)->P))->((P->(P->P))->(P->P)). Instances of A2.
2. P->((P->P)->P). Instances of A1.
3. (P->(P->P))->(P->P). MP: 1,2.
4. P->(P->P). Instances of A1.
5. P->P. MP: 3,4.
 

1. What is propositional logic?

Propositional logic is a type of formal logic that deals with the relationships between propositions or statements. It uses symbols and rules to represent and manipulate logical statements, making it a useful tool for analyzing arguments and reasoning.

2. How is propositional logic different from other types of logic?

Unlike other types of logic, propositional logic does not deal with the content or meaning of propositions, but rather focuses on their logical structure and relationships. It is also limited to dealing with simple statements and does not take into account more complex logical concepts such as quantifiers or variables.

3. What are the basic components of propositional logic?

The basic components of propositional logic include logical symbols, logical connectives, and propositions. Logical symbols, such as "p" and "q", represent propositions, while logical connectives, such as "and", "or", and "not", are used to combine propositions to form more complex statements.

4. How is propositional logic used in scientific research?

Propositional logic is used in scientific research as a tool for analyzing arguments and reasoning. It is particularly useful in identifying logical fallacies and evaluating the validity of scientific theories and hypotheses.

5. What are some real-world applications of propositional logic?

Propositional logic has a wide range of real-world applications, including computer programming, artificial intelligence, and natural language processing. It is also used in fields such as mathematics, philosophy, and linguistics to analyze and represent logical relationships between statements.

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