A Crash Course on Propositional Calculus
Hi guys! Sorry for the longggg delay :D I was busy with schoolwork, and I encountered many catastrophes. So at least enjoy one of the fruits of my labor (tree)! It is my article, required for our Math class magazine :D Enjoy!
Content
A Crash Course in Propositional Calculus
Logic is used in everyday life, from simple deduction that your friend lied to you to the way computers work by means of Boolean logic. Logic is especially used by scientists because it deals with the principles and criteria of validity of inference and demonstration. It helps decide which statements are true, and which are valid/invalid arguments. Mathematical reasoning is applied in law, philosophy, computer science, and many more. It began as a search for universal truths that were unquestionable. But logic alone cannot work without the help of statements called propositions. Propositions are declarative sentences that are either true or false, but not both. Truth value is the truth or falsehood of a proposition. Propositional logic makes use of declarative sentences, while Symbolic logic makes use of variables and symbols to determine the truth of a sentence (i.e. Algebra). Which do you think are propositions? Manny Pacquiao beat Ricky Hatton in two rounds 1 + 1 = 11 When will we get our snacks? 5>6 This sentence is false.
Propositional Calculus works by means of compound propositions. They are made up of propositions linked together by logical operators. These include: Negation (¬) o o “Maria is a schoolgirl.” The negation of this is either “Maria is not a schoolgirl” or “Maria is schoolboy”. Negates/inverses the truth value of a statement.
Conjunction (^) o o The conjunction of “Adam is athletic.” and “Barbie is athletic.” is the sentence “Adam is athletic, and Barbie is also athletic.” The compound proposition is only true if both are true.
Disjunction (v) o o If p and q are propositions, then the disjunction of the two is p v q (p or q; either) True if at least one component is true.
o
Note that p or q is different from “either p or q”.
Conditionals (→) o o o o The usual, if p, then q. Also called implication. Only false when p is true and q is false. Conditionals are independent of cause-and-effect between hypothesis and conclusion. The statement “If the ocean is red, the moon is made of cheese.” is logically true, even if it is in reality false. So beware of politicians and lawyers!
Biconditionals (↔) o o p if and only if q. True if both propositions have the same truth values.
Like in PEMDAS, there is also a hierarchy of evaluation: parenthesis, negation, conjunction, disjunction, conditionals and lastly biconditionals. For example: p T T F F q T F T F p→q T F T T (p→q) ^ p T F T T ((p→q) ^ p) → q T T T F
Truth table for: ((p→q) ^ p) → q
Therefore, the statement “If you have a current password, then you can log onto the network. You have a current password. Therefore, you can log onto the network. ” is false only if you don’t have a current password and you can’t log onto the network.
If the truth value of the whole statement is always false, it is called a contradiction. A contradiction is a compound proposition that is always false. On the other hand, a tautology is a compound proposition that is always true, like the compound proposition p v ¬p. Lastly, a compound proposition is a contingency if it is neither a tautology nor a contradiction.
If two compound propositions have the same truth values, then they are called logically equivalent (p ≡ q). There are many laws on logical equivalences to help simplify a compound proposition. Some of them include: p^T↔p pvF↔p Domination Laws: pvT↔T p^F↔F Idempotent Laws: pvp↔p p^p↔p Double Negation Law: ¬(¬p) ↔ p Commutative Laws: pvq↔qvp p^q↔q^p Associative Laws: (p v q) v r ↔ p v (q v r) (p ^ q) ^ r ↔ p ^ (q ^ r) Distributive Laws: p v (q ^ r) ↔ (p v q) ^ (p v r) p ^ (q v r) ↔ (p ^ q) v (p ^ r) Identity Laws:
Propositional calculus is used in deciding whether an argument is valid or invalid. An argument is a sequence of compound propositions that end with a conclusion. There are also compound propositions that are simplified for easier validation of an argument. The arguments are covered in predicate calculus, which makes common quantifiers like existential Ex and universal Ax quantifiers.
Last challenge! Show that the statements “It is not sunny this afternoon and it is colder than yesterday.”, “We will go swimming only if it is sunny.”, “If we do not go swimming, then we will take a trip.” and “If we take a trip, then we will be home by sunset.” lead to the conclusion “We will be home by sunset.”
Remember, do not believe every argument you come across. Many are either invalid, untrue or both. These rules of logic give the foundations of mathematics!
Bibliography:
Barker, S. (1980). The Elements of Logic. USA: Mc-Graw Hill, Inc. Barnett, R., Byleen, K. & Ziegler, M. (2008). College Mathematics For Business, Economics, Life Science and Social Science. Upper Saddle River, New Jersey: Pearson Education, Inc. Carney, J. (1970). Introduction to Symbolic Logic. Englewood Cliffs, New Jersey: Prentice-Hall Inc.
Answers :D Only the first, second and fourth are propositions (their respective values are true, false and false). The third and fifth statements are not propositions because they are neither true nor false. Let the statement Sunny be q; Colder than yesterday p; Go swimming x; Take a trip y; and Home by sunset z. Then the problem will have the following form: ¬q ^ p, x ↔ q, ¬x → y, and y → z. Therefore, ¬x → z. But, x ↔ q has the same truth value as ¬x ↔ ¬q. So, ¬q → z. But ¬q must be with p (¬q ^ p), which leads to ¬q ^ p → z.
Logic is used in everyday life, from simple deduction that your friend lied to you to the way computers work by means of Boolean logic. Logic is especially used by scientists because it deals with the principles and criteria of validity of inference and demonstration. It helps decide which statements are true, and which are valid/invalid arguments. Mathematical reasoning is applied in law, philosophy, computer science, and many more. It began as a search for universal truths that were unquestionable. But logic alone cannot work without the help of statements called propositions. Propositions are declarative sentences that are either true or false, but not both. Truth value is the truth or falsehood of a proposition. Propositional logic makes use of declarative sentences, while Symbolic logic makes use of variables and symbols to determine the truth of a sentence (i.e. Algebra). Which do you think are propositions? Manny Pacquiao beat Ricky Hatton in two rounds 1 + 1 = 11 When will we get our snacks? 5>6 This sentence is false.
Propositional Calculus works by means of compound propositions. They are made up of propositions linked together by logical operators. These include: Negation (¬) o o “Maria is a schoolgirl.” The negation of this is either “Maria is not a schoolgirl” or “Maria is schoolboy”. Negates/inverses the truth value of a statement.
Conjunction (^) o o The conjunction of “Adam is athletic.” and “Barbie is athletic.” is the sentence “Adam is athletic, and Barbie is also athletic.” The compound proposition is only true if both are true.
Disjunction (v) o o If p and q are propositions, then the disjunction of the two is p v q (p or q; either) True if at least one component is true.
o
Note that p or q is different from “either p or q”.
Conditionals (→) o o o o The usual, if p, then q. Also called implication. Only false when p is true and q is false. Conditionals are independent of cause-and-effect between hypothesis and conclusion. The statement “If the ocean is red, the moon is made of cheese.” is logically true, even if it is in reality false. So beware of politicians and lawyers!
Biconditionals (↔) o o p if and only if q. True if both propositions have the same truth values.
Like in PEMDAS, there is also a hierarchy of evaluation: parenthesis, negation, conjunction, disjunction, conditionals and lastly biconditionals. For example: p T T F F q T F T F p→q T F T T (p→q) ^ p T F T T ((p→q) ^ p) → q T T T F
Truth table for: ((p→q) ^ p) → q
Therefore, the statement “If you have a current password, then you can log onto the network. You have a current password. Therefore, you can log onto the network. ” is false only if you don’t have a current password and you can’t log onto the network.
If the truth value of the whole statement is always false, it is called a contradiction. A contradiction is a compound proposition that is always false. On the other hand, a tautology is a compound proposition that is always true, like the compound proposition p v ¬p. Lastly, a compound proposition is a contingency if it is neither a tautology nor a contradiction.
If two compound propositions have the same truth values, then they are called logically equivalent (p ≡ q). There are many laws on logical equivalences to help simplify a compound proposition. Some of them include: p^T↔p pvF↔p Domination Laws: pvT↔T p^F↔F Idempotent Laws: pvp↔p p^p↔p Double Negation Law: ¬(¬p) ↔ p Commutative Laws: pvq↔qvp p^q↔q^p Associative Laws: (p v q) v r ↔ p v (q v r) (p ^ q) ^ r ↔ p ^ (q ^ r) Distributive Laws: p v (q ^ r) ↔ (p v q) ^ (p v r) p ^ (q v r) ↔ (p ^ q) v (p ^ r) Identity Laws:
Propositional calculus is used in deciding whether an argument is valid or invalid. An argument is a sequence of compound propositions that end with a conclusion. There are also compound propositions that are simplified for easier validation of an argument. The arguments are covered in predicate calculus, which makes common quantifiers like existential Ex and universal Ax quantifiers.
Last challenge! Show that the statements “It is not sunny this afternoon and it is colder than yesterday.”, “We will go swimming only if it is sunny.”, “If we do not go swimming, then we will take a trip.” and “If we take a trip, then we will be home by sunset.” lead to the conclusion “We will be home by sunset.”
Remember, do not believe every argument you come across. Many are either invalid, untrue or both. These rules of logic give the foundations of mathematics!
Bibliography:
Barker, S. (1980). The Elements of Logic. USA: Mc-Graw Hill, Inc. Barnett, R., Byleen, K. & Ziegler, M. (2008). College Mathematics For Business, Economics, Life Science and Social Science. Upper Saddle River, New Jersey: Pearson Education, Inc. Carney, J. (1970). Introduction to Symbolic Logic. Englewood Cliffs, New Jersey: Prentice-Hall Inc.
Answers :D Only the first, second and fourth are propositions (their respective values are true, false and false). The third and fifth statements are not propositions because they are neither true nor false. Let the statement Sunny be q; Colder than yesterday p; Go swimming x; Take a trip y; and Home by sunset z. Then the problem will have the following form: ¬q ^ p, x ↔ q, ¬x → y, and y → z. Therefore, ¬x → z. But, x ↔ q has the same truth value as ¬x ↔ ¬q. So, ¬q → z. But ¬q must be with p (¬q ^ p), which leads to ¬q ^ p → z.
