How to Complete a High School Logical Proof
Within most intermediate algebra classes in a high school setting, you'll be introduced to a type of proof that is elementary to prove other types of proofs (Euclidean Proof). The proof is called a Logical Proof. This article will explain how you can formulate such a proof.
Steps
- 1Learn your logical connectives (including the negation statement). You should familiarize yourself with the truth tables and definitions of negations, conjunctions, disjunctions, conditionals and biconditionals.
- A statement and it's negation have opposite truth values.
- The conjunction p AND q is true only when both parts are true.
- The disjunction p OR q is true, when any part of the compound sentence is true.
- The conditional if p then q or p implies q is false, when a true hypothesis p, leads to a false conclusion q.
- The biconditional p if and only if q is true when p and q are both true or both false.
- 2Learn how to interpret the signs used in what can later be called a logical proof.
- The ~ is the sign for a negation.
- The ^ sign is the sign for a conjunction (meaning AND).
- The "v" sign is the sign for a disjunction (meaning OR).
- An arrow pointing to the right is the sign for a conditional (->, meaning if/then).
- An arrow pointing in both directions is the sign for a biconditional (<->, meaning if and only if).
- 3Define for yourself, of the definition of a tautology. A tautology is a compound sentence which is always true.
- 4Realize that all statements made in a logical proof will end up becoming a logically-equivalent statement (or a statement that is always assumed to be true no matter what condition it ends up being at the end).
- 5Realize that at least two given premises make up a good logical argument, that can lead to one and only one conclusion.
- 6Learn the many laws that can be used to prove the answer to a given proof.
- The Law of Detachment states that when two given premises are true, one a conditional and the other the hypothesis of the conditional, it follows that the conclusion of the conditional is true.
- This law is about the most simplest law that can be used in a proof.
- This law is sometimes called the Law of Modus Ponens.
- The Law of the Contrapositive states that when a conditional premise is proved to be true, it's contrapositive of the premise is also true.
- Contrapositives can be formed by taking any conditional statement, negating it's premises AND negating it's conclusion then reversing the roles of the negated statements.
- The Law of Modus Tollens state that when two premises are true (on conditional and the other a negation of the conclusion of the conditional), what follows will be the negation of the hypothesis of the conditional being true.
- Applying this law, is equivalent to applying both the Law of the Contrapositive and the Law of Detachment as a set of premises.
- The Law of Syllogism (or sometimes called the Chain Rule/Law) states that when two given premises are true, conditionals where one is the consequent of the other, it follows that a conclusion is formed being a conditional using the antecedent of the first and the consequent of the second premise.
- Get used to this premise. You'll use it a lot in life, without even second-guessing yourself.
- The Law of Disjunctive Inference states that when 2 given premises are true, one a disjunction and the other the negation of one of the items, it follows that the other disjunct is true.
- The Law of Double Negation states that the negation of a negation rules a positive statement, and that these are logically-equivalent statements.
- In most books and on most tests, this is not a required step/reason. It can be deduced. But for simplicity, most places want you to deduce this fact quietly without showing this step in the proof.
- De-Morgans Laws have two following statements that can be deduced.
- The negation of a conjunction of two statements is equivalent to the disjunction of the negation of each of the two statements.
- The negation of a disjunction of two statements is equivalent to the conjunction of the negation of each of the two statements.
- The Law of Simplification states that when a single conjunctive premise is true, it follows that each of the individual conjuncts must be true.
- The Law of Conjunction states that when two premises are true, it follows that the conjunction of these premises is true.
- The Law of Disjunctive Addition states that when a simple premise is true, it follows that any disjunction that has this premise as one of its disjuncts must be true.
- The Law of Detachment states that when two given premises are true, one a conditional and the other the hypothesis of the conditional, it follows that the conclusion of the conditional is true.
- 7Produce a two column format. One column will be called "Statements" and the other should be called "Reasons".
- 8Look at your proof-premises you must prove.
- 9Write down each of the two given statements. If there are more than two given statements, you'll need to fill in all remaining Reasons in the proof.
- 10Write down for these given reasons (in the Reasons column), the reason for writing down these items. Therefore, both lines should be represented as "Given". This a mandated step to prove that you can later deduce others and that you understand what could follow.
- Sometimes, three dots in a triangular pattern signify the conclusion you want to prove.
- 11Use each law as you read about earlier, to try to deduce a new fact that can lead you to a final answer. Sometimes, proofs of high school nature shouldn't require more than two to three laws, however, if needed, know your laws.
- 12Write down the law you used to deduce the fact as the Reason given in the Reasons column. You shouldn't need to write down it's description, but the name of the law if sufficient.
- 13Understand the definition of what a valid conclusion is. A valid conclusion is a true statement that is deduced from a set of true premises by using the laws of reasoning.
- 14Make sure that you arrive at the same answer the question proof asked you to arrive at. Otherwise, you will have not given enough proof that you've deduced the right products.
Tips
- If you are still completely baffled, ask your teacher for help.
- Sometimes, after providing two givens being true because of one law, you'll sometimes be given a third or more. Use the statements above and write down the additional givens in order, until all Givens are used up in subsequent steps.
- Throughout the time you are dealing with a logical proof, write down all your laws or note-cards in plain-terms, and mathematical terms. Display these cards within eyesight distance and use them to your advantage.
Warnings
- For most state tests, and for most books who number their rules, most states don't allow the student (for any credit) to write down just the numerical number of the rule.
- Be careful if you ever arrive at what could become an invalid argument (where a set of premises don't always end up being true with two given true statements.
Things You'll Need
- some math background in high-school logic (Algebra)
- ability to formulate a logical truth table for premises
- pen/pencil
- straight-edge
- listing of laws/reasons
- set of premises to prove
Sources and Citations
- Keenan, Edward P., Isidore Dressler, and Ann Gantert Xavier. Integrated Mathematics: Course II. New York, NY. Amsco School Publications, 1999. Print. (ISBN 9781567655155)
- Houghton Mifflin Unified Mathematics Book 2. Boston: Houghton Mifflin, 1985. Print. (ISBN 0395360862)
Article Info
Categories: Mathematics