How to Complete a High School Logical Proof
Learn your logical connectives (including the negation statement)., Learn how to interpret the signs used in what can later be called a logical proof., Define for yourself, of the definition of a tautology., Realize that all statements made in a...
Step-by-Step Guide
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Step 1: Learn 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. -
Step 2: Learn 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). , A tautology is a compound sentence which is always 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.
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. , One column will be called "Statements" and the other should be called "Reasons". ,, If there are more than two given statements, you'll need to fill in all remaining Reasons in the proof. , 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. , Sometimes, proofs of high school nature shouldn't require more than two to three laws, however, if needed, know your laws. , You shouldn't need to write down it's description, but the name of the law if sufficient. , A valid conclusion is a true statement that is deduced from a set of true premises by using the laws of reasoning. , Otherwise, you will have not given enough proof that you've deduced the right products. -
Step 3: Define for yourself
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Step 4: of the definition of a tautology.
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Step 5: Realize 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).
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Step 6: Realize that at least two given premises make up a good logical argument
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Step 7: that can lead to one and only one conclusion.
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Step 8: Learn the many laws that can be used to prove the answer to a given proof.
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Step 9: Produce a two column format.
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Step 10: Look at your proof-premises you must prove.
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Step 11: Write down each of the two given statements.
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Step 12: Write down for these given reasons (in the Reasons column)
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Step 13: the reason for writing down these items.
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Step 14: Use each law as you read about earlier
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Step 15: to try to deduce a new fact that can lead you to a final answer.
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Step 16: Write down the law you used to deduce the fact as the Reason given in the Reasons column.
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Step 17: Understand the definition of what a valid conclusion is.
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Step 18: Make sure that you arrive at the same answer the question proof asked you to arrive at.
Detailed Guide
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.
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). , A tautology is a compound sentence which is always 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.
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. , One column will be called "Statements" and the other should be called "Reasons". ,, If there are more than two given statements, you'll need to fill in all remaining Reasons in the proof. , 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. , Sometimes, proofs of high school nature shouldn't require more than two to three laws, however, if needed, know your laws. , You shouldn't need to write down it's description, but the name of the law if sufficient. , A valid conclusion is a true statement that is deduced from a set of true premises by using the laws of reasoning. , Otherwise, you will have not given enough proof that you've deduced the right products.
About the Author
Aaron Ward
Creates helpful guides on creative arts to inspire and educate readers.
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