Lesson STATEMENT LAWS AND FALLACIES
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The basic laws and fallacies you will more then likely encounter in your study of logic are as follows: LAW OF DISJUNCTION LAW OF CONJUNCTION LAW OF NEGATION LAW OF IMPLICATION LAW OF EQUIVALENCE LAW OF DETACHMENT LAW OF SYLLOGISM LAW OF CONTRA-POSITION FALLACY OF THE INVERSE FALLACY OF THE CONVERSE REFERENCES <a href = "http://people.hofstra.edu/Stefan_Waner/RealWorld/logic/logicintro.html" target = "_blank">http://people.hofstra.edu/Stefan_Waner/RealWorld/logic/logicintro.html</a> <a href = "http://regentsprep.org/Regents/math/geometry/GP1/negatecompound.htm" target = "_blank">http://regentsprep.org/Regents/math/geometry/GP1/negatecompound.htm</a> <a href = "http://faculty.matcmadison.edu/alehnen/weblogic/logsent.htm#Tautologies" target = "_blank">http://faculty.matcmadison.edu/alehnen/weblogic/logsent.htm#Tautologies</a> <a href = "http://it.edgecombe.edu/homepage/killorant/MAT140/Module2/EquivalentStatements.pdf" target = "_blank">http://it.edgecombe.edu/homepage/killorant/MAT140/Module2/EquivalentStatements.pdf</a> <a href = "http://college.cengage.com/mathematics/aufmann/excursions/1e/students/clast1_4.pdf" target = "_blank">http://college.cengage.com/mathematics/aufmann/excursions/1e/students/clast1_4.pdf</a> <a href = "http://alameda.peralta.edu/Projects/20430/Logic...%20Philosophy%20010/Deductive%20argument%20forms.htm" target = "_blank">http://alameda.peralta.edu/Projects/20430/Logic...%20Philosophy%20010/Deductive%20argument%20forms.htm</a> <a href = "http://www.sparknotes.com/math/geometry3/logicstatements/section4.rhtml" target = "_blank">http://www.sparknotes.com/math/geometry3/logicstatements/section4.rhtml</a> <a href = "http://www.rwc.uc.edu/koehler/comath/21.html" target = "_blank">http://www.rwc.uc.edu/koehler/comath/21.html</a> <a href = "http://logictutorial.com/" target = "_blank">http://logictutorial.com/</a> TERMINOLOGY Small letter represent simple statements. Large letters represent simple of compound statements. p, q, or r will most likely be used as simple statements. A, B, or C will most likely be used as compound statements. Symbols most often used in statements are: v, ^, ~, ->, <-> v means "or". This is the disjunction symbol. A v B means "A or B". ^ means "and". This is the conjunction symbol. A ^ B means "A and B". ~ means "not". This is the negation symbol. ~a means "not A". -> means "implies" or "if then" This is the implication symbol. A -> B means "A implies B" or "if A then B". <-> means "is equivalent to" or "if and only if" This is the equivalence symbol. A <-> B means "A is equivalent to B" or "A if and only if B". There are laws associated with each of these symbols and they will be discussed along with a discussion of the statements involved with those laws. LAW OF DISJUNCTION The law of disjunction involves the use of the OR statement. The truth table for the OR statement is: <pre> p q (p v q) T T T T F T F T T F F F </pre> The OR statement is only false if p and if q are both false. If either p or q is true, then the OR statement is true. The law of disjunction states: (p v q) ~q therefore p What this means is: If you are given that (p v q) is true and you are then given that q is false, then p must be true. LAW OF CONJUNCTION The law of conjunction involves the use of the AND statement. The truth table for the AND statement is: <pre> p q (p ^ q) T T T T F F F T F F F F </pre> The AND statement is only true if p and q are both true. If either p or q is false, then the AND statement is false. The law of conjunction states: (p ^ q) q therefore p What this means is: If you are given that (p ^ q) is true and you are then given that q is true, then p must be true. LAW OF NEGATION The law of negation states that: If p is true then ~p is false. If ~p is true, then p is false. The truth table for the law of negation is shown below: <pre> p ~p T F F T </pre> LAW OF IMPLICATION The law of implication involves the use of the IMPLIES statement. The truth table for the IMPLIES statement is: <pre> p q (p -> q) T T T T F F F T T F F T </pre> The implies statement is only false if p is true and q is false. If p and q are both true, then the implies statement is true. If p is false, then the implies statement is true. Note that if p is false, The implies statement is true regardless of whether q is true or false. There are 4 possible statements involving the use of the implies statement. Those are: 1. The implies statement itself. 2. The inverse of the implies statement. 3. The converse of the implies statement. 4. The contra-positive of the implies statement. p -> q is the symbolic form of the IMPLIES statement. It means: If p is true, then q is true. ~p -> ~q is the symbolic form of the INVERSE of the implies statement. It means: If p is not true, then q is not true. q -> p is the symbolic form of the CONVERSE of the implies statement. It means: if q is true, then p is true. ~q -> ~p is the symbolic form of the CONTRA-POSITIVE of the implies statement. It means: if q is not true, then p is not true. The law of implication states: The IMPLIES STATEMENT and the CONTRA-POSITIVE OF THE IMPLIES STATEMENT are equivalent. In symbolic terms, this would appear as: (p -> q) <-> (~q -> ~p) The INVERSE OF THE IMPLIES STATEMENT and the CONVERSE OF THE IMPLIES STATEMENT are equivalent. In symbolic terms, this would appear as: (~p -> ~q) <-> (q -> p) Note that the converse of the implies statement is the contra-positive of the inverse of the implies statement. The law of implication does not state that the implies statement and its inverse are equivalent. The law of implication does not state that the implies statement and its converse are equivalent. The assumption that they are equivalent leads to the fallacy of the inverse and the fallacy of the converse. LAW OF EQUIVALENCE The law of equivalence involves the use of the equivalency statement. The symbolic form of the equivalency statement is: p <-> q Note that <-> translates to "is equivalent to" The truth table for the equivalency statement is: <pre> p q (p <-> q) T T T T F F F T F F F T </pre> The equivalency statement is true if both p and q are true, or if both p and q are false. If p is true and q is false, or if p is false and q is true, then the equivalency statement is false. In other words, as long as the truth table entries for p and q are the same then the equivalency statement is true, and as long as they are different then the equivalency statement is false. This is an important point because it is used to prove that two statements are equivalent to each other. The law of equivalence states: If the truth table entries for two statements are the same under all possible conditions in the truth table, then the two statements are equivalent and can be used inter-changeably. To show you how this law is used, I will prove that: (p v q) is equivalent to ~(~p ^ ~q) (p v q) means that either p is true or q is true or both are true. (~p ^ ~q) means both p and q are false. ~(~p ^ ~q) means "it is not true that both p and q are false" which means that both p and q are not false. The truth table for this proof is shown below: <pre> p q ~p ~q (~p ^ ~q) ~(~p ^ ~q) (p v q) ~(~p ^ ~q) <-> (p v q) - - -- -- --------- ---------- ------- ---------------------- T T F F F T T T T F F T F T T T F T T F F T T T F F T T T F F T </pre> The last column is the equivalency statement. ~(~p ^ ~q) <-> (p v q) means: ~(~p ^ ~q) is equivalent to (p v q) This means that: In order for the statement (p v q) to be true, either p is true or q is true or both are true. p and q cannot both be false. In order for the statement ~(~p and ~q) to be true, then both p and q cannot be false. The two statements are equivalent. If both p and q are not false, then either p is true or q is true or both are true. If either p is true or q is true or both are true, then both p and q cannot be false. The truth table for the equivalency statement supports this. It supports this because it is true for all possible conditions in the truth table. The equivalency statement itself is the last column in the truth table. Since these two statements are equivalent, they can be used inter-changeably. You can either say (p v q), or you can say ~(~p ^ ~q). They mean the same thing. LAW OF DETACHMENT The law of detachment involves the use of the implies statement. The law of detachment states: p->q p therefore q What this means is: If you are given that p->q is true and you are then given that p is true, then q must also be true. LAW OF SYLLOGISM The law of syllogism involves the use of the implies statement. The law of syllogism states: p -> q q -> r therefore p -> r What this means is: If you are given that p -> q is true and you are then given that q -> r is true, then p -> r must also be true. LAW OF CONTRA-POSITION The law of contra-position involves the use of the implies statement. The law of contra-position states: p->q ~q therefore ~p What this means is: If you are given that p -> q is true and you are given that q is not true, then p must not be true. The proof of the law of contra-position stems from the truth table for the implies statement. That proof will be detailed in a separate lesson involving proof of the law of contra-position. FALLACY OF THE INVERSE The fallacy of the inverse involves the use of the implies statement. The fallacy of the inverse states: p->q ~p therefore ~q What this means is: If you are given that p->q is true and you are then given that p is false, then q must be false. Unfortunately, this is not the case. If p is false, there is no way to determine if q is true or false. Therefore a statement like: p->q ~p therefore ~q is invalid by the fallacy of the inverse. The proof of the fallacy of the inverse stems from the truth table for the implies statement. That proof will be detailed in a separate lesson involving proof of the fallacy of the inverse. FALLACY OF THE CONVERSE The fallacy of the converse involves the use of the implies statement. The fallacy of the converse states: p->q q therefore p What this means is: If you are given that p->q is true and you are then given that q is true, then p must be true. Unfortunately, this is not the case. If q is true, there is no way to determine if p is true or not. Therefore a statement like: p->q q therefore p is invalid by the fallacy of the converse. The proof of the fallacy of the converse stems from the truth table for the implies statement. That proof will be detailed in a separate lesson involving proof of the fallacy of the converse.
