CmSc 175 Discrete mathematics | ||||||||||||||||||||||||||||||||||||||||||
Lesson 04: Valid and Invalid Arguments
Definition: An argument is a sequence of statements, ending in a
conclusion. Verbal form of an argument: (2) Socrates is a human being Here (1) and (2) are the assumptions, and (3) is the conclusion. Abstract (logical) form of an argument - using variables: (2) P Definition: A logical argument is valid, if the conclusion is true
whenever the assumptions are true. There are three ways to test an argument for validity. The argument is valid if the conclusion is true whenever the assumptions
are true. "All assumptions" means the conjunction of all the assumptions. Let A1, A2, … An be the assumptions, and B - the conclusion. must be a tautology - true for all assignments of values to its variables, i.e. its column in the truth table must contain only T: If the argument is valid, then we have (A1 Λ A2 Λ… Λ An) → B ≡ T. In order to find the negation we have first to represent the conditional statement as a disjunction
(A1 Λ A2 Λ… Λ An) → B ≡ ~( A1 Λ A2 Λ… Λ An) V B ≡ ~A1 V ~A2 V …. V ~An V B. The negation is: ~((A1 Λ A2 Λ… Λ An) → B) ≡ ~(~A1 V ~A2 V …. V ~An V B) ≡ A1 Λ A2 Λ …. Λ An Λ ~B The argument is valid if A1 Λ A2 Λ …. Λ An Λ ~B ≡ F There are two ways to show that a logical form is a tautology or a contradiction: They are illustrated in the examples below. Examples: 1. Consider the argument: P → Q P \ Q (the symbol \ means "therefore").Testing its validity: P Q P→ Q --------------------- T T T T F F F T T F F T Here and in the other truth tables below, the critical rows are in boldface,
There is only one critical row - the first one, where both the premises ( P
and P→ Q) are true. The premises are P and P→ Q. The statement to be considered is: (P Λ (P→ Q)) → Q We can show that it is a tautology in two ways: P Q P→ Q P Λ (P→ Q) (P Λ (P→ Q)) → Q -------------------------------------------------- T T T T T T F F F T F T T F T F F T F T the conjunction of all assumptions and the negation of the conclusion is a contradiction: Again, this can be done using truth tables, as shown below, or by using the logical equivalences to transform the expression and reduce it to F. P Q ~Q P→ Q P Λ (P→ Q) (P Λ (P→ Q) Λ ~Q) ----------------------------------------------------------- T T F T T F T F T F F F F T F T F F F F T T F F Crucial fact about a valid argument: the truth of its conclusion follows
necessarily from the logical form alone Thus in the argument (2) P Q is true whenever (1) and (2) are true, no matter what is the nature of the statements P and Q. 2. Consider the argument P → Q Q \ P We can show that this argument is invalid by examining the truth tables P Q P→ Q --------------------- T T T T F F F T T here the assumptions are true, however the conclusion (in red) is false. F F T Note, that we cannot use proof by contradiction to show that an argument is invalid, Aristotle
(384 – 322 B.C) was the first to study patterns of
arguments, which he called syllogisms. Syllogisms are inference rules, rules to make valid arguments, rules for deductive reasoning. 3. 1. Modus Ponens and Modus Tollens Modus ponens (method of affirming) This is the well known already argument (2) P Modus ponens uses a conditional statement: P → Q, i.e.
if P is true, then Q is true. Modus Tollens (method of denying) (2) ~Q Modus tollens is based on ~Q → ~P and this is the contrapositive of
P → Q. Examples: If today is Monday, tomorrow is Tuesday. If it is Sunday we go fishing. If today is Monday, tomorrow is Tuesday. If it is Sunday we go fishing Examples of invalid arguments
The argument would be valid if the inverse of the conditional statement If P then Q had been used as an assumption. (The inverse of "If P then Q" is "If ~P then ~Q")
The argument would be valid if the converse of the conditional statement If P then Q had been used as an assumption. (The converse of "If P then Q" is "If Q then P") To show that the arguments are invalid we use truth tables: Let Q = We go fishing P Q ~P ~Q P→Q ----------------------------------- T T F F T T F F T F F T T F T F F T T T The T values of the premises are in boldface. The first and the third rows are the critical rows for the second argument. 3. 2. Disjunctive syllogism (2) ~P Example: During the weekend we either go fishing or we play cards Therefore, this weekend we were playing cards 3. 3. Hypothetical syllogism (2) Q → R Example: If we win the game we will get much money. Therefore, if we win the game we will go on a trip to China In the truth table below the critical rows are in boldface, the premises are in blue,
P Q R P → Q Q → R P → R --------------------------------------- T T T T T T T T F T F F T F T F T T T F F F T F F T T T T T F T F T F T F F T T T T F F F T T T The value of the conclusion in the critical rows is T Examples are: converse error, inverse error. If I read a book, I need my glasses Therefore I don't need my glasses Where is the error? If you are a college student, you don't need to study. Therefore you don't need to study.
An argument is a sequence of statements. All but the final one are called premises Syllogisms are arguments with two premises only. An argument is a valid argument if the conclusion is true whenever the premises are true.
The conclusion must be true in all rows where all the premises are true. Note, that we cannot prove that an argument is invalid by a
tautology or a contradiction, Fallacies are either invalid arguments, or valid arguments based on false premises. Learning Goals
Exam-like problems
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