CmSc 175 Discrete mathematics


Lesson 04: Valid and Invalid Arguments

Learning goals
Exam-like questions and problems


  1. Definitions
  2. Definition: An argument is a sequence of statements, ending in a conclusion.
    All the statements but the final one (the conclusion) are called premises (or assumptions, hypotheses)

    Verbal form of an argument:

    (1) If Socrates is a human being then Socrates is mortal.
    (2) Socrates is a human being

    Therefore       (3) Socrates is mortal

    Here (1) and (2) are the assumptions, and (3) is the conclusion.

    Abstract (logical) form of an argument - using variables:

    (1) If P then Q
    (2) P

    Therefore       (3) Q

    Definition: A logical argument is valid, if the conclusion is true whenever the assumptions are true.
    An argument is invalid if it is not valid.

  3. Testing an argument for its validity
  4. There are three ways to test an argument for validity.

    1. Critical rows
      1. Identify the assumptions and the conclusion and assign variables to them.
      2. Construct a truth table showing all possible truth values of the assumptions and the conclusion.
      3. Find the critical rows - rows in which all assumptions are true
      4. For each critical row determine whether the conclusion is also true.
        1. If the conclusion is true in all critical rows, then the argument is valid
        2. If there is at least one row where the assumptions are true,
          but the conclusion is false
          , then the argument is invalid

    2. Using tautologies
    3. The argument is valid if the conclusion is true whenever the assumptions are true.
      This means: If all assumptions are true, then the conclusion is true.

      "All assumptions" means the conjunction of all the assumptions.

      Let A1, A2, An be the assumptions, and B - the conclusion.
      For the argument to be valid, the statement

      If (A1 Λ A2 Λ Λ An) then B
      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:

      (A1 Λ A2 Λ Λ An) → B ≡ T

    4. Using contradictions
    5. If the argument is valid, then we have (A1 Λ A2 Λ Λ An) → B ≡ T.
      This means that the negation of (A1 Λ A2 Λ Λ An) → B should be a contradiction -
      containing only F in its truth table.

      In order to find the negation we have first to represent the conditional statement as a disjunction
      and then to apply the laws of De Morgan.

      (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:

    1. by constructing the truth table
    2. by logical transformations applying the logical equivalences (logical identities)

    They are illustrated in the examples below.

    Examples:

    1. Consider the argument:

    P → Q

    P

        \ Q

    (the symbol \ means "therefore").

    Testing its validity:

    1. by examining the truth table in order to find the critical rows:
    2.   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,
      the premises are in blue and the conclusion - in red color.

      There is only one critical row - the first one, where both the premises ( P and P→ Q) are true.
      In that row the value of Q is true, hence the argument is a valid argument.

    3. By showing that the statement "If all premises are true then the conclusion is true" is a tautology:
    4. 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:

      1. Using truth tables
      2.   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
        
      3. Using the logical equivalences to transform the expression and reduce it to T.

    5. Using contradiction
    6. We can prove the argument by showing that
      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
    and the truth of the assumptions

    Thus in the argument

    (1) If P then Q
    (2) P

    Therefore       (3) Q

    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
    of the assumptions and the conclusion. The critical rows are in boldface.

      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,
    because there may be critical rows where the conclusion would be true.

  5. Syllogisms (patterns of arguments, inference rules)
  6. Aristotle (384 322 B.C) was the first to study patterns of arguments, which he called syllogisms.
    Here is a description of Aristotle's work on 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

    (1) If P then Q
    (2) P

    Therefore       (3) Q

    Modus ponens uses a conditional statement: P → Q, i.e. if P is true, then Q is true.
    The second assumption in the argument states that P is true.
    Hence we conclude that Q is true.

    Modus Tollens (method of denying)

    (1) If P then Q
    (2) ~Q

    Therefore       (3) ~P

    Modus tollens is based on ~Q → ~P and this is the contrapositive of P → Q.
    If ~Q is true, the ~P is true.
    The second assumption states that ~Q is true.
    Hence we conclude that ~P is true (i.e. P is false)

    Examples:

    1. Modus ponens
    2. If today is Monday, tomorrow is Tuesday.
      Today is Monday.
      Therefore tomorrow is Tuesday.

      If it is Sunday we go fishing.
      It is Sunday
      Therefore we go fishing

    3. Modus tollens
    4. If today is Monday, tomorrow is Tuesday.
      Tomorrow is not Tuesday.
      Therefore today is not Monday.

      If it is Sunday we go fishing
      We do not go fishing
      Therefore it is not Sunday

    Examples of invalid arguments

    1. Inverse error
    2.  

      If P then Q

      If it is Sunday we go fishing

       

      ~P

      It is not Sunday

      Therefore

      ~Q

      We do not go fishing

      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")

    3. Converse error
    4.  

      If P then Q

      If it is Sunday we go fishing

       

      Q

      We go fishing

      Therefore

      P

      It is Sunday

      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

    P = It is Sunday
    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 third and the fourth rows are the critical rows for the first argument.
    In the third row however the conclusion ~Q is false (in red).

    The first and the third rows are the critical rows for the second argument.
    In the third row however the conclusion P is false (in red).

    3. 2. Disjunctive syllogism

    (1) P V Q
    (2) ~P

    Therefore       (3) Q

    Example:

    During the weekend we either go fishing or we play cards
    This weekend we did not go fishing

    Therefore, this weekend we were playing cards

    3. 3. Hypothetical syllogism

    (1) P → Q
    (2) Q → R

    Therefore       (3) P → R

    Example:

    If we win the game we will get much money.
    If we have money we will go on a trip to China.

    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,
    and the conclusion is in red color.

     
        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

  7. Fallacies
    1. Using incorrect syllogism, incorrect argument.
    2. Examples are: converse error, inverse error.

      If I read a book, I need my glasses
      I am not reading a book

      Therefore I don't need my glasses

      Where is the error?

    3. The argument is correct, however the premises are false.
    4. If you are a college student, you don't need to study.
      You are a college student

      Therefore you don't need to study.

  8. Inference rules
  9. A → B, A , therefore B

    Modus ponens

    A → B, ~B, therefore ~A

    Modus tollens

    A, therefore A V B

    Disjunctive addition

    A, B, therefore A Λ B

    Conjunctive addition

    A Λ B, therefore A
    A Λ B, therefore B

    Conjunctive simplification

    A V B, ~A, therefore B

    A V B, ~B, therefore A

    Disjunctive syllogism

    A → B, B → C, therefore A → C

    Hypothetical syllogism

    A V B, A → R, B → R, therefore R

    A → B, ~A → B, therefore B

    Dilemma, proof by division into cases

    ~P → F, therefore P

    Law of contradiction

    A B, therefore A → B, B → A

    Equivalence elimination

    A → B, B → A, therefore A B

    Equivalence introduction

    A, ~A, therefore B

    Inconsistency law

     

  10. Summary
    1. Arguments
    2. An argument is a sequence of statements. All but the final one are called premises
      the last one is the conclusion.

    3. Syllogisms
    4. Syllogisms are arguments with two premises only.
      Two important syllogisms based on the conditional statement and its contrapositive are
      Modus ponens and Modus tollens.

    5. Valid and invalid arguments
    6. An argument is a valid argument if the conclusion is true whenever the premises are true.
      Otherwise the argument is invalid.

    7. Test for validity
      • We can show that an argument is valid by:

        1. examining the critical rows in the truth table of the premises and the conclusion.

        2. The conclusion must be true in all rows where all the premises are true.

        3. showing that the expression "If premises then conclusion" is a tautology:
          • by constructing its truth table
          • by transforming to "T"
        4. showing that the conjunction of all premises and the negation of the conclusion
          is a contradiction:
          • by constructing its truth table
          • by transforming to "F"

      • We can show that an argument is invalid by examining the critical rows
        in the truth table of the premises and the conclusion.
        The conclusion must be false in at least one critical row.

      • Note, that we cannot prove that an argument is invalid by a tautology or a contradiction,
        because in some critical rows the conclusion may be true, in other it may be false.

      • Fallacies
      • Fallacies are either invalid arguments, or valid arguments based on false premises.


Learning Goals

  • Given a set of premises, be able to derive a conclusion using the inference rules
    (having the table with the inference rules at hand).
  • Be able to apply modus ponens, modus tollens, the disjuctive syllogism and the hypothetical syllogism
    (see the shadowed rows in the table) without having the list of inference rules at hand.
  • Given an argument, be able to determine if it is valid or invalid

Exam-like problems

  1. Assuming that the premises are true, determine whether the argument is valid or invalid
    • Premises:
      a. If it is not raining we go fishing
      b. It is not rainingg
      Conclusion : We go fishing

      Solution: valid, Modus Ponens

    • Premises:
      a. If it is not raining we go fishing
      b. We don't go fishing
      Conclusion : It is raining

      Solution: valid, Modus Tollens

    • Premises:
      a. If problems are hard then we study all day.
      b. Problems are not hard.
      Conclusion : We don't study all day

      Solution: invalid, inverse error

    • Premises:
      a. If problems are hard then we study all day.
      b. We study all day.
      Conclusion : Problems are hard

      Solution: invalid, converse error

    • Premises:
      a.If John is playing, the team will win.
      b.If the team does not win, the trip will be postponed.
      c.John is not playing.
      Conclusion: The trip is postponed

      Solution: invalid, innverse error

      Note: Let me know if you don't see where the inverse error is.

  2. Other problems, with a sample solution

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