ARGUMENTS

• An argument is a compound proposition of the form (p1∧p2∧p3∧· · ·∧pn) → q. The propositions p1, p2, p3, . . . , pn are the premises of the argument and q is the conclusion. Arguments can be written in column or standard form
• An argument is valid if all its premises are true implies that the conclusion is true. Otherwise, we say that the argument is invalid. An error in reasoning that leads to an invalid argument is known as a fallacy.
• An argument is valid if the conclusion necessarily follows from the premises, and invalid if it is not valid.
• I If a statement is a tautology, then the argument is valid, otherwise, the

  argument is invalid.
• Procedure in determining the validity of an argument:

  1. Write the arguments in symbols.

  2. Write the argument as a conditional statement; use a conjunction (∧)

  between/among the premises and the implication (→) for the conclusion.

  3. Set up and construct a truth table for the symbolic form.

  4. If all truth values under → are Ts or 1s (that is, the last column is a

  tautology), then the argument is valid, otherwise, it is invalid.
• Common Forms of Valid Arguments

  1. Law of Detachment

  (Modus Ponens)

  p → q

  p

  ∴ q
• 2. Law of Contraposition

  (Modus Tollens)

  p → q

  ¬q

  ∴ ¬p
• 3. Law of Disjunctive Syllogism

  p ∨ q

  ¬p

  ∴ q
• 4. Law of Hypothetical Syllogism

  (Law of Transitivity)

  p → q

  q → r

  ∴ p → r
• 1. Fallacy of the converse

  p → q

  q

  ∴ p
• 2. Fallacy of the inverse

  p → q

  ¬p

  ∴ ¬q
• 3. Fallacy of the inclusive or

  p ∨ q

  p

  ∴ ¬q
• To analyze an argument with an Euler diagram:

  1. Draw an Euler diagram based on the premises of the argument

  2. The argument is valid if the diagram cannot be drawn to make the conclusion

  false

  3. The argument is invalid if there is a way to draw the diagram that makes the

  conclusion false

  4. If the premises are insufficient to determine the location of an element or a set mentioned in the conclusion, then the argument is invalid.
• In addition to these categorical style premises of the form ”all ”it is also common to see premises that are conditionals.