Argument Forms

- ™ - Some famous deductively valid (and invalid) argument forms Modus Ponens P>Q P If poison was the cause, the cook did it. Poison was the cause, Q The cook did it Modus Tollens P>Q -~Q If poison was the cause, the cook did it. The cook did not do it. ~p Poison was not the cause. Affirming the Consequent {invalid) P>Q Q If poison was the cause, the cook did it. The cook did it. Poison was the cause. P Wrong! Denying the Antecedent (invalid) If poison was the cause, the cook did it. Poison was not the cause. P>Q ~P ~Q* Wrong! The cook did not do it. Hypothetical Syllogism P5>Q Q>R If the cook was in love with the maid, the cook did it. PR If the cook did it, poison was the cause. If the cook was in love with the maid, poison was the cause. Disjunctive Syllogism Pv ~P The cook did it or the maid did it. The maid did not do it. The cook did it. Q Constructive Dilemma (general form) It was done in the kitchen or the dining room. PvQ PSR QS RvS If it was done in the kitchen, the cook did it. If it was done in the dining room, the maid did it. The cook did it or the maid did it. Constructive Dilemma (special form) It was done in the kitchen or the dining room. PvQ P33R Q>R R If it was done in the kitchen, the cook did it. If it was done in the dining room, the cook did it. Either way, the cook did it. Double-Negation Elimination It is not the case that the cook didn’t do it. —P The cook did it. P (Predicate logic) Universal Instantiation Vx(Fx —» Gx) Fa All men are mortal. Socrates is a man. Ga All FsareGs _aisF aisG Socrates is a mortal. Existential Generalization gisF _aisG Fa Ga x(Fx & Gx) SomeFisG A few important points about English connectives Socrates is a man. Socrates is rational. Some man is rational. . ‘Only if is the opposite of ‘if (they are not the same thing!) If today is Tuesday, we have a logic class. T > L = We have a logic class if today is Tuesday. = Today is Tuesday only if we have a logic class # We have a logic class only if today is Tuesday. Today is Wednesday if and only if tomorrow is Thursday = Today is Wednesday if tomorrow is Thursday + Today is Wednesday only if tomorrow is Thursday ‘Unless’ = “if not’ = “or otherwise’. Unless today is Saturday or Sunday, we have a Spanish dass = If today is not Saturday or Sunday, we have a Spanish class = We have a Spanish class, or otherwise today s Saturday or Sunday The cook did it unless the gardener did it = The cook did it or the gardener did it The inclusive ‘or’ and exclusive ‘or’. Valid? Depends. PvQ P Chomsky is a linguist or an anarchist Chomsky s 3 ingusst Chomsky s not an anarchust ~Q o The inclusive ‘or’. P or Q = either P or Q, including the case in which both P and Q. Used in logic. P/ Q The exdlusive ‘or’. P or Q = either P or Q, excluding the case in which both P and Q PvaQ