L1 txt Thu Sep 15 13 26 02 2005 1 Review of Logic and Rules Gerald Jay Sussman 15 September 2005 L1 txt Thu Sep 15 13 26 02 2005 2 General Caveat If you think that your paper is vacuous use the first order functional calculus It then becomes logic and as if by magic the obvious is hailed as miraculous Paul Halmos c 1967 L1 txt Thu Sep 15 13 26 02 2005 3 A great book Introduction to Logic by Patrick Suppes L1 txt Thu Sep 15 13 26 02 2005 4 If Alfred studies then he receives good grades If Alfred does not study then he enjoys college If Alfred does not receive good grades then he does not enjoy college Therefore Alfred receives good grades Do we believe this argument if so why if not why not L1 txt Thu Sep 15 13 26 02 2005 5 Logic is the study of arguments An argument is a sequence of statements beginning with premises and ending with a conclusion The conclusion is said to follow from or is supported by the premises Arguments may be good or bad convincing or not convincing A good argument is called valid A valid form of argument is one that never allows the deduction of a false conclusion from true premises An invalid form of argument is one that sometimes allows the deduction of a false conclusion from true premises L1 txt Thu Sep 15 13 26 02 2005 6 A Valid Argument If Socrates is human then Socrates is mortal Socrates is human Socrates is mortal An Invalid Argument If Socrates is human then Socrates is mortal Socrates is mortal Socrates is human L1 txt Thu Sep 15 13 26 02 2005 7 I have a dog named Socrates L1 txt Thu Sep 15 13 26 02 2005 8 Sentential Connectives Truth functions Implication a b If a then b a implies b a only if b Negation a Not a It is not the case that a Conjunction a b a and b Disjunction a V b a or b Equivalence a b a if and only if b L1 txt Thu Sep 15 13 26 02 2005 9 Truth Tables define Truth Functions A A T F F T A B A B T T T F T F T F F F F F A B A V B T T T F T T T F T F F F L1 txt Thu Sep 15 13 26 02 2005 A B A B T T T F T T T F F F F T A B A B T T T F T F T F F F F T 10 L1 txt Thu Sep 15 13 26 02 2005 11 Analysis of arguments by sentential interpretation Primitive sentences S Alfred studies E Alfred enjoys college G Alfred receives good grades Premises A S G B S E C G E S E G A B C T T T T T T T T F F T F T F T T T T T F F F T T F T T F T T F T F F T F F F T F F T F F F T F T Note that for every row where the conclusion G is false at least one of the premises A B or C is false L1 txt Thu Sep 15 13 26 02 2005 12 Sentential Derivation 1 If Alfred studies then he receives good grades PREMISE 2 If Alfred does not study then he enjoys college PREMISE 3 If Alfred does not receive good grades then he does not enjoy college PREMISE 4 Alfred does not receive good grades PREMISE hypothetical 5 Alfred does not enjoy college MP 3 4 6 Alfred studies MT 2 5 NE 7 Alfred does not study MT 1 4 8 Alfred receives good grades RAA 4 6 7 QED L1 txt Thu Sep 15 13 26 02 2005 13 Sentential Derivation with Dependencies The Suppes Formalism 1 All y greek y human y Premise 2 All x human x mortal x Premise 3 greek G 4 greek G human G 5 human G 6 human G mortal G 7 mortal G 8 greek G mortal G 1 2 Premise 3 US 1 G y 1 MP 4 3 1 3 US 2 G x 2 MP 6 5 1 2 3 CP 7 3 1 2 9 All z greek z mortal z UG 8 z G 1 2