STATEMENT LOGIC: TRUTH TABLES (Charles Sanders Peirce, )

Transcription

1 STATEMENT LOGIC: TRUTH TABLES (Charles Saders Peirce, ) Statemet logic is the logic of statemets. I statemet logic, statemets as opposed to predicates, subject terms, ad quatifiers are the basic uit o which to do logic. SYMBOLIZING ENGLISH ARGUMENTS Symbolizig a argumet helps us to see its logic ad to determie its validity. To symbolize a argumet we eed to symbolize the statemets that compose it. Toward that ed, let s distiguish two kids of statemets: atomic ad compoud. Atomic statemet: a statemet that does ot have ay other statemet as a compoet Peter likes to play soccer. William adores his mother. Fraces s favorite city is Siea. We symbolize a atomic statemet with a capital letter; use a differet letter for each differet atomic statemet. A B C Compoud statemet: a statemet that has at least oe atomic statemet as a compoet Peter likes to play soccer ad William adores his mother. Either Peter likes to play soccer or William adores his mother. If Fraces s favorite city is Siea, the Fraces s favorite city is i Italy. Fraces s favorite city is Siea if ad oly Fraces s favorite city is i Italy. It s false that William is a lousy chess player. We symbolize a compoud statemet by assigig a differet capital letter to each differet atomic statemet that is a compoet. A ad B. Either A or B. If C, the D. C if ad oly if D. It s false that E. 1

2 Logical operators (key logical words): OPERATOR NAME TRANSLATES TYPE OF COMPOUND ~ tilde ot egatio dot ad cojuctio v vee or disjuctio arrow if, the coditioal double arrow if ad oly if bicoditioal A ad B. A B Either A or C. A v C If A, the B. A B A if ad oly if B. A B It s false that D. ~D Some Commets o Types of Compouds Negatios Stylistic variatios: It is ot the case that Peter has a broke thumbail. It is false that Peter has a broke thumbail. It is ot true that Peter has a broke thumbail. Peter fails to have a broke thumbail. Peter does ot have a broke thumbail. All of these are symbolized the same way: ~P Negatios ca ivolve other logical operators: ~P ~(P B) ~(P v B) ~(P B) ~(P B) I each of these cases, the mai operator is the tilde. 2

3 Cojuctios Stylistic variatios: Peter has a broke thumbail, but he s brave about it. Peter has a broke thumbail; however, he s brave about it. While Peter has a broke thumbail, he s brave about it. Although Peter has a broke thumbail, he s brave about it. Peter has a broke thumbail, yet he s brave about it. Peter has a broke thumbail; evertheless, he s brave about it. Peter has a broke thumbail though he s brave about it. All of these are symbolized the same way: P B Not every use of ad is properly traslated with the dot. I climbed Mt. Baker ad looked iside the sulfur coe. I got i my truck ad tured the key. The poit: The word ad sometimes meas ad the, idicatig temporal order; the dot does ot idicate temporal order. Aother example Peter ad William are brothers. Rya ad Christie are married. The poit: The word ad sometimes idicates a relatioship; the dot does ot idicate ay relatioship. Paretheses Oftetimes, you eed to use paretheses i order to make it clear what is beig said. It s false that both Obama ad McCai are presidet. A = Obama is presidet B = McCai is presidet Icorrect traslatio: ~A B Correct traslatio: ~(A B) 3

4 Cojuctios ca ivolve other logical operators A (B C) A (B v C) (~A B) C I each of these cases, the mai operator is the dot. Disjuctios Stylistic variatios Either Peter broke the widow or William did. Peter broke the widow or William did. Peter broke the widow ad/or William did. Peter broke the widow or William did (or both). All of these are symbolized i the same way: A v B Iclusive vs. exclusive disjuctio Two rules to remember: Rule #1. Assume iclusive disjuctio uless explicitly stated otherwise For example: Either he loves Mary or he loves Shelly, but ot both. Either he scored 21 poits or he scored 22 poits, but ot both. Rule #2. To express exclusive disjuctio, use this symbolizatio: (A v B) ~(A B) either or. Neither you or I kow for sure exactly whe we are goig to die. (Y = You kow for sure exactly whe you are goig to die; I = I kow for sure exactly whe I am goig to die.) Permissible symbolizatios: ~Y ~I ~(Y v I) 4

5 Disjuctios ca ivolve other logical operators A v (B C) A v (B C) ~B v (C D) I each of these cases, the mai operator is the vee. Coditioals Stylistic variatios (page 286) Sufficiet coditio For example (1) If Pat is a bachelor, the Pat is umarried. If A, the B. (2) Pat s beig a bachelor is a sufficiet coditio for Pat s beig umarried. A is a sufficiet coditio for B. Symbolized the same way: A B Necessary coditio (1) If there s a fire i the buildig, the there is air i the buildig. If A, the B. (2) There beig air i the buildig is a ecessary coditio for there beig a fire i the buildig. B is a ecessary coditio for A. Symbolized the same way: A B uless There are two ways to traslate statemets with uless : Use the coditioal Use disjuctio I like the coditioal, but it s easier to memorize the disjuctio. 5

6 Illustratio (1) We will lose o Saturday uless Dykstra has a super game. (L: We will lose o Saturday; D: Dykstra has a super game) L uless D Note: (1) meas the same thig as (2) Dykstra s havig a super game is a ecessary coditio for our ot losig o Saturday D is a ecessary coditio for ~L (3) Alterative: if Dykstra does ot have a super game, the we will lose Symbolize: ~L D If ~D, the L So, wheever you see somethig of the form L uless D A atural way to symbolize it as: ~L D However, it might be easier to simply remember to replace uless with a vee: L v D Coditioals ca ivolve other logical operators (A v B) C D (E ~F) G (H L) I each case, the mai operator is the arrow. Bicoditioals Stylistic variats A if ad oly if B A just i case B A is a ecessary ad sufficiet coditio for B 6

7 Cojuctio of A B ad B A. For example It will rai if ad oly if the atmospheric coditios are just right for rai. R A (R A) (A R) Aother example There s fire if ad oly if there s air, combustio, ad fuel. F (A (C U)) (F (A (C U))) (((A C) U) F) Bicoditioals ca ivolve other logical operators. Lear whe it s the mai oe. 7

8 STATEMENT LOGIC Our Symbolic Laguage Vocabulary: (1) paretheses, (2) logical operators (3) statemet capital letters A expressio i statemet logic is ay strig usig this vocabulary. A well-formed formula (WFF) is a grammatically correct expressio. What couts as a WFF? Let lowercase letters, e.g. p, q, r, etc., stad for statemet variables, which ca stad for ay statemet. 1. Capital letters (which stad for atomic statemets) are WFFs. 2. If p is a WFF, the so is ~p; 3. If p ad q are WFFs, the so is (p q). 4. If p ad q are WFFs, the so is (p v q). 5. If p ad q are WFFs, the so is (p q). 6. If p ad q are WFFs, the so is (p q). Nothig is a WFF uless it ca be demostrated to be oe by 1-6. (~B) Is it a WFF, strictly speakig? (M ~~N) y ((Q S) T) ~(~W v ~Z) ~(m h) (~E ~F ~~G) (~U (W)) ((~H ~~F) ~(~K ~N)) y y y (A B C) (E (~F G) ((L v M) ~S) (~P v Q v ~R) y 8

9 Permissible departures from strict grammar i our symbolic laguage Droppig paretheses (without creatig ambiguity) Usig brackets Is it a permissible departure? ~G ~H y (A v B) (C v D) y [~Z ~W v ~~Y] ~J ~K y (~Q v ~R v ~~S) ~A (~C F) y Idicatig a argumet i our symbolic laguage Comma (, ) used to separate premises Triple-dot ( :. ) used to idicate coclusio So, for example, A, A B :. B Name of Form K L, L N :. K N E v F, ~E :. F C D, ~D :. ~C O P, ~O :. ~P G v H, G I, H J :. I v J O P, P :. O 9