5.10.6.Logic Carneades.org https://www.youtube.com/watch?v=3ZDY81utzzU&list=PLz0n_SjOttTcjHsuebLrl0fjab5fdToui B.J.7.30.22.p.63.65.70.78.88.95.97.98. Logic =================== Unresolved =================== Logic - subdivisions - order of operation - beginning terms - proposition - arguments - diametric - negation - predicate - self contradictory - tautology -------------------- --{subdivisions] - - Categorical - - Propositional - - formal - - Informal - --------------- - -{Categorical] - - - deals with - - - subjects - - - predicates - - - particular topics - - - proposition's - - - affirmation - - - denial --{Propositional] - - - compares: - - - propositions - - - statements - - - whole sentences - --{Formal] - - - deals with formal statements - - - codifies language - - - follows a set of rules - --{Informal] - - deals with - - content of subjects - - loose guidelines --{beginning terms] - - truth table - - order of operations - - letters - ------------------- - --{truth table] - - a list o possible outcomes of a statement - --{order of operations] - - 1. parentheses - - 2. negation - --{letters] - - - represent statements -- - - P,Q,R,S - --{world] - - - a possile outcome of a state --{proposition] - - a clear statement - - subdivision 1 - - subdivision 2 - - ------------- - --{subdivision 1] - - - true - - - false - - ------- - - --{true] - - - - tautology - - - - always true. - - - - rule of logic - - - - assumes the basic variables stand firm. - - - - true in all possible worlds - - --{false] - --{Subdivision 2] - - compond propositions - - contradictory propositions - -------------------------- - --{compond proposition] - - - combines two or more operations - - - subdivision 1 - - - subdivision 2 - - - ------------- - - --{subdivision 1] - - - - consistent - - - - inconsistent - - - - ------------- - - - --{consistant] - - - - - two propostions are true in the same world - - - - - possible - - - - - not necessary - - - - - neither completely true nor, completely false - - - --{inconsistent] - - - - logically equivalent - - - - never true - - - - truth table - - - - -------------------- - - - --{logically equivalent] - - - - - Some logically equivalent propositions are inconsistent. - - - - - logically equivalent.all false propositions are completely false on truth tables . - - - - - It's possible for 2 logically equivalent propositions to always be false, then they would be inconsistent. - - - --{never true] - - - - - Inconsistent propositions will never share trues but can share false. - - - - - one world.both false.inconsitent.not contradictory - - - --{inconsistent truth table] - - - - P = q P & ~Q - - - -------------- - - - t t t t f ft <(x)signifies negation - - - t f f t t tf - - - f[f]f * f[f]ft < These two are false in the same world. They're not contradictory but inconsistent. - - - f t f f t tf - - --{subdivision 2] - - - conjunction - - - disjunction - - - equivication - - - implication - - --------------- - - --{conjunction] - - - - symbols - - - - table - - - - ------- - - - --{symbols] - - - - - "and" - - - - - "." - - - - - "^" - - - - - "&" - - - --{table] - - - - true - - - - false - - - - conjunction table - - - - ----------------- - - - --{true] - - - - - if all statements are truel - - - - ."It is raining and I have an umbrella." - - - --{false] - - - - - if a part of a statement is false, then the entire statement is false. - - - --{conjunction tabl]e - - - - P ^ Q - - - ----- - - - t t t - All is true, therefore the statement is true. - - - f f f - All is false, therefore the statement is false. - - - t f f - One part is false, therefore the statement is false. - - - f f t - One part is false, therefore the statement is false. - - --{disjunction] - - - - symbols - - - - table - - - --------- - - - --{symbols] - - - - "or" - - - - connects two statements with the word "or" - - - - the inclusive "or" - - - - "this or that or BOTH" - - - - "v" - - - --{table] - - - - true - - - - false - - - - disjunction table - - - ------------------- - - - --{true] - - - - if P is true - - - - if Q is true - - - - if BOTH are true - - - --{false] - - - - if both are false - - - - disjunction table - - - - P v Q - - - ----- - - - t t t - If P is true and Q is true, then v is true. - - - f f f - If P is false and Q is false, then v is false - - - t t f - If only one of the statements are true then thedisjunction is true. - - - f t t - If only one of the statements are true then the disjunction is true. - - --{equivication] - - - - = biconditional statement - - - - = logical equivalance - - - - ≠ necessarly consistant - - - - symbols - - - - equivication table - - - ----------------------- - - - - = biconditional statement - - aka "biconditional statement" - - - = logical equivalance - - - - two propositions with the same true values - - - - tautology - - - - ≠ necessarly consistant - - - - logically equivalent propositions are not necessarily consistent propositions. - - - - all false.inconsistent - - - - logically equivalent but inconsistent propositions are completely false on truth tables . - - - --{symbols - - - = "iff" - - - - = "≡" - - - - "≡" = material equivalance - - - - if a "≡" is between two propositions then the final result is true. - - - - P if and only if Q - - - . "Something happens if and only if something else happens." - - - - true - - - - if all parts are matching. - - - - equicalance table - - - - P iff Q - - - ------- - - - t t t - The statement is true because P and Q have matching values. - - - f f f - The statement is true because P and Q have matching values. - - - t f f - The statement is false because the values don't match. - - - f t t - The statement is false because - - - the values don't match. - - - - salva vertitate - - - - Two propositions are naturally - - - equivalant in all possible worlds. - - - - 'With (or by) unharmed truth' - - - - - when two expressions can be - - - interchanged without altering their - - - truth values - - - implication - - - symbol - - - false - - - implication table - - -------- - - - symbol - - - ">" - - - "If (...), then(...)." - - - false - - - If P is true And Q is false - - - implication table - - - P > Q - - ----- - - t t t - - f f f - If P is false, then the statement is false. - - t f f - If P is true and Q is false, then the statement is false. - - f t t - If P is false, then the statement is true. - --{contradictory propositions] - - two propositions with opposite truth values. - - always inconsistent Arguments - - --{argument] - - subdivisions - - valid - - other (invalid?) - ------------------ - --{valid] - - - one world.true premise.true conclusion. - - - premise > conclusion - - - rules (not all of them) - ----------------------------------------- - - - one world.true premise.true conclusion. - - - A valid argument means there is no possible world where premises are true and the conclusion is false. - - - premise > conclusion - - - a valid argument is the same as > between the - - premises and conclusion - - - rules - - - disjunctive syllogism - - - hypothetical syllogism - - - modus ponen - - - modus tollens(?) - - ------------------------ - - --{disjunctive syllogism] - - - = DS - - - - deny 1/2 - - - - To make a disjunctive syllogism, make a disjunctive proposition. Deny a part of it. Conclude that the other part must happen. - - - - "John will either go to France or Ohio tomorrow. He does not go to France, therefore he will go to Ohio." - - - - P = John go goes to France - - - - Q = John goes to Ohio - - - ----------------------------- -- - - PvQ ~P Q - - - - P v Q ~P Q - - - ---------------- - - - t t t ft t - - - t t f ft f - - - f t t tf t - - - f f f tf f - - - - statement is false. - - --{hypothetical syllogism] - - - - a rule of implication - - - - consolidation - - - - a consoliaton of multiple conditions into one big conditional - - - P>Q Q>R P>R - - - - P = Jerry buys a cake. - - - - Q = Jerry eats a cake. - - - - R = Jerry is poisoned. - - - - P > Q Q > R P > R - - - --------------------- - - - t[t]t t[t]t t[t]t x - - - t t t t f f t f f - - - t f f f t t t t t - - - t f f f t f t f f - - - f[t]t t[t]t f[t]t x - - - f t t t f f f t f - - - f[t]f f[t]t f[t]t x - - - f[t]f f[t]f f[t]f x - - - there are 4/8 possible worlds where Jerry is poisoned by a cake. - - --{Modus tollens] - - - deny 2nd half to deny 1st - - - The point of modus tollens is to deny the second part of the statement so you can deny the first. - - - P>Q ~P ~Q - - - p - He won in Waterloo - - - Q - He conquered the world - - - ~ - (negation) - - - P > Q ~P ~Q - - --------------- - - t t t ft ft - - t f f ft tf - - f t t tf ft - - f t f tf tf x a valid argument - - - x 1. If he Waterloo, He would have conquered the world. - - 2. He did not conquer the world. - - 3. He did not win Waterloo. - - - - - - - - - - - --{other] - - (invalid) - - - not explored - - - - - - - - - - ----------- - --{diametric} - - - completely and absolutely when compared to an opposition - - - absolutely opposite - -- {negation] - - - opposite statement - - - the statement - - - "It is raining." - - - the negation - - - "It is not raining." - - - ~ - - - P = "It is raining." - - - ~p = "It is not raining." - --{predicate] - - - what is being affirmed or denied in a proposition - - self-contrdictary - - always false