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The Logic of Boolean Connectives

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The Logic of Boolean Connectives Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Announcements For 09.15.11 1 HW1 was due on Tuesday The Logic of Boolean Connectives • If you didn't turn it in, it's late 2 HW2 & HW3 are due next Tuesday (09.20) Truth Tables, Tautologies & Logical Truths • Electronic HW must be submitted before class • Written HW must be handed in at beginning of class William Starr • Otherwise, it's late 3 Optional sections have been scheduled • Wednesday 1:25-2:10, Uris 307 09.15.11 • Thursday 1:25-2:10, Uris G22 William Starr j Phil 2621: Minds & Machines j Cornell University 1/45 William Starr j Phil 2621: Minds & Machines j Cornell University 2/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Outline Introduction Truth Functions • Last class, we learned the meaning of ^; _; : in terms 1 Introduction & Review of truth functions • We also saw that truth functions allowed us to do 2 Truth Tables something useful: • Figure out the truth value of a complex sentence from 3 Logical Truths & Tautologies the truth values of its atomic parts, and vice versa • For example, we know that :(Cube(a) _ Cube(b)) is true in a world where neither a nor b are cubes, since: 4 Equivalence • If :(Cube(a) _ Cube(b)) is true, then Cube(a) _ Cube(b) is false 5 Consequence • If Cube(a) _ Cube(b) is false, then Cube(a) and Cube(b) are false William Starr j Phil 2621: Minds & Machines j Cornell University 3/45 William Starr j Phil 2621: Minds & Machines j Cornell University 5/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction Review Truth Tables The Boolean Connectives Truth Table for : Truth Table for _ • These calculations | ones like the one we just went P :P through | are a bit clunky P Q P _ Q true false • There's a more elegant method: Truth tables false true true true true • As it turns out, truth tables will also provide us with a true false true helpful way to understand 3 core logical concepts: false true true false false false 1 Logical Consequence Truth Table for ^ 2 Logical Truth P Q P ^ Q 3 Logical Equivalence • : flips the value true true true • Today, we'll learn all about truth tables & these true false false • ^ takes the applications! false true false `worst' value false false false • _ takes the `best' value William Starr j Phil 2621: Minds & Machines j Cornell University 6/45 William Starr j Phil 2621: Minds & Machines j Cornell University 7/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence The Basics The Basics Step 1: The Reference Columns Step 2: Inside Out • We are going to construct a truth table for Truth Table for (1) (1) Cube(a) _:Cube(a) Cube(a) Cube(a) _: Cube(a) First, some columns: Truth Table for (1) t f 1 Reference Columns: f t Cube(a) Cube(a) _: Cube(a) columns for each atomic sub-sentence of (1) t f • Third, fill column beneath innermost connective :: 2 A column for (1) itself • In the first row, Cube(a) is t so :Cube(a) is f Second, fill the reference columns w/truth values. • In the second row, Cube(a) is f so :Cube(a) is t • One row for each unique logical possibility William Starr j Phil 2621: Minds & Machines j Cornell University 10/45 William Starr j Phil 2621: Minds & Machines j Cornell University 11/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence The Basics The Basics Step 3: The Main Connective Summary: In General Truth Table for (1) How to Construct a Truth Table for Any Sentence P Cube(a) Cube(a) _: Cube(a) 1 Reference Columns: Draw a column for each atomic t t f sub-sentence of P, these columns are called the f t t reference columns and are filled with every possible combination of truth-values for the sub-sentences • Last, fill columns beneath outermost connective _: 2 Inside Out: Draw a column for P itself. Then fill in • In the first row, Cube(a) is t and :Cube(a) is f, so the column below P's innermost connective. Repeat their disjunction is t for the next innermost connective, until you get to the • In the first row, Cube(a) is f and :Cube(a) is t, so main connective. their disjunction is t 3 Main Connective: Fill in the column under the main • This column lists every logically possible truth value connective. This row lists the possible truth values of P for (1) William Starr j Phil 2621: Minds & Machines j Cornell University 12/45 William Starr j Phil 2621: Minds & Machines j Cornell University 13/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Reference Columns Reference Columns There is More to It... How Many Rows? Let's figure it out: • We have 2 atomic sub-sentences • When you have more than one atomic sub-sentence, • Each can have 2 different truth values (t,f) filling in the reference columns requires more thought • So there 22 = 4 possible combinations of truth values • Remember that each row of the reference columns lists and atomic sub-sentences a unique logical possibility • Therefore, a table for a formula with 2 atomic • Also remember that there is supposed to be a row for sub-sentences needs 4 rows every unique possibility You Always Need 2n Rows • Okay, well how many rows would we need for a formula with 2 atomic sub-sentences? In general, if there are n atomic sub-sentences of P then there will be 2n possible assignments of truth values to those atomic sub-sentences, in which case the truth table for P should have 2n rows. William Starr j Phil 2621: Minds & Machines j Cornell University 15/45 William Starr j Phil 2621: Minds & Machines j Cornell University 16/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Another Example Another Example Step 1 Steps 2 & 3 • Let's construct a truth table for: Truth Table for (2) (2) :(Cube(a) ^ Cube(b)) • We need 2 reference columns: Cube(a) Cube(b) :(Cube(a) ^: Cube(b)) • In them, we need a row for each of the 4 logical t t t f f possibilities t f f t t • Cube(a) can be t and Cube(b) t; Cube(a) t and f t t f f Cube(b) f, and so on. f f t f t Truth Table for (2) • Next, the innermost connective :: • Cube(a) Cube(b) :(Cube(a) ^ :Cube(b)) It will flip each value of Cube(b) t t • Now, the next innermost ^: t f • ^ takes worst value of the pair f t • Finally, the main connective :: f f • : flips the value of the conjunction we just computed William Starr j Phil 2621: Minds & Machines j Cornell University 17/45 William Starr j Phil 2621: Minds & Machines j Cornell University 18/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Building Reference Columns Yet Another Example A Helpful Routine Step 1 • You need to list each possibility exactly once when We'll construct a table for: filling in reference columns (3)( Cube(a) ^ :Cube(b)) _:Cube(c) • Here's a helpful routine for this: Let A = Cube(a); B = Cube(b); C = Cube(c) 1 In the innermost ref. column, you alternate t's and f's Table for (3) 2 In the next innermost column, you double that A B C (A ^ :B) _:C • First, the 8 rows of the alternation, and so on for any more rows t t t reference columns: Truth Table for (2) t t f 1 Alternate on innermost Cube(a) Cube(b) :(Cube(a) ^ :Cube(a)) t f t column t t t f f f t t 2 Double this alternation t f on the next f t f t f 3 Double again f f f f t f f f Let's put this routine to work William Starr j Phil 2621: Minds & Machines j Cornell University 19/45 William Starr j Phil 2621: Minds & Machines j Cornell University 20/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Yet Another Example You Try It Steps 2 & 3 In Class Exercise Table for (3) • Next, rows for innermost connectives (:) A B C (A ^ :B) _:C t t t f f f f • :B: flips value of B Break into groups of 4-6 and construct a truth table for: t t f f f t t • :C: flips value of C t f t t t t f • Now, the row for the next t f f t t t t innermost connective (^) f t t f f f f f t f f f t t • ^ takes lowest value (4)( B _:C) ^ :A f f t f t f f f f f f t t t • Finally, the row for the main connective (_): • _ takes highest value William Starr j Phil 2621: Minds & Machines j Cornell University 21/45 William Starr j Phil 2621: Minds & Machines j Cornell University 22/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Boole Truth Tables An Introduction Summary • LPL contains a program called Boole, which is for • Okay, we've learned how to draw these pretty tables, but how do they help us do logic? constructing truth tables 1 Truth tables probe logical possibility • Now that you've done a one by hand, you can 2 This underlies several important concepts: appreciate how nice of a tool this is! • Logical truth • Let's run through the basics of Boole by using it to • Logical consequence construct a table for (3) • Logical equivalence • Let's see how William Starr j Phil 2621: Minds & Machines j Cornell University 24/45 William Starr j Phil 2621: Minds & Machines j Cornell University 25/45 Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Introduction & Review Truth Tables Logical Truths & Tautologies Equivalence Consequence Logical Truth Tautologies The Basics Logical Truth Tautology Truth Table for (1) P is a logical truth if and only if it is logically necessary.
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