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LOGICAL CONNECTIVES and or if…..then if and only if not

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s : John studies. p : John passes. h : John is happy This symbol replaces the word “AND” John passes AND John is happy ph

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s : John studies. p : John passes. h : John is happy John does NOT study: s This symbol negates the statement it precedes ~ ~

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s : John studies. p : John passes. h : John is happy This symbol replaces the word “OR” John studies OR John does NOT pass s~ p

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s : John studies. p : John passes. h : John is happy This symbol replaces the connective “ if … then ” IF John studies THEN John passes. sp

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s : John studies. p : John passes. h : John is happy This symbol replaces the words “ if and only if ” John is happy IF AND ONLY IF John studies. hs

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When using the connective The direction of the arrow is important. causeeffect

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If Herman lives in Cleveland then Herman lives in Ohio. co Herman lives in Ohio if Herman lives in Cleveland. oc Herman lives in Cleveland only if Herman lives in Ohio.

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John is happy ONLY IF he studies. John is happy IF he studies. John is happy IF AND ONLY IF he studies.

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m: Herman takes math h: Herman is happy a: Herman gets an A b: Herman bribes the teacher Herman takes math and he is not happy.m~h If Herman gets an A then he is happy. ah Herman takes math if and only if he bribes the teacher. mb

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( 3 + 5 ) x 2 is different from 3 + ( 5 x 2 ) similarly is different from When a statement has two or more connectives, one of the connectives is said to be dominant. This determines the grouping (placement of parentheses)

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A SIMPLE STATEMENT has one verb. The following four statements are simple: Today is Friday f It is hot h We will swim s We will picnic p

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Today is Friday f It is hot h We will swim s We will picnic p or and Now we have two COMPOUND STATEMENTS

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Finally, we can connect these two compound statements: Today is Friday f It is hot h We will swim s We will picnic p or andIf then Is the dominant connective. It is last one placed.

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l: The teacher likes Herman g: Herman does good work a: Herman gets an A The teacher likes Herman and if Herman does good work then Herman gets an A. If the teacher likes Herman and Herman does good work then Herman gets an A. l ga l ga

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statement DOMINANT CONNECTIVE

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( 3 + 5 ) x 2 is different from 3 + ( 5 x 2 ) Without parentheses we have a convention to interpret: 3 + 5 x 2 to mean (3) + ( 5 x 2 ) When an arithmetic sentence contains + and signs and NO PARENTHESES to indicate otherwise we assume that + is dominant

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There is a similar convention in logic: If a statement contains two or more connectives and there are NO PARENTHESES to show you how to group symbols, then we will agree that the DOMINANT CONNECTIVE is whatever is highest on this chart:

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If it rains then we go to a movie and if it doesn’t rain then we swim. ( r m ) (~r s ) In this statement, the parentheses are necessary because “and” is dominant

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If Herman does not study then he will fail math and his life will be ruined. ~sfr Because is the dominant connective, the parentheses are correctly placed…BUT … they are unnecessary

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If Herman does not study then he will fail math and his life will be ruined. ~sfr fr Because this means

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Interpret the following statement: ~pqr~sr This is highest on the chart. This is dominant

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Interpret the following statement: ~pqr~sr Now we have two simpler statements to deal with. ~pqr In the pink statement, is highest on the chart

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Interpret the following statement: ~pqr ~sr Now we have two simpler statements to deal with. In the blue statement, is highest on the chart ~sr

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Interpret the following statement: ~pqr~sr

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