SlideShare ist ein Scribd-Unternehmen logo
1 von 17
Downloaden Sie, um offline zu lesen
Logical Connectives
Truth Functions
• Declarative sentences (statements) are either true or
  false but not both. They cannot be neither.
• Whether a sentence is true or false determines its truth
  value.
• Functions take input values to unique output values.
  That is, the input values determine the output values.
• There are some ways of combining sentences into
  longer ones so that the truth value of the longer
  sentence is determined by the truth value of the parts
  that were combined to form it.
• So, we’ll call the connectives that combine sentences
  this way truth functions.
And
• ‘And’ is often a truth functional connective.
  Inserting ‘and’ between two statements gives a
  longer sentence the truth of which is determined
  by the truth of the parts.
• For example, consider: (A) Al admires aardvarks.
  (B) Barb bakes bologna. The truth of ‘Al admires
  aardvarks and Barb bakes bologna’ is determined
  by the truth of A and of B.
• If A and B are both true then the longer sentence
  is true. If either of A or B is false then so is the
  longer sentence.
A Truth Table for ‘And’
• We can express all four        A   B   A and B
  possibilities for A and B in   T   T   T
  table form.                    T   F   F

• Here’s how to read the         F   T   F
  table. The second              F   F   F
  (horizontal) row says that
  when A is true and B is
  true then A and B is true.
  The bottom row says that
  when A is false and B is
  false then A and B is false.
Conjunction
• We’ll use a dot ‘∙’ or an ampersand ‘&’ to represent the
  truth functional connective that is captured by the truth
  table on slide four.
• We’ll call sentences formed with the dot (or ampersand)
  conjunctions. We’ll call the parts conjuncts. For example,
  ‘A&B’ is a conjunction and ‘A’ is its left conjunct.
• Not every instance of ‘and’ in English can be symbolized
  with the dot. Contrast ‘Al and Barb are chefs’ (It means: Al
  is a chef & Barb is a chef.) with ‘Al and Barb are enemies.’ (It
  doesn’t mean: Al is an enemy & Barb is an enemy.)
• Some words besides ‘and’ can be symbolized with the dot
  or ampersand. For example, ‘but’ and ‘also’ form
  conjunctions.
Or
• ‘Or’ is often used as a truth functional
  connective.
• Using the sentences from slide three, the truth of
  ‘A or B’ can be determined if we know the truth
  value of A and the truth value of B.
• If Al doesn’t admire aardvarks and Barb doesn’t
  bake bologna then ‘A or B’ is false. Otherwise it’s
  true.
• We’ll symbolize the truth functional (inclusive)
  ‘or’ with a wedge ‘ ’ and we’ll call the resulting
  sentences disjunctions. The parts are disjuncts.
Truth Table for Disjunction
• The four possibilities for          A   B   A   B
  sentences A and B are               T   T   T
  represented by the left two
  (vertical) columns.                 T   F   T
• The top row of Ts and Fs is the     F   T   T
  possibility where A is true and     F   F   F
  B is true. On that possibility ‘A
     B’ is true.
• The bottom row is the
  possibility where A and B are
  both false. In that case ‘A B’
  is also false.
• Disjunction differs from
  conjunction on the middle two
  rows.
Examples
• Lenny and Manny left for Bermuda. (L=Lenny left for
  Bermuda. M=Manny left for Bermuda.) Translation:
  L&M
• Either Lenny or Manny left for Bermuda. Translation:
  L M
• Lenny left for Bermuda and either Manny left for
  Bermuda or Nancy stayed. (N=Nancy stayed.)
  Translation: L&(M N)
• Lenny left for Bermuda but Manny did too.
  Translation: L&M
• Either Manny left for Bermuda and Nancy stayed or
  else Lenny left for Bermuda. Translation: (M&N) L
Negation
• Prefixing a statement with ‘it is   A       A
  not the case that’ flips the        T   F
  truth value.
                                      F   T
• The symbol to represent that
  truth function is ‘ ’. The
  symbol is called ‘tilde’ or just
  ‘squiggle’.
• Unlike the other connectives,
  tilde doesn’t connect two
  sentences. It just flips the
  value of a single sentence.
• Also unlike the other
  connectives, we don’t usually
  put parenthesis around a
  negation.
Examples
• Manny left but Lenny did not leave.
  Translation: M& L
• Either Nancy did not stay and Lenny left or
  Manny didn’t leave. Translation: ( N&L) M
• Neither Lenny nor Manny left. Translation:
    (L M) [Alternate Translation: L& M]
• It’s not true that both Manny and Lenny left.
  Translation: (M&L) [or M L]
If..Then…
• Some uses of ‘if..then…’ are truth functional and
  some are not. We’ll only care about the truth
  functional ones.
• Consider: If you whistle loudly then the dog will
  come. There’s really just one scenario that shows
  the sentence to be false: whistle loudly and have
  the dog not come. If you whistle and the dog
  comes then the sentence was true. If you don’t
  whistle then no matter whether the dog comes
  or not, you didn’t show the sentence wrong.
Truth Table for Conditional
• We’ll use the arrow ⟶         A   B   A⟶B
  (or the horseshoe ) to        T   T   T
  represent the truth           T   F   F
  functional ‘if...then…’;      F   T   T
  and we’ll call statements
                                F   F   T
  formed with the arrow
  (or the horseshoe)
  ‘conditionals.’
• The top two rows of the
  truth table for conditional
  are uncontroversial. The
  bottom two are less
  obvious.
Order Matters
• Notice on the table that the order of A and B matter. A
  true and B false yields a different value than A false and
  B true. So, unlike conjunction and disjunction where
  order doesn’t matter, we have a different names for
  the different parts of the conditional.
• For A⟶B, A is called the ‘antecedent’ and B is called
  the ‘consequent’.
• There are many English expressions that can be
  translated as conditionals. The trick to symbolizing
  them correctly is to identify the antecedent and the
  consequent.
Examples of Conditionals
• Lenny left if Manny left. Translation: M⟶L
• Lenny left only if Manny left. Translation: L⟶M
  [Think about this one.]
• Lenny left in case Manny left. Translation: M⟶L
• Lenny left provided that Manny left. Translation:
  M⟶L
• Lenny left unless Manny left. Translation: M⟶L
• Whenever Lenny leaves, Manny leaves.
  Translation: L⟶M
If and Only If
• ‘A only if B’ is translated A⟶B because when we say ‘A only
  if B’ we are saying that A can’t be true without B; so, if A is
  true then so is B. Hence, A⟶B.
• ‘A if B’ is the same as ‘if B then A and so is translated B⟶A.
• So, if we say ‘A if B and A only if B’ or in other words ‘A if
  and only if B’ we could translate that as ‘(B⟶A)&(A⟶B)’.
  We’ll use the double arrow ⟷ (or the triple bar ) as an
  abbreviation for that. Also, we’ll call sentences formed by
  using the double arrow ‘biconditionals’.
• Sometimes you see ‘iff’ between two sentences. It’s not a
  typo. It’s an abbreviation for ‘if and only if.’ So ‘A iff B’
  would be translated as ‘A⟷B’.
Truth Table for Biconditional
• Think of the biconditional as       A   B   A⟷B
  saying that two sentences have
  matching truth value. The longer    T   T   T
  sentence is true when both parts    T   F   F
  are true or when both parts are
  false.                              F   T   F
• Notice that order doesn’t matter;   F   F   T
  so we don’t have special names
  for the letter that comes first.
  Unlike regular conditional, A⟷B
  is equivalent to B⟷A.
• The sentence ‘A⟷B ‘is just an
  abbreviation for the more
  complicated sentence
  ‘(B⟶A)&(A⟶B)’ so next I’ll show
  you where the values in the table
  to the right came from.
Justification for the Biconditional Table
• The table gives the value of           A   B   (B⟶A)   &   (A⟶B)
  ‘(B⟶A)&(A⟶B)’ in each of the
  four possibilities for A and B.        T   T   T       T   T
• The left two columns list the          T   F   T       F   F
  possibilities for A and B.             F   T   F       F   T
• The sentence we care about is a
  conjunction of two conditionals.       F   F   T       T   T
  To figure out its truth value we
  need to know the truth value of
  the conditionals. Then we can
  use it to figure out the truth value
  of the conjunction.
• The 3rd column gives the values
  for B⟶A and the 5th column
  gives values for A⟶B. The 4th
  column was computed last using
  the values from columns 3 and 5.

Weitere ähnliche Inhalte

Was ist angesagt?

TRIANGLE INEQUALITY THEOREM
TRIANGLE INEQUALITY THEOREMTRIANGLE INEQUALITY THEOREM
TRIANGLE INEQUALITY THEOREMMichaellaApale
 
proposition, types and difference between proposition and sentence
proposition, types and difference between proposition and sentenceproposition, types and difference between proposition and sentence
proposition, types and difference between proposition and sentencezainulla
 
Abbreviated Truth Tables
Abbreviated Truth TablesAbbreviated Truth Tables
Abbreviated Truth Tablesdyeakel
 
2.5.5 Triangle Inequalities
2.5.5 Triangle Inequalities2.5.5 Triangle Inequalities
2.5.5 Triangle Inequalitiessmiller5
 
Properties of Parallelogram
Properties of ParallelogramProperties of Parallelogram
Properties of ParallelogramCipriano De Leon
 
Grade 8-if-then-statement
Grade 8-if-then-statementGrade 8-if-then-statement
Grade 8-if-then-statementAnnalizaTenioso
 
Applications of trignometry
Applications of trignometryApplications of trignometry
Applications of trignometryAbhijit Jadhav
 
Congruent triangles theorem
Congruent triangles theoremCongruent triangles theorem
Congruent triangles theoremMadhavi Mahajan
 
Triangle congruence (Group 1) Grade 8
Triangle congruence  (Group 1) Grade 8Triangle congruence  (Group 1) Grade 8
Triangle congruence (Group 1) Grade 8Kaye Abordo
 
Conditional Statements
Conditional StatementsConditional Statements
Conditional Statementsmicdsram
 
Introduction to Postulates and Theorems
Introduction to Postulates and TheoremsIntroduction to Postulates and Theorems
Introduction to Postulates and Theoremsneedmath
 
Regression Analysis Research Presentation
Regression Analysis Research PresentationRegression Analysis Research Presentation
Regression Analysis Research PresentationDianaWilbur
 
Labeling Trigonometry Triangles
Labeling Trigonometry TrianglesLabeling Trigonometry Triangles
Labeling Trigonometry TrianglesPassy World
 

Was ist angesagt? (20)

Angle Pairs
Angle PairsAngle Pairs
Angle Pairs
 
Congruent Triangles
Congruent TrianglesCongruent Triangles
Congruent Triangles
 
5.2 Venn Diagrams
5.2   Venn Diagrams5.2   Venn Diagrams
5.2 Venn Diagrams
 
TRIANGLE INEQUALITY THEOREM
TRIANGLE INEQUALITY THEOREMTRIANGLE INEQUALITY THEOREM
TRIANGLE INEQUALITY THEOREM
 
Symbolic logic
Symbolic logicSymbolic logic
Symbolic logic
 
Trigonometry
TrigonometryTrigonometry
Trigonometry
 
proposition, types and difference between proposition and sentence
proposition, types and difference between proposition and sentenceproposition, types and difference between proposition and sentence
proposition, types and difference between proposition and sentence
 
Abbreviated Truth Tables
Abbreviated Truth TablesAbbreviated Truth Tables
Abbreviated Truth Tables
 
2.5.5 Triangle Inequalities
2.5.5 Triangle Inequalities2.5.5 Triangle Inequalities
2.5.5 Triangle Inequalities
 
Properties of Parallelogram
Properties of ParallelogramProperties of Parallelogram
Properties of Parallelogram
 
Grade 8-if-then-statement
Grade 8-if-then-statementGrade 8-if-then-statement
Grade 8-if-then-statement
 
Applications of trignometry
Applications of trignometryApplications of trignometry
Applications of trignometry
 
Congruent triangles theorem
Congruent triangles theoremCongruent triangles theorem
Congruent triangles theorem
 
Verbals[1]
Verbals[1]Verbals[1]
Verbals[1]
 
Triangle congruence (Group 1) Grade 8
Triangle congruence  (Group 1) Grade 8Triangle congruence  (Group 1) Grade 8
Triangle congruence (Group 1) Grade 8
 
Conditional Statements
Conditional StatementsConditional Statements
Conditional Statements
 
Introduction to Postulates and Theorems
Introduction to Postulates and TheoremsIntroduction to Postulates and Theorems
Introduction to Postulates and Theorems
 
Regression Analysis Research Presentation
Regression Analysis Research PresentationRegression Analysis Research Presentation
Regression Analysis Research Presentation
 
Labeling Trigonometry Triangles
Labeling Trigonometry TrianglesLabeling Trigonometry Triangles
Labeling Trigonometry Triangles
 
Proposition (Logic)
Proposition (Logic)Proposition (Logic)
Proposition (Logic)
 

Andere mochten auch

Discrete mathematic question answers
Discrete mathematic question answersDiscrete mathematic question answers
Discrete mathematic question answersSamet öztoprak
 
Language Part I
Language Part ILanguage Part I
Language Part Idyeakel
 
Mistakes in Reasoning
Mistakes in ReasoningMistakes in Reasoning
Mistakes in Reasoningdyeakel
 
Validity
ValidityValidity
Validitydyeakel
 
Identify and Reconstruct Arguments
Identify and Reconstruct ArgumentsIdentify and Reconstruct Arguments
Identify and Reconstruct Argumentsdyeakel
 
Conspiracy Theories and Explanations
Conspiracy Theories and ExplanationsConspiracy Theories and Explanations
Conspiracy Theories and Explanationsdyeakel
 
Truth tables presentation
Truth tables presentationTruth tables presentation
Truth tables presentationMujtaBa Khan
 
6.3 Truth Tables For Propositions
6.3   Truth Tables For Propositions6.3   Truth Tables For Propositions
6.3 Truth Tables For PropositionsNicholas Lykins
 
Rhetorical devices
Rhetorical devicesRhetorical devices
Rhetorical devicesdyeakel
 
Truth tables
Truth tablesTruth tables
Truth tableswalkerlj
 
CATEGORICAL SYLLOGISM
CATEGORICAL SYLLOGISMCATEGORICAL SYLLOGISM
CATEGORICAL SYLLOGISMashelle14
 

Andere mochten auch (11)

Discrete mathematic question answers
Discrete mathematic question answersDiscrete mathematic question answers
Discrete mathematic question answers
 
Language Part I
Language Part ILanguage Part I
Language Part I
 
Mistakes in Reasoning
Mistakes in ReasoningMistakes in Reasoning
Mistakes in Reasoning
 
Validity
ValidityValidity
Validity
 
Identify and Reconstruct Arguments
Identify and Reconstruct ArgumentsIdentify and Reconstruct Arguments
Identify and Reconstruct Arguments
 
Conspiracy Theories and Explanations
Conspiracy Theories and ExplanationsConspiracy Theories and Explanations
Conspiracy Theories and Explanations
 
Truth tables presentation
Truth tables presentationTruth tables presentation
Truth tables presentation
 
6.3 Truth Tables For Propositions
6.3   Truth Tables For Propositions6.3   Truth Tables For Propositions
6.3 Truth Tables For Propositions
 
Rhetorical devices
Rhetorical devicesRhetorical devices
Rhetorical devices
 
Truth tables
Truth tablesTruth tables
Truth tables
 
CATEGORICAL SYLLOGISM
CATEGORICAL SYLLOGISMCATEGORICAL SYLLOGISM
CATEGORICAL SYLLOGISM
 

Ähnlich wie Logical Connectives (w/ ampersands and arrows)

Logical connectives
Logical connectivesLogical connectives
Logical connectivesdyeakel
 
Maths P.P.T. Connectives
Maths P.P.T. ConnectivesMaths P.P.T. Connectives
Maths P.P.T. ConnectivesAmberSinghal1
 
Logical Operators in Brief with examples
Logical Operators in Brief with examplesLogical Operators in Brief with examples
Logical Operators in Brief with examplesMujtaBa Khan
 
Truth tables complete and p1 of short method
Truth tables complete and p1 of short methodTruth tables complete and p1 of short method
Truth tables complete and p1 of short methodNat Karablina
 
Report in math (conjunctions and disjunctions)
Report in math (conjunctions and disjunctions)Report in math (conjunctions and disjunctions)
Report in math (conjunctions and disjunctions)Omegaxis26
 
CPSC 125 Ch 1 sec 1
CPSC 125 Ch 1 sec 1CPSC 125 Ch 1 sec 1
CPSC 125 Ch 1 sec 1David Wood
 

Ähnlich wie Logical Connectives (w/ ampersands and arrows) (7)

Logical connectives
Logical connectivesLogical connectives
Logical connectives
 
Maths P.P.T. Connectives
Maths P.P.T. ConnectivesMaths P.P.T. Connectives
Maths P.P.T. Connectives
 
Logic.pptx
Logic.pptxLogic.pptx
Logic.pptx
 
Logical Operators in Brief with examples
Logical Operators in Brief with examplesLogical Operators in Brief with examples
Logical Operators in Brief with examples
 
Truth tables complete and p1 of short method
Truth tables complete and p1 of short methodTruth tables complete and p1 of short method
Truth tables complete and p1 of short method
 
Report in math (conjunctions and disjunctions)
Report in math (conjunctions and disjunctions)Report in math (conjunctions and disjunctions)
Report in math (conjunctions and disjunctions)
 
CPSC 125 Ch 1 sec 1
CPSC 125 Ch 1 sec 1CPSC 125 Ch 1 sec 1
CPSC 125 Ch 1 sec 1
 

Kürzlich hochgeladen

Anyhr.io | Presentation HR&Recruiting agency
Anyhr.io | Presentation HR&Recruiting agencyAnyhr.io | Presentation HR&Recruiting agency
Anyhr.io | Presentation HR&Recruiting agencyHanna Klim
 
Chicago Medical Malpractice Lawyer Chicago Medical Malpractice Lawyer.pdf
Chicago Medical Malpractice Lawyer Chicago Medical Malpractice Lawyer.pdfChicago Medical Malpractice Lawyer Chicago Medical Malpractice Lawyer.pdf
Chicago Medical Malpractice Lawyer Chicago Medical Malpractice Lawyer.pdfSourav Sikder
 
NASA CoCEI Scaling Strategy - November 2023
NASA CoCEI Scaling Strategy - November 2023NASA CoCEI Scaling Strategy - November 2023
NASA CoCEI Scaling Strategy - November 2023Steve Rader
 
Upgrade Your Banking Experience with Advanced Core Banking Applications
Upgrade Your Banking Experience with Advanced Core Banking ApplicationsUpgrade Your Banking Experience with Advanced Core Banking Applications
Upgrade Your Banking Experience with Advanced Core Banking ApplicationsIntellect Design Arena Ltd
 
Q2 2024 APCO Geopolitical Radar - The Global Operating Environment for Business
Q2 2024 APCO Geopolitical Radar - The Global Operating Environment for BusinessQ2 2024 APCO Geopolitical Radar - The Global Operating Environment for Business
Q2 2024 APCO Geopolitical Radar - The Global Operating Environment for BusinessAPCO
 
Live-Streaming in the Music Industry Webinar
Live-Streaming in the Music Industry WebinarLive-Streaming in the Music Industry Webinar
Live-Streaming in the Music Industry WebinarNathanielSchmuck
 
PDT 88 - 4 million seed - Seed - Protecto.pdf
PDT 88 - 4 million seed - Seed - Protecto.pdfPDT 88 - 4 million seed - Seed - Protecto.pdf
PDT 88 - 4 million seed - Seed - Protecto.pdfHajeJanKamps
 
IIBA® Melbourne - Navigating Business Analysis - Excellence for Career Growth...
IIBA® Melbourne - Navigating Business Analysis - Excellence for Career Growth...IIBA® Melbourne - Navigating Business Analysis - Excellence for Career Growth...
IIBA® Melbourne - Navigating Business Analysis - Excellence for Career Growth...AustraliaChapterIIBA
 
Borderless Access - Global B2B Panel book-unlock 2024
Borderless Access - Global B2B Panel book-unlock 2024Borderless Access - Global B2B Panel book-unlock 2024
Borderless Access - Global B2B Panel book-unlock 2024Borderless Access
 
UNLEASHING THE POWER OF PROGRAMMATIC ADVERTISING
UNLEASHING THE POWER OF PROGRAMMATIC ADVERTISINGUNLEASHING THE POWER OF PROGRAMMATIC ADVERTISING
UNLEASHING THE POWER OF PROGRAMMATIC ADVERTISINGlokeshwarmaha
 
Developing Coaching Skills: Mine, Yours, Ours
Developing Coaching Skills: Mine, Yours, OursDeveloping Coaching Skills: Mine, Yours, Ours
Developing Coaching Skills: Mine, Yours, OursKaiNexus
 
Borderless Access - Global Panel book-unlock 2024
Borderless Access - Global Panel book-unlock 2024Borderless Access - Global Panel book-unlock 2024
Borderless Access - Global Panel book-unlock 2024Borderless Access
 
Graham and Doddsville - Issue 1 - Winter 2006 (1).pdf
Graham and Doddsville - Issue 1 - Winter 2006 (1).pdfGraham and Doddsville - Issue 1 - Winter 2006 (1).pdf
Graham and Doddsville - Issue 1 - Winter 2006 (1).pdfAnhNguyen97152
 
Data skills for Agile Teams- Killing story points
Data skills for Agile Teams- Killing story pointsData skills for Agile Teams- Killing story points
Data skills for Agile Teams- Killing story pointsyasinnathani
 
AMAZON SELLER VIRTUAL ASSISTANT PRODUCT RESEARCH .pdf
AMAZON SELLER VIRTUAL ASSISTANT PRODUCT RESEARCH .pdfAMAZON SELLER VIRTUAL ASSISTANT PRODUCT RESEARCH .pdf
AMAZON SELLER VIRTUAL ASSISTANT PRODUCT RESEARCH .pdfJohnCarloValencia4
 
Team B Mind Map for Organizational Chg..
Team B Mind Map for Organizational Chg..Team B Mind Map for Organizational Chg..
Team B Mind Map for Organizational Chg..dlewis191
 
Fabric RFID Wristbands in Ireland for Events and Festivals
Fabric RFID Wristbands in Ireland for Events and FestivalsFabric RFID Wristbands in Ireland for Events and Festivals
Fabric RFID Wristbands in Ireland for Events and FestivalsWristbands Ireland
 
Lecture_6.pptx English speaking easyb to
Lecture_6.pptx English speaking easyb toLecture_6.pptx English speaking easyb to
Lecture_6.pptx English speaking easyb toumarfarooquejamali32
 
Talent Management research intelligence_13 paradigm shifts_20 March 2024.pdf
Talent Management research intelligence_13 paradigm shifts_20 March 2024.pdfTalent Management research intelligence_13 paradigm shifts_20 March 2024.pdf
Talent Management research intelligence_13 paradigm shifts_20 March 2024.pdfCharles Cotter, PhD
 

Kürzlich hochgeladen (20)

Anyhr.io | Presentation HR&Recruiting agency
Anyhr.io | Presentation HR&Recruiting agencyAnyhr.io | Presentation HR&Recruiting agency
Anyhr.io | Presentation HR&Recruiting agency
 
Chicago Medical Malpractice Lawyer Chicago Medical Malpractice Lawyer.pdf
Chicago Medical Malpractice Lawyer Chicago Medical Malpractice Lawyer.pdfChicago Medical Malpractice Lawyer Chicago Medical Malpractice Lawyer.pdf
Chicago Medical Malpractice Lawyer Chicago Medical Malpractice Lawyer.pdf
 
NASA CoCEI Scaling Strategy - November 2023
NASA CoCEI Scaling Strategy - November 2023NASA CoCEI Scaling Strategy - November 2023
NASA CoCEI Scaling Strategy - November 2023
 
Upgrade Your Banking Experience with Advanced Core Banking Applications
Upgrade Your Banking Experience with Advanced Core Banking ApplicationsUpgrade Your Banking Experience with Advanced Core Banking Applications
Upgrade Your Banking Experience with Advanced Core Banking Applications
 
Q2 2024 APCO Geopolitical Radar - The Global Operating Environment for Business
Q2 2024 APCO Geopolitical Radar - The Global Operating Environment for BusinessQ2 2024 APCO Geopolitical Radar - The Global Operating Environment for Business
Q2 2024 APCO Geopolitical Radar - The Global Operating Environment for Business
 
Live-Streaming in the Music Industry Webinar
Live-Streaming in the Music Industry WebinarLive-Streaming in the Music Industry Webinar
Live-Streaming in the Music Industry Webinar
 
PDT 88 - 4 million seed - Seed - Protecto.pdf
PDT 88 - 4 million seed - Seed - Protecto.pdfPDT 88 - 4 million seed - Seed - Protecto.pdf
PDT 88 - 4 million seed - Seed - Protecto.pdf
 
IIBA® Melbourne - Navigating Business Analysis - Excellence for Career Growth...
IIBA® Melbourne - Navigating Business Analysis - Excellence for Career Growth...IIBA® Melbourne - Navigating Business Analysis - Excellence for Career Growth...
IIBA® Melbourne - Navigating Business Analysis - Excellence for Career Growth...
 
Borderless Access - Global B2B Panel book-unlock 2024
Borderless Access - Global B2B Panel book-unlock 2024Borderless Access - Global B2B Panel book-unlock 2024
Borderless Access - Global B2B Panel book-unlock 2024
 
UNLEASHING THE POWER OF PROGRAMMATIC ADVERTISING
UNLEASHING THE POWER OF PROGRAMMATIC ADVERTISINGUNLEASHING THE POWER OF PROGRAMMATIC ADVERTISING
UNLEASHING THE POWER OF PROGRAMMATIC ADVERTISING
 
Developing Coaching Skills: Mine, Yours, Ours
Developing Coaching Skills: Mine, Yours, OursDeveloping Coaching Skills: Mine, Yours, Ours
Developing Coaching Skills: Mine, Yours, Ours
 
Borderless Access - Global Panel book-unlock 2024
Borderless Access - Global Panel book-unlock 2024Borderless Access - Global Panel book-unlock 2024
Borderless Access - Global Panel book-unlock 2024
 
Graham and Doddsville - Issue 1 - Winter 2006 (1).pdf
Graham and Doddsville - Issue 1 - Winter 2006 (1).pdfGraham and Doddsville - Issue 1 - Winter 2006 (1).pdf
Graham and Doddsville - Issue 1 - Winter 2006 (1).pdf
 
Data skills for Agile Teams- Killing story points
Data skills for Agile Teams- Killing story pointsData skills for Agile Teams- Killing story points
Data skills for Agile Teams- Killing story points
 
AMAZON SELLER VIRTUAL ASSISTANT PRODUCT RESEARCH .pdf
AMAZON SELLER VIRTUAL ASSISTANT PRODUCT RESEARCH .pdfAMAZON SELLER VIRTUAL ASSISTANT PRODUCT RESEARCH .pdf
AMAZON SELLER VIRTUAL ASSISTANT PRODUCT RESEARCH .pdf
 
WAM Corporate Presentation Mar 25 2024.pdf
WAM Corporate Presentation Mar 25 2024.pdfWAM Corporate Presentation Mar 25 2024.pdf
WAM Corporate Presentation Mar 25 2024.pdf
 
Team B Mind Map for Organizational Chg..
Team B Mind Map for Organizational Chg..Team B Mind Map for Organizational Chg..
Team B Mind Map for Organizational Chg..
 
Fabric RFID Wristbands in Ireland for Events and Festivals
Fabric RFID Wristbands in Ireland for Events and FestivalsFabric RFID Wristbands in Ireland for Events and Festivals
Fabric RFID Wristbands in Ireland for Events and Festivals
 
Lecture_6.pptx English speaking easyb to
Lecture_6.pptx English speaking easyb toLecture_6.pptx English speaking easyb to
Lecture_6.pptx English speaking easyb to
 
Talent Management research intelligence_13 paradigm shifts_20 March 2024.pdf
Talent Management research intelligence_13 paradigm shifts_20 March 2024.pdfTalent Management research intelligence_13 paradigm shifts_20 March 2024.pdf
Talent Management research intelligence_13 paradigm shifts_20 March 2024.pdf
 

Logical Connectives (w/ ampersands and arrows)

  • 2. Truth Functions • Declarative sentences (statements) are either true or false but not both. They cannot be neither. • Whether a sentence is true or false determines its truth value. • Functions take input values to unique output values. That is, the input values determine the output values. • There are some ways of combining sentences into longer ones so that the truth value of the longer sentence is determined by the truth value of the parts that were combined to form it. • So, we’ll call the connectives that combine sentences this way truth functions.
  • 3. And • ‘And’ is often a truth functional connective. Inserting ‘and’ between two statements gives a longer sentence the truth of which is determined by the truth of the parts. • For example, consider: (A) Al admires aardvarks. (B) Barb bakes bologna. The truth of ‘Al admires aardvarks and Barb bakes bologna’ is determined by the truth of A and of B. • If A and B are both true then the longer sentence is true. If either of A or B is false then so is the longer sentence.
  • 4. A Truth Table for ‘And’ • We can express all four A B A and B possibilities for A and B in T T T table form. T F F • Here’s how to read the F T F table. The second F F F (horizontal) row says that when A is true and B is true then A and B is true. The bottom row says that when A is false and B is false then A and B is false.
  • 5. Conjunction • We’ll use a dot ‘∙’ or an ampersand ‘&’ to represent the truth functional connective that is captured by the truth table on slide four. • We’ll call sentences formed with the dot (or ampersand) conjunctions. We’ll call the parts conjuncts. For example, ‘A&B’ is a conjunction and ‘A’ is its left conjunct. • Not every instance of ‘and’ in English can be symbolized with the dot. Contrast ‘Al and Barb are chefs’ (It means: Al is a chef & Barb is a chef.) with ‘Al and Barb are enemies.’ (It doesn’t mean: Al is an enemy & Barb is an enemy.) • Some words besides ‘and’ can be symbolized with the dot or ampersand. For example, ‘but’ and ‘also’ form conjunctions.
  • 6. Or • ‘Or’ is often used as a truth functional connective. • Using the sentences from slide three, the truth of ‘A or B’ can be determined if we know the truth value of A and the truth value of B. • If Al doesn’t admire aardvarks and Barb doesn’t bake bologna then ‘A or B’ is false. Otherwise it’s true. • We’ll symbolize the truth functional (inclusive) ‘or’ with a wedge ‘ ’ and we’ll call the resulting sentences disjunctions. The parts are disjuncts.
  • 7. Truth Table for Disjunction • The four possibilities for A B A B sentences A and B are T T T represented by the left two (vertical) columns. T F T • The top row of Ts and Fs is the F T T possibility where A is true and F F F B is true. On that possibility ‘A B’ is true. • The bottom row is the possibility where A and B are both false. In that case ‘A B’ is also false. • Disjunction differs from conjunction on the middle two rows.
  • 8. Examples • Lenny and Manny left for Bermuda. (L=Lenny left for Bermuda. M=Manny left for Bermuda.) Translation: L&M • Either Lenny or Manny left for Bermuda. Translation: L M • Lenny left for Bermuda and either Manny left for Bermuda or Nancy stayed. (N=Nancy stayed.) Translation: L&(M N) • Lenny left for Bermuda but Manny did too. Translation: L&M • Either Manny left for Bermuda and Nancy stayed or else Lenny left for Bermuda. Translation: (M&N) L
  • 9. Negation • Prefixing a statement with ‘it is A A not the case that’ flips the T F truth value. F T • The symbol to represent that truth function is ‘ ’. The symbol is called ‘tilde’ or just ‘squiggle’. • Unlike the other connectives, tilde doesn’t connect two sentences. It just flips the value of a single sentence. • Also unlike the other connectives, we don’t usually put parenthesis around a negation.
  • 10. Examples • Manny left but Lenny did not leave. Translation: M& L • Either Nancy did not stay and Lenny left or Manny didn’t leave. Translation: ( N&L) M • Neither Lenny nor Manny left. Translation: (L M) [Alternate Translation: L& M] • It’s not true that both Manny and Lenny left. Translation: (M&L) [or M L]
  • 11. If..Then… • Some uses of ‘if..then…’ are truth functional and some are not. We’ll only care about the truth functional ones. • Consider: If you whistle loudly then the dog will come. There’s really just one scenario that shows the sentence to be false: whistle loudly and have the dog not come. If you whistle and the dog comes then the sentence was true. If you don’t whistle then no matter whether the dog comes or not, you didn’t show the sentence wrong.
  • 12. Truth Table for Conditional • We’ll use the arrow ⟶ A B A⟶B (or the horseshoe ) to T T T represent the truth T F F functional ‘if...then…’; F T T and we’ll call statements F F T formed with the arrow (or the horseshoe) ‘conditionals.’ • The top two rows of the truth table for conditional are uncontroversial. The bottom two are less obvious.
  • 13. Order Matters • Notice on the table that the order of A and B matter. A true and B false yields a different value than A false and B true. So, unlike conjunction and disjunction where order doesn’t matter, we have a different names for the different parts of the conditional. • For A⟶B, A is called the ‘antecedent’ and B is called the ‘consequent’. • There are many English expressions that can be translated as conditionals. The trick to symbolizing them correctly is to identify the antecedent and the consequent.
  • 14. Examples of Conditionals • Lenny left if Manny left. Translation: M⟶L • Lenny left only if Manny left. Translation: L⟶M [Think about this one.] • Lenny left in case Manny left. Translation: M⟶L • Lenny left provided that Manny left. Translation: M⟶L • Lenny left unless Manny left. Translation: M⟶L • Whenever Lenny leaves, Manny leaves. Translation: L⟶M
  • 15. If and Only If • ‘A only if B’ is translated A⟶B because when we say ‘A only if B’ we are saying that A can’t be true without B; so, if A is true then so is B. Hence, A⟶B. • ‘A if B’ is the same as ‘if B then A and so is translated B⟶A. • So, if we say ‘A if B and A only if B’ or in other words ‘A if and only if B’ we could translate that as ‘(B⟶A)&(A⟶B)’. We’ll use the double arrow ⟷ (or the triple bar ) as an abbreviation for that. Also, we’ll call sentences formed by using the double arrow ‘biconditionals’. • Sometimes you see ‘iff’ between two sentences. It’s not a typo. It’s an abbreviation for ‘if and only if.’ So ‘A iff B’ would be translated as ‘A⟷B’.
  • 16. Truth Table for Biconditional • Think of the biconditional as A B A⟷B saying that two sentences have matching truth value. The longer T T T sentence is true when both parts T F F are true or when both parts are false. F T F • Notice that order doesn’t matter; F F T so we don’t have special names for the letter that comes first. Unlike regular conditional, A⟷B is equivalent to B⟷A. • The sentence ‘A⟷B ‘is just an abbreviation for the more complicated sentence ‘(B⟶A)&(A⟶B)’ so next I’ll show you where the values in the table to the right came from.
  • 17. Justification for the Biconditional Table • The table gives the value of A B (B⟶A) & (A⟶B) ‘(B⟶A)&(A⟶B)’ in each of the four possibilities for A and B. T T T T T • The left two columns list the T F T F F possibilities for A and B. F T F F T • The sentence we care about is a conjunction of two conditionals. F F T T T To figure out its truth value we need to know the truth value of the conditionals. Then we can use it to figure out the truth value of the conjunction. • The 3rd column gives the values for B⟶A and the 5th column gives values for A⟶B. The 4th column was computed last using the values from columns 3 and 5.