Truth Tables for Propositions

Август 23, 2022

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Truth Tables for Propositions

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2. Today’s Plan Truth Tables for Propositions What is a truth table and how to build a
3. LAST WEEK: Summary of Previous Lecture Every compound proposition has the property of bivalence. Propositional Logic
4. TODAY: we will look at the case where we don t know the value of the
5. 1. HOW TO CONSTRUCT A TRUTH TABLE Equation tells how many lines a truth table need
6. 6
7. STEPS Determine number of lines required Make sure you consistently record the truth table (last version
8. If Cecilia goes to the party and Dave is not going to the party, then Erik
9. 3. CLASSIFYING STATEMENTS TAUTOLOGY (all the values that fall under the main operator are true) SELF
10. 10
11. If George goes to the party Harry is gonna go, and George is gonna go then
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13. 4. COMPARING STATEMENTS (comparing two compound statements) There are 4 possibilities: LOGICALLY EQUIVALENT (identical values under
14. 2. CONTRADICTORY (opposite values under the main operator) 14
15. 3. CONSISTENT (at least one line where both statements are true) 15
16. 4. INCONSISTENT (there is no line where both statements are true) – they cannot be both
17. INTERMIDIATE SUMMARY 3 types of statements: Tautology (all true values) Self Contradictory (all false values) Contingent
18. ANALOGY WITH CSI CASE - TWO WITNESSES LOGICAL EQUIVALENCE – they both have the same story
19. EXERCISES Tautology, self contradictory, or contingent statements? 19
20. 20
21. 21
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Today’s Plan
Truth Tables for Propositions
What is a truth table and how

to build a truth table
Computing compound propositions
Classifying propositional statements
Comparing propositional statements
Exercises

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LAST WEEK: Summary of Previous Lecture
Every compound proposition has the property

of bivalence.
Propositional Logic allows us to create compound propositions We saw that if you know the value of its elements you can calculate the truth value of the whole proposition by looking at the main operator
Example:

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TODAY: we will look at the case where we don t

know the value of the elements contained in the proposition.
We look at how to build Truth tables and how to use them to calculate all possible truth values of a proposition
Looking at all possible truth values affords us to do two things:
1. classify specific types of statements. Some of those are always good and always true statements (these are called tautological) some of those are not etc..
2. compare compound statements. So, assess complex arguments

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1. HOW TO CONSTRUCT A TRUTH TABLE
Equation tells how many lines

a truth table need to be in order to adequately capture all possible truth values
L = number of lines
= numerical
n = number of propositional variables (unique propositional variables)

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6

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STEPS
Determine number of lines required
Make sure you consistently record the truth

table (last version T/F; T/F; T/F)
Compute the table using the truth functional rules

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If Cecilia goes to the party and Dave is not going

to the party, then Erik will go to the party.
Under which conditions these statements will be truth?

2. ANALYSE COMPOUND PROPOSITIONS

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3. CLASSIFYING STATEMENTS
TAUTOLOGY (all the values that fall under the main

operator are true)
SELF CONTRADICTORY (all false values under the main operator – all possible conditions are always false: avoid)
CONTINGENT (mixed values under the main operator, at least one true and one false

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10

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If George goes to the party Harry is gonna go, and

George is gonna go then Harry is gonna go.

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12

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4. COMPARING STATEMENTS (comparing two compound statements)
There are 4 possibilities:
LOGICALLY

EQUIVALENT (identical values under the main operator)

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2. CONTRADICTORY (opposite values under the main operator)
14

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3. CONSISTENT (at least one line where both statements are true)
15

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4. INCONSISTENT (there is no line where both statements are true)

– they cannot be both truth at the same

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INTERMIDIATE SUMMARY
3 types of statements: Tautology (all true values) Self Contradictory (all

false values) Contingent (mixed values) When you compare such statements you get: Logically equivalent (identical values) Contradictory (opposite truth values) Consistent (at least one line where both are true) Inconsistent (no line where both are true)

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ANALOGY WITH CSI CASE -
TWO WITNESSES
LOGICAL EQUIVALENCE – they

both have the same story
CONTRADICTION – one of the them is telling truth; the other is telling a false story
CONSISTENCY - both could possibly be lying but is a possibility that they can both say the truth
INCONSISTENCY – both could be lying but there is no way in which both are telling truth

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