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Truth Functional Logic. Compound Statements. What is a Truth Table?. A truth table is a way of representing a statement’s meaning symbolically. Each compound statement has a single identifying characteristic. Negation. Every claim has a negation or contradictory claim.

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truth functional logic

Truth Functional Logic

Compound Statements

what is a truth table
What is a Truth Table?
  • A truth table is a way of representing a statement’s meaning symbolically.
  • Each compound statement has a single identifying characteristic.
negation
Negation
  • Every claim has a negation or contradictory claim.
  • “Ginger is a dog” has as its negation statement, “Ginger is not a dog.”
  • “Ginger is a dog” = P
  • “Ginger is not a dog” = ~P
truth table for negation
Truth Table for Negation
  • Every statement has two possible truth values, T & F. (True or false)
  • P ~P Negation means

T F opposite truth value

F T If “P” is true, then

~P is false; “P” is false,

then ~P is true.

conjunction statement
Conjunction Statement
  • A conjunction is a compound claim asserting both the simpler claims contained in it.
  • Thus, a conjunction is true only if both of the claims are also true.
  • A conjunction = P & Q
truth table conjunction
Truth Table-Conjunction
  • A conjunction has two claims; each have two possible truth values and thus the compound statement has four possible truth values.

P Q P & Q Since a conjunction is

T T T true only when both first

T F F are true, the first case is

F T F the key case.

F F F

disjunction
Disjunction
  • A disjunction is a compound claim asserting either or both claims contained in it. Thus, a disjunction is false only if both simpler claims are false.
  • P Q P v Q

T T T

T F T

F T T

F F F ↔ This is the key case.

conditional
Conditional
  • A conditional asserts the second claim on the condition that the first is true. A conditional thus is false if and only if the first claim is true and the second is false.
  • P Q P →Q

T T T

T F F ↔ This is key case.

F T T

F F T

truth functional logic 2

Truth Functional Logic-2

Arguments and Truth Tables

validity of arguments
Validity of Arguments
  • The validity of an argument guarantees that if its premises are true, then its conclusion must be true.
  • Thus, an argument is invalid if there is any case where its premises are true and its conclusion false.= Key case.
truth table for arguments
Truth Table for Arguments
  • If there are two variables, S and P, then you need four lines; if three variables you need eight lines.
  • You will need a column for each premise and the conclusion and for each variable, e.g. S and ~S.
truth table argument 2
Truth Table-Argument-2

“If building this requires a small Philips screwdriver, then I will not be able to build it. It does require a small Philips screwdriver. Thus, I will not be able to build it.

Symbolize: S → ~B, S, Thus, ~B

truth table argument 3
Truth Table Argument-3
  • Build the truth table as follows:
  • S B S → ~B S ~B
  • T T F F T F *

T F T T T T

F T T F F F *

F F T T F T

There is no line where the premises are true and the conclusion is false and thus the argument is valid.

truth table argument 4
Truth Table-Argument-4
  • Martin is not buying a new car {since} he said he would buy a new car or take a Hawaiian vacation. He is now in Maui.
  • Symbolize: C v H

H

~ C

truth table argument 5
Truth Table-Argument- 5
  • Build a truth table as follows:

C H C v H H ~ C

T T TTF ↔ Invalid

T F T F F

F T T T T

F F F F T

truth table argument 6
Truth Table Argument-6
  • “If you want to over-clock your processor you must make both hardware and software changes. But you either can’t do hardware or can’t do software. So you won’t be over-clocking your processor.
  • Symbolize: O→ (H & S)

~H v ~ S

~ O

build the truth table
Build the Truth Table
  • Because you have three variables you will need eight lines.
  • First column alternates four true with four false
  • Second column- alternates pairs of true and false.
  • Third column- alternate one true and one false all the way down.
the truth table
The Truth Table

O H S 0 →( H & S) (~H v ~S) ~0

T T T T F F

T T F F T F

T F T F T F

T F F F T F

F T T T F T

F T F F T T

F F T F T T

F F F F T T

There is no case where the conclusion is false and the premises are all true- so it is a valid argument.