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Propositional Logic – The Basics (2). Truth-tables for Propositions. Assigning Truth. True or false? – “This is a class in introductory-level logic.”. “This is a class in introductory-level logic, which does not include a study of informal fallacies.”.

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Propositional Logic – The Basics (2)

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Propositional Logic – The Basics (2)

Truth-tables for Propositions

Assigning Truth

True or false? –

“This is a class in introductory-level logic.”

“This is a class in introductory-level logic, which does not include a study of informal fallacies.”

“This is a class in introductory-level logic, which does not include a study of informal fallacies.”

L● ~F

“This is a class in introductory logic, which includes a study of informal fallacies.”

“This is a class in introductory logic (T), which includes a study of informal fallacies (F).”

L● F

TF

F

Propositional Logic and Truth

The truth of a compound proposition is a function of:

• The truth value of it’s component, simple propositions, plus

• the way its operator(s) defines the relation between those simple propositions.

p ● qp v q

T FTF

F

T

Truth Table Principles and Rules

Truth tables enable you to determine the conditions under which you can accept a particular statement as true or false.

Truth tables thus define operators; that is, they set out how each operator affects or changes the value of a statement.

Truth and the Actual World

Some statements describe the actual world - the existing state of the world at “time x”; the way the world in fact is.

“This is a logic class and I am seated in SOCS 203.”

- Actually and currently true on a class day.

- Possibly true, but not “currently” true on Monday, Wednesday or Friday.

Truth and Possible Worlds

Some statements describe possible worlds - particular states of the world at “time y”; a way the world could be..

“This is a history class and I am seated in SOCS 203.”

Possibly true, but not currently true.

Actually true, if you have a history class here and it is a history class day/time.

A truth table describes all possible combinations of truth values for a statement. It will, in fact, even tell you if a statement could not possibly be true in any world.

Constructing Truth Tables

1. Write your statement in symbolic form.

2. Determine the number of truth-value lines you must have to express all possible conditions under which your compound statement might or might not be true.

Method: your table will represent 2n power, where n = the number of propositions symbolized in the statement.

3. Distribute your truth-values across all required lines for each of the symbols (operators will come later).

Method: Divide by halves as you move from left to right in assigning values.

Constructing Truth Tables - # of Lines

For statement forms, there are only two symbols. Thus, these require lines numbering 22 power, or 4 lines.

Constructing Truth Tables – Distribution across all Symbols

Under “p,” divide the 4 lines by 2. In rows 1 & 2 (1/2 of 4 lines), enter “T.” In rows 3 & 4, (the other ½ of 4 lines), enter “F.”

TTFF

TTFF

Constructing Truth Tables – Distribution across all Symbols

Under “q,” divide the 2 “true” lines by 2. In row 1 (1/2 of 2 lines), enter “T.” In row 2, (the other ½ of 2 lines), enter “F.”

Repeat for lines 3 & 4, inserting “T” and “F” respectively.

TTFF

TTFF

TF

TF

TF

TF

Constructing Truth Tables – Operator Definitions

Thinking about the corresponding English expressions for each of the operators, determine which truth value should be assigned for each row in the table.

TTFF

TTFF

T

TF

TF

T

FF

FFF

TF

TF

T

Constructing Truth Tables - # of Lines

Remember that you are counting each symbol, not how many times symbols appear.

2 symbols: 1 appearance of “p” and 2 appearances of “q”

Exercises - 1

Using the tables which define the operators, determine the values of this statement.

( M > P ) v ( P > M )

TTFF

TFTT

TFTF

TTTT

TFTF

TTFT

TTFF

Exercises – 2

Using the tables which define the operators, determine the values of this statement.

TTTTFFFF

TTFFTTTT

TTFFTTFF

TTFFTFTF

FFFFTTTT

TTTTFFFF

TTTTTFTF

TFTFTFTF

FFFFFFFF

FFFTFFFT

TTFFTTFF

TTTFTTTF

TFTFTFTF

TTFFTTFF