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Propositional Logic – The Basics (2)

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Propositional Logic – The Basics (2)

Truth-tables for Propositions

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

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 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.

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.

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

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

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

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

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

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

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