Propositions and Truth Values Notes

Propositions and Truth Values

Definitions

• A proposition makes a claim (either an assertion or a denial) that may be either true or false. It must have the structure of a complete sentence.

• Any proposition has two possible truth values:

  • T = true

  • F = false

• A truth table is a table with a row for each possible set of truth values for the propositions being considered.

Negation (Opposites)

• The negation of a proposition $p$ is another proposition that makes the opposite claim of $p$.

  • If $p$ is true (T), not $p$ is false (F).

  • If $p$ is false (F), not $p$ is true (T).

• Symbol: $\sim$

Example: Negation

• Proposition: Amanda is the fastest runner on the team.

• Negation: Amanda is not the fastest runner on the team.

• If the negation is false, the original statement must be true, meaning that Amanda is the fastest runner on the team.

Double Negation

• The double negation of a proposition $p$, not not $p$, has the same truth value as $p$.

Example: Radiation and Health

• Original statement: "My opinion is that it’s unlikely that there is no association."

• Analysis:

  • $p$ = it’s likely that there is an association (between low-level radiation and cancer)

  • "unlikely" gives us: it’s unlikely that there is an association, which is $\sim p$.

  • "no association" transforms the last statement into the original statement, it’s unlikely that there is no association, which is $\sim \sim p$.

• Conclusion: the scientist believes it likely that there is an association between low-level radiation and cancer among younger workers, because the double negation has the same truth value as the original proposition.

Logical Connectors

• Propositions are often joined with logical connectors—words such as and, or, and if…then.

• Example:

  • $p$ = I won the game.

  • $q$ = It was fun.

  • I won the game and it was fun.

  • I won the game or it was fun.

  • If I won the game, then it was fun.

And Statements (Conjunctions)

• Given two propositions $p$ and $q$, the statement $p$ and $q$ is called their conjunction.

• It is true only if $p$ and $q$ are both true.

• Symbol: $\land$

Example: And Statements

• a. The capital of France is Paris and Antarctica is cold.

  • Both propositions are true, so their conjunction is also true.

• b. The capital of France is Paris and the capital of America is Madrid.

  • The first proposition is true, but the second is false. Therefore, their conjunction is false.

Two Types of Or

• An inclusive or means “either or both.”

• An exclusive or means “one or the other, but not both.”

• In logic, assume or is inclusive unless told otherwise.

The Logic of Or (Disjunctions)

• Given two propositions $p$ and $q$, the statement $p$ or $q$ is called their disjunction.

• It is true unless $p$ and $q$ are both false.

• Symbol: $\lor$

Example: Smart Cows?

• Statement: Airplanes can fly or cows can read.

• The statement is a disjunction of two propositions: (1) airplanes can fly; (2) cows can read.

• The first proposition is true, while the second is false, which makes the disjunction $p \lor q$ true.

• That is, the statement Airplanes can fly or cows can read is true.

The Logic of If . . . Then Statements (Conditionals)

• A statement of the form if p, then q is called a conditional proposition (or implication).

• It is true unless $p$ is true and $q$ is false.

  • Proposition $p$ is called the hypothesis.

  • Proposition $q$ is called the conclusion.

Alternative Phrasings of Conditionals

The following are common alternative ways of stating if p, then q:

• $p$ is sufficient for $q$

• $p$ will lead to $q$

• $p$ implies $q$

• $q$ is necessary for $p$

• $q$ if $p$

• $q$ whenever $p$

Variations on the Conditional

Logical Equivalence

• Two statements are logically equivalent if they share the same truth values.

• Original and Contrapositive are logically equivalent.

• Converse and Inverse are logically equivalent.

Example: Logical Equivalence

Consider the true statement if a creature is a whale, then it is a mammal.

• Conditional: If a creature is a whale, then it is a mammal. (True)

• Converse: If a creature is a mammal, then it is a whale. (False, because most mammals are not whales.)

• Inverse: If a creature is not a whale, then it is not a mammal. (False, for example, dogs are not whales, but they are mammals.)

• Contrapositive: If a creature is not a mammal, then it is not a whale. (True, because all whales are mammals.)

• The original proposition and its contrapositive have the same truth value and are logically equivalent.

• Similarly, the converse and inverse have the same truth value and are logically equivalent.