Conditional Statements and Their Negations

If-Then

Example: "If it rains, then the streets will be wet."

Explanation: The "if" introduces a sufficient condition (it rains), and the "then" introduces a necessary condition (the streets will be wet). This means rain is enough to guarantee wet streets, but wet streets alone do not guarantee rain; other factors (e.g., sprinklers) could also cause wet streets.

Sufficient and Necessary Conditions

Example: "Having a ticket is necessary to enter the concert." Rephrased: "If you enter the concert, then you have a ticket."

Explanation: The necessary condition is what must be true for the sufficient condition to occur. Having a ticket is required (necessary), but having a ticket alone does not guarantee entry (since other factors like arriving on time may also apply).

And

Example: "If it's sunny and warm, we'll go to the beach."

Explanation: Both conditions (sunny and warm) must be true for the result (going to the beach) to occur. If either condition is false, the result will not happen.

Or (Inclusive)

Example: "If you bring a pencil or a pen, you can take the test."

Explanation: In inclusive "or," at least one condition must be true, but both being true is also acceptable.

Or (Exclusive)

Example: "You can have either the cake or the ice cream, but not both."

Explanation: This type requires exactly one condition to be true—both true or both false will violate the statement.

Nor

Example: "If neither John nor Mary comes, the party will be cancelled."

Explanation: "Nor" combines two negative conditions. For the result (party cancellation) to occur, both John and Mary must be absent.

Either/Or

Example: "Either you clean your room, or you can't go to the movies."

Explanation: This conditional presents two mutually exclusive options. If one occurs, the other cannot.

Unless

Example: "Unless it rains, we will have a picnic."

Equivalent to: "If it doesn't rain, we will have a picnic." Explanation: "Unless" statements indicate that one condition prevents the other. This is best translated as "If not X, then Y."

Negating "Unless" Statements

Example: "We will have a picnic unless it rains."

Negation: "If it doesn't rain, we will have a picnic."

Explanation: To negate an "unless" statement, convert it to its positive form first (as above), then negate the new form.

Only If

Example: "The team will win only if they score more goals."

Explanation: "Only if" introduces a necessary condition. The necessary condition follows "only if," while the sufficient condition appears before it.

If and Only If (IFF)

Example: "A number is even if and only if it's divisible by 2."

Explanation: "Iff" indicates a bi-conditional relationship, meaning both conditions are both necessary and sufficient. Each implies the other.

Whenever/Every Time

Example: "Whenever it snows, schools close."

Explanation: "Whenever" introduces a sufficient condition. The result (schools closing) always follows the stated condition (snow).

All/Every

Example: "All mammals are warm-blooded."

Explanation: Introduces a universal statement that links a sufficient condition to a necessary one.

Without

Example: "Without sunlight, plants cannot grow."

Explanation: Introduces a necessary condition. Its translation is "If no sunlight, then no plant growth."

Not

Example: "If it's not raining, we'll go for a walk."

Explanation: Simple negation directly denies a condition.

Negating 'All' Statements

Original: "All cats are black."

Negation: "Some cats are not black."

Explanation: The negation introduces at least one counterexample.

Negating 'Some' Statements

Original: "Some birds can't fly."

Negation: "No birds can't fly." (Or: "All birds can fly.")

Explanation: Requires denying any exceptions.

Negating 'None' Statements

Original: "None of the students passed the exam."

Negation: "At least one student passed the exam."

Explanation: Negated by introducing one counterexample.

Negating 'Either/Or' Statements

Original: "Either John or Mary will attend the meeting."

Negation: "Neither John nor Mary will attend the meeting."

Explanation: Requires denying both options

Even if

Example: "Even if it rains, the game will go on."

Explanation: This highlights that the condition (rain) has no effect on the result (game continuing).

Whether or not

Example: "Whether or not you agree, we're going ahead with the plan."

Explanation: The outcome remains the same regardless of the stated condition.

Regardless of whether

Example: "Regardless of whether it's cold, we'll have the picnic."

Explanation: Similar to "whether or not," this signals that the condition does not affect the outcome.

As long as

Example: "As long as you have a ticket, you can enter the venue."

Explanation: This indicates that the stated condition is the only requirement for the outcome to occur.

Provided that

Example: "You can use the car provided that you fill up the tank."

Explanation: "Provided that" functions as "if and only if," signaling a requirement for the condition to hold true.

On condition that

Example: "I'll lend you the money on condition that you pay it back within a month."

Explanation: Similar to "provided that," this signals a strict condition for the result to occur.

In the event that

Example: "In the event that it rains, the outdoor concert will be cancelled."

Explanation: This introduces a sufficient condition; if the condition occurs, the outcome will follow.

None/No

Example: "No students are allowed in the faculty lounge."

Explanation: "None" statements set a strict rule that no element from the first category can be part of the second.

Negating "If-Then" Statements

Original: "If it's sunny, then we'll go to the beach."

Negation: "It's sunny, but we won't go to the beach."

Explanation: The proper negation shows that the sufficient condition occurs without the necessary condition following.

“Than either”

Example: “Mary's LSAT score is higher than either Jane or John,"

Explanation: The phrase "than either" or "than does either" indicates that a condition is true for both of the alternatives, not just one. This means it's higher than both of them, not just one.