Logical Connective - Implication (If…Then…)
What is the truth table for the logical connective “If X then Y” (X → Y)?
X Y X → Y
T T T
T F F
F T T
F F T
Logical Connective - Disjunction (Or)
What is the truth table for the logical connective “X or Y” (X ∨ Y)?
X Y X ∨ Y
T T T
T F T
F T T
F F F
Logical Connective - Conjunction (And)
What is the truth table for the logical connective “X and Y” (X ∧ Y)?
X Y X ∧ Y
T T T
T F F
F T F
F F F
What is a tautology? What is a classic example?
A tautology is a proposition that is ALWAYS true, regardless of the truth values of its components. It is a necessary truth.
Example: “X or Not X” is always True.
What is a contradiction? What is a classic example?
A contradiction is a proposition that is ALWAYS false, regardless of the truth values of its components.
Example: “X and Not X” is always False.
What is a contingent statement?
A contingent statement is one that is sometimes true and sometimes false, depending on the truth values of its components. (Most everyday statements are contingent).
What is a valid argument?
An argument is valid if there is no possible way for all of its premises (hypotheses) to be true and its conclusion to be false at the same time. Validity is about the logical structure, not the actual truth of the premises.
What is the counter-intuitive result of the Monty Hall paradox?
In the Monty Hall problem, switching doors after the host reveals a goat doubles your probability of winning from 1/3 to 2/3. This is counter to the intuitive feeling that the odds should be 50/50.
What is the Correspondence Theory of Truth?
A proposition is true if and only if it corresponds to, or matches, some objective fact in reality.
Core Idea: Truth is a connection between language/thought and the physical world.
What is the Coherence Theory of Truth?
A proposition is true if and only if it is consistent with, or coheres with, the rest of our previously accepted beliefs and knowledge.
Core Idea: Truth is about internal consistency within a system of beliefs, rather than a connection to an external reality.
What is the Pragmatic Theory of Truth in logic?
It’s the view that a logical statement or belief is considered “true” if it is useful and has practical success.
In logic, this means a rule or system is “true” if it reliably helps us reason correctly, solve problems, and make accurate predictions.
What is the barber’s rule that causes a paradox?
“I shave everyone who does not shave themselves.”
This creates a contradiction because we can’t determine if the barber shaves himself or not.
What is the contradiction with russel’s paradox?
If YES, he shaves himself: Then he does shave himself, so he should NOT be shaved by the barber (according to the rule). But he is the barber! So he must not shave himself. Contradiction.
If NO, he doesn’t shave himself: Then he does not shave himself, so he should be shaved by the barber (according to the rule). So he must shave himself. Contradiction.
What does Russell’s Paradox prove?
You can’t just define any set with any property. Some definitions (like “set of all sets that don’t contain themselves”) lead to logical contradictions.
What is the Liar Paradox?
A statement that says: “This sentence is false.” If it’s true, then it must be false. If it’s false, then it must be true. It can’t be either without creating a contradiction.
Why is the Liar Paradox a problem?
If TRUE: Then it’s telling the truth by saying it’s false. So it must be FALSE.
If FALSE: Then it’s lying. Since it says “I am false,” a lie would mean it’s actually TRUE.
It creates an endless loop where it can’t be consistently true or false.
