8/9 8. Reasoning Errors and Informal Fallacies Free Online FlashCards

What does $\forall$ mean?

The universal quantifier $\forall$ means “for every,” “all,” or “each.”

What does $\exists$ mean?

The existential quantifier $\exists$ means “there exists,” “some,” or “at least one.”

Why does the domain of discourse matter?

A quantified statement’s truth can change when its domain changes, because the variables range over different objects.

How is “some student passed” formalized?

“Some student passed” is $\exists x\,(Student(x)\land Passed(x))$; the witness must be both a student and someone who passed.

What is a quantifier’s scope?

The scope is the formula governed by a quantifier. In $\forall x\,[P(x)\rightarrow Q(x)]$, the quantifier governs the entire conditional.

How do bound and free variables differ?

A bound variable occurs within a quantifier’s scope; a free variable is not governed by a quantifier and leaves the formula generally open.

How do you negate a universal quantifier?

Negation switches the quantifier and negates the predicate: $\neg(\forall x\,P(x))\equiv\exists x\,\neg P(x)$.

How do you negate an existential quantifier?

Negation switches the quantifier and negates the predicate: $\neg(\exists x\,P(x))\equiv\forall x\,\neg P(x)$.

What does $\forall x\,\exists y\,L(x,y)$ mean?

$\forall x\,\exists y\,L(x,y)$ allows the liked person to vary with $x$; each person likes at least one person.

What does $\exists y\,\forall x\,L(x,y)$ mean?

$\exists y\,\forall x\,L(x,y)$ requires one single person whom everyone likes.

What refutes “All $A$ are $B$”?

One valid counterexample: an object in the domain that satisfies $A$ but not $B$. Its form is $\exists x\,(A(x)\land\neg B(x))$.

What establishes an existential claim?

A witness is one object in the domain that satisfies the predicate. For example, $5$ witnesses that some integer is positive and prime.