Recap: Alyssa Ney's Metaphysics: Preparatory Background

Propositional Logic

AND: P∧Q

P and Q.

OR: P∨Q

P or Q.

IMPLIES: P⟶Q

If P then Q.

NEGATION: ¬P

Not P.

IFF: P⟷Q

If P then Q and if Q then P.

THEREFORE: ∴Q

Therefore Q.

Modus Ponens (Affirming the Antecedent): 

1. P⟶Q

2. P

3. ∴Q

Modus Tollens (Denying the Consequent):

1. P⟶Q

2. ¬Q

3. ∴¬P

Disjunctive syllogism:

1. P∨Q

2. ¬P

3. ∴Q

First-order Predicate Logic

Existential Quantifier: ∃x

This can be read in the following ways (pg 20):

There exists an x such that...

There is at least one x such that...

Some x is...

Something is...

e.g. There is at least one philosopher = ∃xPx (There is an x such that x is a philosopher.)

e.g. There is at least one happy philosopher = ∃x(Px∧Hx) (There is an x such that it is a philosopher and happy.)

We can call a line of symbolic logic a sentence.

Note 1: Sentences with the existential quantifier do not always imply that something exists. (pg 26)

e.g. Fa ⟶ ∃x∃Px (If a is F, then there is at least one thing that is P.)

You're only committed to something being P if there is an a that is F.

Note 2: For proper syntax, make sure your variables are bound to quantifiers. (pg 21)

e.g. ∃xFx ∧ Gx

The variable x in Fx is bound to a quantifier, but the variable x in Gx is not. To fix this, use parentheses:

∃x(Fx∧Gx)

Universal Quantifier: ∀x (For all x's...)

e.g. All philosophers are happy = ∀x(Px⟶Hx) (For all x, if x is a philosopher, then x is happy.) (pg 23)

Note 3: ∀x does not entail that x's exist, only that if they do exist, then all of them have the features stated.

Existential Quantifier Introduction (EI)

Fa ⟶ ∃xFx (If a is F, then something is F.)

e.g. Humility is a virtue. Therefore there is something that is a virtue. (pg 25)

Existential Quantifier Elimination (EE)

∃xFx ⟶ Fa (There is something that is F and we will label a as one of those things.)

Universal Quantifier Introduction (UI)

Fa ⟶ ∀xFx (If an arbitrary a is F, then all a's are F. So all of something is F.)

e.g. Show that a is an arbitrary apple. If a is affected by gravity, then all apples are affected by gravity. So all of something is affected by gravity. (example validated by Chat GPT)

Universal Quantifier Elimination (UE)

∀xFx ⟶ Fa (If everything is F, then a is F.)

 

e.g. ∀xIx (Idealism: Everything is an idea in a mind). Therefore, Disneyland is an idea in a mind. (pg 27)

Mostly, EI and UE are used.

(The quantifier characters show up as slightly bigger or smaller versions. On the back end it's all one symbol; I don't know what causes the change.)

There's more to this chapter but these are the highlights that interest me.