PHIL 201 Notes

Context:

• Socrates - dies 399 BC
• Plato - writes 15 years after that
• Aristotle - even younger than that

What did Socrates write? We know because of comedies written about him, and ideas attributed here. He would say he couldn't teach you anything, only talk about stuff. He talked about the "good life, not a teacher." He was the enemy of the "sophist," who takes money for teaching.

What did Plato do? Dialogues about Socrates, and other stuff. People copied his works for 1500 years, so we have crystal clear knowledge of what he did. He also ran the Academy in Athens (where the "right people"–young men–could get together and learn). He coined the term Academy in Greek. He wrote about "The Good".

Aristotle went to Plato's Academy. He created Formal Logic, all by himself. Here's some of his stuff.

• All Greeks are mortal. Socrates is a greek. Therefore, Socrates is mortal.
• Some snakes are poisonous. There's a snake right next to your shoe. Therefore, you should move away slowly.

Differences between the two:

• The first is a "Universal" and the second is a "Particular" (all/some).
• The first is "deductively valid". That means that every argument of this form is valid, so long as each of the propositions are true.

Modern logic came about because of von Neuman, Alonzo Church, and Alan Turing. The same guys who created computers.

This class is about new ways of organizing arguments, not new and clever arguments. We will also learn how to evaluate arguments:

• Soundness: no false positives. A sound method could be as simple as never answering.
• Completeness: all correct positives. A complete method would always say yes.
• Effective: always gives you an answer.

This logic is:

1. Utterly precise
2. Artificial
3. Simple (but not necessarily easy)
4. Limited

The textbook: "The Logic Book" by Merrie Bergman.

Sentences \(p\) and \(q\) are logically equivalent if there is no way to make one True and the other False.

EG:

• It is either not Wednesday or not raining. (or both)
• It is not Wednesday and raining.
• (FYI, DeMorgan's Law)

Quiz on Wednesday. What will be on the quiz? Definitions mostly. Definitions we covered up till the quiz. Which may include chapter 2.

A list of sentences is consistent if there is Some Way for all to be True at once. Satisfiability!! EG:

• If today is Wednesday, then it's not raining.
• Today is Wednesday.
• It is raining.

No matter how you did it, you couldn't make all of these true. So this list of sentences is not consistent.

P₁, P₂, …, P_n ∴ C is Logically Valid iff the list {P₁, P₂, …, P_n, ¬C} is inconsistent.

• A = it is raining
• B = it is wednesday
• ¬B = not - it is raining

A	B	¬A	¬¬A	A∧B	A∨B	A→B	A≡B
T	T	F	T	T	T	T	T
T	F	F	T	F	T	F	F
F	T	T	F	F	T	T	F
F	F	T	F	F	F	T	T

In "King George's English", you may repeat "not" several times to emphasize just how much "not" you mean. EG "I will not not not not not go to class!". Also, in English, "or" can mean "one or the other, but not both" (exclusive or). It can also mean "one or the other or both" (inclusive or). The only way you can tell is by context. In logic, we define or to be inclusive, when we use the ∨ symbol.

For if/then, we say A → B, where A is the antecedent, and B is the consequent. If has a lot of meanings in English, but in logic, we use implication. It's like a promise. When you tell your friend "if I finish the paper, then I'll come over to your place", the only thing that would offend them is if you finished the paper, but did not come over. Any other situation is pretty much fine.

He says to use plus for exclusive or (if you ever use it), but I prefer ⊕. And we don't really use exclusive or.

• Tomorrow will Rain unless it Snows
  • This is R ∨ S.
  • Unless ≡ Or

Quiz.

No class, professor absent.

No class, labor day.

I skipped. We went over the quiz.

John will sing and Don dance unless Alice reads.

• Sentential logic: J & D ∨ A. But which?
  • (J & D) ∨ A
  • J & (D ∨ A)
  • They are both valid, and different.
  • To see this, let J=F, D=T, A=T. The first one is True, and the second is False.
  • Now we're getting side tracked on unless, vs logical implication, etc.
  • Unless = inclusive or. We get it. Really. Please stop.
• You need to rely on the grammar of the sentence. "Don dance" depends on "John will sing" for a verb, so you would expect that those two are closer than the Alice part.
• In general, showing your work (e.g. by parenthesizing the original sentence), leads to partial credit, but more commonly, it just leads to getting the answer correct more frequently.

If I buy Apples then if I get Bananas I'll get Cherries.

• A → (B → C)
• Or is it (A → B) → C?
• You can tell that they're different because the first is T when A=T, and you can make the second false by then making B=T and C=F.
• It's easy to resolve this one.

If I buy Apples and Bananas, then I'll buy Cherries.

• (A & B) → C

Are those last two sentences the same?

• ~(A & B) ∨ C
• ~A ∨ ~B ∨ C
• ~A ∨ (B → C)
• A → (B → C)
• Yaas!
• Or you could do

Al dislikes bananas, passionately. Al detests bananas.

• Are they equivalent?
• Ch. 1: maaaybe?
• Ch. 2: no. A ≡ B!!

I missed.

~(A&B), ~(B&C), ∴ ~A → C

Not a valid argument. A, B, and C could all be false. Then, you have both premises true, but the conclusion is false.

They're talking about a "tree" method. I missed that. So he's going to do a tree here. I'll take notes.

Set of sentences (not argument, we negated conclusion).

• ~(A&B)
• ~(B&C)
• ~(~A→C)

Now we try to satisfy all 3 with an tree.

• ~A will make ~(A&B) true
  • ~B will make ~(B&C) true
    • ~A and ~C will make ~(~A→C) true
  • ~C will make ~(B&C) true
    • ~A and ~C will make ~(~A→C) true
• ~B will make ~(A&B) true
  • No need to go further.

We can see that in the first branch, we have ~A, ~B, ~C as a way to make all sentences true. Which means it's a counterexample. Yay.

(A → (B & C)) → (A → C)

Is it true?

We can try to make the opposite true.

~((A → (B & C)) → (A → C))

• To make the not true, we make the inside false.
  • To make the inside false, we make the antecedent true and the consequent false. That is, (A → (B & C)) and (A → C).
    • Not sure where we go from here.

There are logic sentences on the board.

• \((A \lor B) \to C\)
• \(D \to E\)
• \(E \to B\)
• ∴ ~C → ~D (negate to ~(~C→~D))
• (A ∨ B) → C
  • ~(A ∨ B)
    • ~A, ~B. Now, D → E
      • ~D: E → B
        • ~E : ~(~C → ~D)
          • ~C, D: FALSE
        • B : ~(~C → ~D)
          • ~C, D: FALSE
      • E: E → B
        • ~E: FALSE
        • B: FALSE
  • C: Now, D → E
    • more stuff
• Should start with the tree that doesn't branch.
• Then go to sentences that look like they'll have conflicts.
• A → (B ∨ C)
• B → (D & E)
• ~(A → D)
• A, ~D: go to A → (B ∨ C)
  • ~A FALSE
  • B ∨ C
    • B: B → (D & E)
      • ~B FALSE
      • D & E
        • D, E: FALSE (~D)
    • C: B → (D & E)
      • ~B: TRUE
      • D & E

Set A, ~B, C, ~D

A ≡ (B ∨ C) C ≡ (A & ~B)

• A, (B ∨ C)
  • B: C ≡ (A & ~B)
    • C, (A & ~B)
      • A, ~B: FALSE
    • ~C, ~(A & ~B)
      • ~A, B: FALSE
  • C:
    • C, (A & ~B)
      • A, ~B: TRUE

Set A, C, ~B

These aren't actually my notes for today's class. There's a quiz in class today, so I'm using today's notes as a place to take notes on the content of the past week. We have three major topics: soundness, completeness, and effectiveness. Particularly, we've been talking about these properties of the Tree Method for Validity in Sentential Logic.

Review day before FOL quiz! Let's get ready to rumble.

A counter example in FOL is a universe of discourse, and probably some other.

Here is a premise and conclusion:

• \(\forall x \exists y Wxy\)
• \(\exists x \forall y Wyx\)

Or the list:

• \(\forall x \exists y Wxy\)
• \(~\exists x \forall y Wyx\), which is \(\forall x ~\forall y Wyx\).

To do the tree:

1. Since there's no way to get a constant, create \(a\) substituted for \(x\).
2. Now we have:
   • \(\exists y Way\)
   • \(~\forall y Wya\)
3. Now, we create \(b\) substituted for \(y\) in the first one:
   • \(Wab\)
   • \(\exists y ~Wya\)

I got a bit lost in this tree stuff. But here is the counter example, in a way that I believe he'd like it on the quiz:

• \(UD = \{0,1\}\)
  • \(a\) is a name for 0
  • \(b\) is a name for 1
• \(W: \{<0,1>, <1,0>\}\)

Or you could also use \(W: \{<0,0>, <1,1>\}\).

Translations?

"Nobody wrote to everyone taller than them."

• ~ ∃ x (∀ y (Tyx → Wxy))
• Here, Tyx means y is taller than x.
• And, Wxy means x wrote to y.
• It appears that he'd like us to go quantifier by quantifier, substituting a new sentence inside so that we can show how we went. For the above:
• Nobody wrote to everyone taller than them.
• ~ ∃ x (x wrote to everyone taller than them)
• ~ ∃ x (∀ y (if y is taller than x, then x wrote to y))
• ~ ∃ x (∀ y (Tyx → Wxy))
• This is a pretty darn solid way to do it.
• Also, liberally apply parentheses.

Sally never writes to me. ~Wsm

Sally never writes to anyone: ~ ∃ x Wsx. Or just ∀ x ~Wsx

Need to translate and do trees. The FOL trees will a bit difficult to learn in time.

List of rules:

• Never work inside a sentence when doing a tree.
• Never create a constant for a universal (unless there's no other place to get one).

The quiz is not today. Go figure. It was postponed due to a study group of 8 people who said they weren't ready. Interesting. So, we'll do some more examples. Maybe I'll get the FOL tree algorithm down.

Don't forget universe of discourse. It's important for statements to have meaning when you interpret them. It's less important for logic class, but still important to have in mind.

Vocab:

• Gx: x is greek
• Px: x is a poet
• Mx: x is mortal

Argument:

• All Greeks are Poets.
• Some Poets are Immortal.
• Some Greeks are immortal.

Translation:

• ∀ x (Gx → Px)
• ∃ x (Px ∧ ~Mx)
• ∃ x (Gx ∧ ~Mx)

Now the discussion is getting lost in the silly, unimportant details of translation.

Tree:

1. Start with the second sentence. Why? Because that will give us a constant. We define \(a\) such that \(Pa \land ~Ma\).
2. Go to the second sentence. Now, we apply the universal to \(a\), and get \(Ga \to Pa\).
3. Go to the third sentence. Do the "rotation', and we get a universal, so we can apply it to \(a\): \(~(Ga \land ~Ma)\).
4. Now we have checked off all our existentials and applied all our universals, so we can start doing a SL tree!
   • We have Pa, ~Ma (from the first sentence)
     • ~Ga (from third sentence). If we did the second sentence, we would see that this branch is left open, so we have a contradiction.
     • ~~Ma (from third sentence) contradiction, so this closes.

The contradiction:

• UD = {a}
• P = {a}
• G = ∅
• M = ∅

Notes on how FOL trees work:

• You close a branch when you encounter anything and its negation (i.e. a contradiction).
• You know you have a counter example when everything that can be checked off (sentences and existentials) has been, and all universals have been applied to all constants, and there are no contradictions.
• You don't check off universals. You simply mark the substitutions next to it.
• You do check off existentials, once you have created a constant for them.