Kinds of Truths

In-Class Assignment #2

The null set is a proper subset of every set? FALSE “A is a subset of B” = “You can’t find a member of A that isn’t also in B” So the null set is a subset of every set. “A is a proper subset of B” = “A is a subset of B and not vice versa.” But null set is a subset of itself and vice versa.

Names for Sets

Extensive Notation Bracket symbols

Extensive Notation Names of the set’s members John , Paul , George , Ringo

Extensive Notation Order doesn’t matter Paul , Ringo , John , George

Extensive Notation Any name will do Paul , Ringo , Dr. Winston O’Boogie , George

Intensive Notation Variable (your choice: x, y, z, etc.) x

Intensive Notation Up and down line x

Intensive Notation Condition that uniquely picks out the set’s members x x is a member of The Beatles

Intensive Notation Condition that uniquely picks out the set’s members x x had the most #1 British albums

Intensive Notation Condition that uniquely picks out the set’s members x x sang a song on Rubber Soul

Intensive Notation Condition that uniquely picks out the set’s members x x = John or x = Paul or x = George or x= Ringo

Converting Notations Write names for the following set in extensive notation: { x | 3x = 4 } Wrong answers: 4/3 ‘4/3’ x ⊆ { 4/3 } ‘{ 4/3 }’ { x | 4/3} { 3x = 4 } 4

Converting Notations Write names for the following set in extensive notation: { x | 3x = 4 } Right answer: { 4/3 } { 1 1/3 } { 8/6 } { 2 – 2/3 }

Converting Notations Write names for the following set in extensive notation: { x | x is the name of the instructor of PHIL 2000 } Wrong answers: Michael ‘Michael’ { Michael } ‘{ Michael }’

Converting Notations Write names for the following set in extensive notation: { x | x is the name of the instructor of PHIL 2000 } Right answer: { ‘Michael’ } { ‘Michael Johnson’ } { ‘Dr. Johnson’ }

Converting Notations Write names for the following set in intensive notation: { } Wrong answers: The null set the empty set Ø { Ø } { x | x is empty } { x | x is { } }

Converting Notations Write names for the following set in intensive notation: { } Right answers: { x | x is a member of { } } { x | x = 0 & x = 1 } { x | x > 5 & x < 5 } { x | x rhymes with ‘orange’ & x ≠ ‘orange’ }

Converting Notations Write names for the following set in intensive notation: { Canada, Kennedy Town, Run Run Shaw Tower } Wrong answers: Places Places on Earth The set whose members are Canada, K-Town, and Run Run Shaw Twr. { x | x = Canada, x | x = K-Town, x | x = Run Run Shaw Tower }

Converting Notations Write names for the following set in intensive notation: { Canada, Kennedy Town, Run Run Shaw Tower } Right answers: { x | x is a member of { Canada, Kennedy Town, Run Run Shaw Tower } } { x | x = Canada OR x = K-Town OR x = Run Run Shaw Tower } { x | x is a place where the police are looking for Michael }

Power Sets What is the power set of { A, B, C } ? STEP 1: Do the calculation. How many members does the set have? Three. So the power set has 23 members = 8. STEP 2: Make a chart with 1 column for each member and 1 row for each subset.

A B C -

Power Set STEP 3: Put set brackets around each row. These are the subsets of the original set.

A B C -

Power Set STEP 4: Put set brackets around all the subsets. This is the power set.

A B C - , , , , , , ,

What We’re Skipping

Chapter 3 Real numbers and the power set of the natural numbers The power set theorem The continuum hypothesis

Two Distinctions

Analytic/ Synthetic Distinction Examples of analytic truths: geometric terms, kinship terms, animal terms (boar, sow, piglet, drift, pork) True in virtue of meaning? Relation to definitions? Synthetic truths: “truth depends on actual facts”

A priori/ A posteriori Epistemological distinction A priori “can be known prior to the experience of facts”

A Priori Knowledge

A priori/ A posteriori Epistemological distinction A priori “can be known prior to the experience of facts” Examples: analytic truths Some experience necessary: learning the concepts Other examples: Cartesian truths “I exist” true in virtue of meaning? A priori known vs. knowable: computing sums w/ calculator A posteriori “can only be known as a result of relevant experiences”

Synthetic A Priori? Truth depends on actual facts, not just word meanings/ can be known without investigating actual facts Examples? “All triangles have interior angles that sum to π radians”? “The real numbers can’t be paired one-to-one with the integers”? “The future will resemble the past”? Universal Grammar?

The Knowledge of Babies “In a few domains, babies seem to have intuitions that guide their expectations about how important entities in the world (e.g., objects, people) act and interact. For example, babies appear to be born knowing that objects cannot magically appear or disappear, that they cannot pass through each other, and that they cannot move unless contacted by another object. These expectations hold for objects, but not for non-object entities like substances (e.g., liquid, sand).” --Kristy vanMarle

Explanation 1: “Innate Ideas from God” God gives us some knowledge at birth. Problem: methodological naturalism. The problem of epistemological evil– why doesn’t God give us more or better knowledge?

Explanation 2: “Preconditions of Experience” Experience is a product of the “things in themselves” AND the way our mind structures them. Our mind imposes a Euclidean space-time structure on experience. Problem?: space-time isn’t part of external reality. Problem: empirically false.

Explanation 3: Evolution Just as we are born with various innate physical traits that are the product of evolution, so too are we born with innate mental traits that are the product of evolution.

Explanation 3: Evolution Problem: evolution satisfices, doesn’t optimize. In fact, many innate principles are only approximately true– consider optical illusions.