Concepts Knowledge (“knowing that__”) as justified true belief Truth value Belief Justification Counterexample Sorites Paradox Mathematical Induction
What’s the point of this discussion? We confuse truth with notions like belief, knowledge and justification. That makes us reluctant to accept the account of truth value that figures in classical logic. If we get clear about what knowledge is--and isn’t--then the claims we make about truth value won’t seem that crazy. We will also use this discussion as an excuse to talk about some other important concepts along the way.
Propositional Knowledge Propositional knowledge is knowing that as distinct from… Knowing who or Knowing how x knows that P
Knowledge as Justified True Belief (the “JTB” account of knowledge) What is justification? What is truth? What is belief?
Truth Value There are just two truth values: true and false (“bivalence”) Truth value does not admit of degree Truth value is not relative to persons, places, times, cultures or circumstances How do we know? We stipulate that this is how we’ll understand truth value! We idealize…
Idealization Idealization is the process by which scientific models assume facts about the phenomenon being modeled that are not strictly true. Often these assumptions are used to make models easier to understand or solve. Examples of idealization –In geometry, we assume that lines have no thickness. –In physics people will often solve for Newtonian systems without friction. –In economic models individuals are assumed to be maximally rational self-interested choosers.
Defending bivalence Is our idealized notion of truth value close enough to the messy real world idea of truth and falsity? To make the case that it is, we’ll consider some apparent counterexamples –Where truth value seems to be a matter of degree –Where truth value seems to be relative And respond to them. First…how to respond to putative counterexamples…
Counterexample A case that shows a general claim to be false E.g. claim: for all numbers a, b, x, if a > b then ax > bx. True? NO! The case in which x = 0 is a counterexample! And there are lots more.
But not everything that looks like a counterexample really is one E.g. claim: All monkeys have tails. Apparent counterexample: Chimpanzees don’t have tails. NOT A COUNTEREXAMPLE! Chimps aren’t monkeys--they’re apes. Rebutting apparent counterexamples
Defending our idealized account of truth value We’ll consider apparent counterexamples to our claims about truth value which purport to show that: –Some propositions have truth values that are “between” true and false –Some propositions are neither true nor false –The truth value of some propositions is relative to persons, places, cultures, etc. We’ll respond to these counterexamples in various ways in order to show that our account of truth value isn’t completely off the wall.
Bivalence: “2-valuedness” Claim: there are just two truth-values, true and false-- nothing else, nothing in between, no almost-true or almost-false. Apparent Counterexamples: –Conjunctions –Vagueness
Apparent Counterexample –For Sale: 1996, 4-door Nissan Sentra. New clutch, low mileage [um, it’s almost true--everything except the low mileage] –Response: we treat this as a conjunction and stipulate that a conjunction is true only if all its conjuncts are true.
Conjunction: “and” statement My car is a 1996 and it’s got four doors and it’s a Nissan Sentra and it’s got a new clutch and it’s got low mileage. False! It’s got 209,173 miles on it. If we want to get more specific, we can ask: is it a 1996? Does it have 4 doors, etc.
Vagueness Truth and falsity are all-or-nothing, like the oddness and evenness of numbers. Counterexamples? –Vagueness, e.g. “Stealing is wrong.” –Response: This isn’t a complete thought. We clarify and spell out details to eliminate vagueness where possible… –And ignore recalcitrant cases like the dread Sorites Paradox.
The Sorites Paradox We agree that 100,000 grains of sand are a heap… And that one grain of sand is not a heap… And…
Sorites Paradox We agree that removing one grain of sand from a heap won’t make it stop being a heap…
The Sorites Paradox a.k.a the Paradox or the Heap or the Bald Man 1.A 100,000 grain collection is a heap 2.If a k-grain collection is a heap then a (k - 1)-grain collection is a heap 3.Therefore, a 9,999-grain collection is a heap [by 1, 2] 4.Therefore, a 9,998-grain collection is a heap [by 2, 3]… Uh-oh! n.Therefore, a one-grain collection is a heap [by 2, n - 1]
A Big Problem The Sorites argument, which leads to the ridiculous conclusion that one grain of sand is a heap, is a proof by mathematical induction. To say that the argument is no good would seem to commit us to rejecting mathematical induction… And that would be VERY BAD!
Mathematical Induction Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.
Mathematical Induction A proof by mathematical induction consists of two steps: The basis (base case): showing that the statement holds for a natural number, n, e.g. when n = 1 The induction step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n. This proves that the statement holds for all values of n.
Mathematical Induction 1.P holds for 1 [by base step] 2.If P holds for some natural number n then it holds for n + 1 [by induction step] 3.So P holds for 2 [by 1, 2] 4.So P holds for 3 [by 2, 3] 5.So P holds for 4 [by 2, 4] … So the dominos all fall!
We want to show that for any natural number n, the sum of numbers 1 + … + n = Call the proposition that 1 + … + n = “P” P is true for n = 1 since P is true for n = 2 since 1 + 2 = 3 and P is true of n = 3 since 1 + 2 + 3 = 6 and And so on... But “and so on” is not a proof! Example of Math Induction
This is how you prove it We want to prove P: 1 + … + n = Base Step: we show that P holds where n = 1: Induction Step: we show that if P holds for a number n then it holds for n + 1 –Suppose P holds for n, i.e. 1 + … + n = –We do some algebra to show that P holds for n + 1, i.e. that 1 + … + n + (n + 1) = We’re done! This shows that P holds for all n’s! See how it’s done here: https://www.khanacademy.org/math/precalculus/seq_induction/p roof_by_induction/v/proof-by-induction https://www.khanacademy.org/math/precalculus/seq_induction/p roof_by_induction/v/proof-by-induction
Sorites is a Math Induction Argument! Basis: A 100,000 grain collection is a heap. Induction step: If an k-grain collection is a heap then an (k - 1)- grain collection is a heap. So all the dominoes fall…and there seems no way to avoid the conclusion that a one-grain collection is a heap! What should we do???
We run away fast! Sorites We’ll ignore the Sorites in this class...So now for some easier problems. (For further discussion see http://plato.stanford.edu/entries/sorites-paradox/)http://plato.stanford.edu/entries/sorites-paradox/) Sorites seeking to impale a wet philosopher on the Horns of a Dilemma
An easier problem We claim that truth value is not relative to persons, times, places, etc. Counterexamples? “True-for” sentences –“For the ancient Greeks, the earth was at the center of the universe.” Context-dependent sentences –I like chocolate
Response to “True-for” “True-for” is an idiom: it means “believed by” Example: “For the ancient Greeks, the earth was the center of the universe. Translation: “The ancient Greeks believed that the earth was the center of the universe” Compare to the “historical present” e.g. “Socrates is in the Athens Jail awaiting execution.”
I like chocolate I don’t like chocolate Not a counterexample! the truth value of these context-independent sentences isn’t relative: 1.Alice likes chocolate 2.Bertie doesn’t like chocolate A B Context Dependence
Response to context-dependence For any utterance of a context-dependent sentence, there’s a context-independent sentence that makes the same statement. 1.[uttered by Alice] “I like chocolate.” 2.Alice likes chocolate We’ll say that truth value belongs to propositions expressed by context- independent sentences. Given this restriction, truth value is not relative to persons, places, times, etc. A
What’s the point? In doing formal logic we will make some idealizing assumptions about truth value that seem crazy. The point of considering and responding to apparent counterexamples is to argue that these assumptions aren’t so crazy. We argue for the legitimacy of this idealization
What is truth? But we still haven’t answered the Big Question
Correspondence Theory of Truth Reality (“the World,” the way things are) Truth Value
Roses are red. Our working definition: Truth is correspondence with reality True!
Does this tell us anything? Not really. Because we haven’t made sense the idea of “correspondence” So, as with sorites, we’ll leave this sit for further philosophy classes…
Propositional Attitudes Ways in which people are related to propositions Propositions are expressed by that clauses X _____ that p [hopes, is afraid, believes]
Belief We call beliefs “true” or “false” in virtue of the truth value of the propositions believed. By “belief” we don’t mean “mere belief” Believing doesn’t make it so - denial doesn’t make it not so. We may believe with different degrees of conviction.
Proposition Person Reality Believing doesn’t make it so! The relation between propositions and reality is completely separate from the relation between persons and propositions! Belief: a propositional attitude Propositional Attitude Truth Value
Controversial Beliefs God exists. God doesn’t exist. People disagree. Who’s to say? No one knows.
Who’s to say??!!? That’s a different question from the true or false question! A proposition is either true or false--even if we don’t (or can’t) know which. –Example: No one now knows, or can know, whether Lucy, an early hominid who lived 3.18 million years ago had exactly 4 children or not. But “Lucy had exactly 4 children” is either true or false.
So when there’s a genuine disagreement, someone is wrong… …but it’s alright to be wrong ! Atheists Welcome
Justification Having good reasons for what you believe
“Reasons” for belief Causal: what causes a person to hold a belief Pragmatic: the beneficial effects of holding a belief Evidential: evidence for the truth of a belief
Justification X is justified in believing that p if x has good enough evidential reasons for believing that p Knowledge doesn’t require certainty Justification is relative to persons
The JTB Account of Knowledge x knows that p: 1.x believes that p 2.x’s belief that p is justified 3.p is true
Sources of knowledge Sense perception Introspection Memory Reason Expert testimony Reliable… but not infallible!
Knowledge doesn’t require certainty! I think, therefore I am Now what?
Truth and Justification TrueFalse Justified KNOWLEDGE Not Justified
Truth and Justification TrueFalse Justified KNOWLEDGE Not Justified e.g. lucky guesses
Truth and Justification TrueFalse Justified KNOWLEDGE Not Justified e.g. lucky guesses e.g. unlucky guesses
Truth and Justification TrueFalse Justified KNOWLEDGE e.g. “Smoking gun” example Not Justified e.g. lucky guesses e.g. unlucky guesses