Aug 28-Sep 1 - Introduction and the Idea of Proof
 
1. Overview
 
  1. •Epistemology = theory of knowledge
    1. •What do we mean when we claim to know something?
    2. •How do we achieve such a state?
 
• Reasoning – from given to new knowledge
    1. I.Deductive Reasoning – how do we reason in logic and mathematics?
    2. II.Probabilistic Reasoning – how do we learn from experience?
    3. III.Skepticism – can we ever achieve knowledge?
 
2. Deductive Proof
 
• Argument = attempt to show that a conclusion follows from/is supported by given premises – that the premises are reasons to accept the conclusion.
 
NB: in logic classes, an argument = premises+conclusion, and a proof = attempt to show that the conclusion follows from the premises.
 
• Inductively Valid Argument = successfully shows that the premises are ‘merely’ good reasons for believing the conclusion.
 
• Deductive or Valid Argument = successfully shows that the conclusion must be true if premises are.
    1. Q1 How do we tell if a proof is valid?
    2. Q2 How do we find valid proofs?
    3. Q3 Why are there any deductive arguments?
      1. – TTT p7
 
• Counterexample = a situation in which the premises are true but the conclusion is false: that the premises do not necessitate the conclusion.
 
• Sound Argument = valid argument with true premises.
 
3. The Socratic Investigation of Deductive Proof
 
 
• We will begin work on Glymour’s 3 questions by seeking criteria that …
    1. … include all valid proofs – sufficient conditions
    2. … exclude all invalid proofs – necessary conditions
 
4. Euclid’s First Proposition: for every line segment AB there is an equilateral triangle with AB as one side (TTT 11-16)
 
 
• What, according to Glymour are the characteristics of the proof?
 
• Is the proof valid?
 
5. St Anslem’s (First) Ontological Proof: the ‘greatest conceivable being’ exists (TTT 16-18)
 
 
• Premise 1: The conception of the greatest conceivable being (‘GCB’) exists in our minds.
 
(‘Even the fool … understands [the idea of the GCB] …’)
 
• Premise 2: Anything conceivable can be conceived as existing in mind-independent reality.
 
(‘…if it is in the understanding alone, it can be thought of as existing also in reality …’)
 
• Premise 3: The conception of X as something that exists in reality is greater than the, otherwise identical, conception of X as something that exists in the mind alone.
 
(‘… and this is greater.’)
 
• Step (i): Suppose (for argument’s sake) that the GCB exists only in our minds; by 2, we can conceive the GCB existing in reality.
 
• Step (ii): By 3, our conception of the GCB existing in reality is greater than the GCB supposed to exist only in our minds.
 
• Step (iii): But it is impossible by definition for there to be a conception greater than that of the GCB.
 
• Therefore, the GCB cannot exist only in our minds.
 
 
• Something has gone wrong – substitute anything you like for ‘being’ (and ‘B’) and the argument works just as well (or badly) … ice cream, car, soccer team. But just where does the argument break down?
 
 
6. The Proof that √2 is irrational
 
• Step (i): Suppose (for argument’s sake) that is rational.
 
• Step (ii): Hence √2 = n/m, where n and m have no common factors.
 
• Step (iii): Hence, n2 = 2m2 – why can you infer that n is even?
 
• Step (iv): Given that n and m have no common factors, what can you conclude about whether m is even or odd?
 
• Step (v): Since n is even, n = 2c, for some integer c – so, by (iii), 4c2 = 2m2. What can you conclude about whether m is even or odd?
 
• Therefore: You should find a contradiction between (iv) and (v), which implies that (i) cannot be correct after all.