Oct 2-6 - Frege’s Logic
1. Syntax – the formal language of the theory
• Connectives: and, or, not, if then.
• Predicates: __ is red, __ loves __, __ = __, __ is greater than __
• Names: Fred, Bertha, a, 3
• Functions: the mother of __, __ + 1, __ + __
• Quantifiers: ∀ (all), ∃ (some)
• Variables: x, y, z
• Examples:
All unicorns are hoofed
Some cars are red
No cars are red but some cars are blue
Everyone loves Allan and Allan loves everyone
Everyone loves someone
Someone loves everyone
Allan loves Bertha’s mother
Bertha’s maternal grandmother loves Charles
2. Proof Theory
• A proof is a series of sentences in the language of Frege’s system, each of which is either (a) a premise; (b) a logical truth; or (c) follows from earlier steps in the proof by Frege’s rules of inference.
• Such a proof shows that the premises necessitate the conclusion – thus we have an answer to Glymour’s first question.
• If some premises necessitate a conclusion then there is a proof that they do – Gödel. Moreover, there is a method that is guaranteed to find – in principle – the proof. If they don’t necessitate the conclusion, then there is no method that is guaranteed to show – even in principle – that that they don’t. Thus we have an answer to Glymour’s second question.
3. Semantics – what makes sentences true or false?
• Examples:
Everyone loves Allan
Everyone loves someone
Someone loves everyone
∃x ∃y ¬(x=y)
∃x ∃y ∃z (¬ x=y ∧ ¬ x=z ∧ ¬ y=z)
4. What Makes Deductive Arguments Possible?
• E.g., Festino:
The laws of language stipulate the meanings of the sentences
The laws of set theory determine whether the premises necessitate the conclusion