Proof theory (next topic)

Model theory

• Focuses on the structure of phrases and sentences (syntax).

• Defines "good inference/argument" in terms of patterns of reasoning.

• Easy to prove that an inference/ argument is good.

• Notation: P₁ ,...,P_n ⊢ C

• Focuses on the meanings of words, phrases, and sentences (semantics).

• Defines "good inference/argument" in terms of entailment/validity.

• Easy to prove that an inference/ argument is bad.

• Notation: P₁ ,...,P_n ⊨ C

• In practice, mathematicians usually call ⊢ entailment  and don't use ⊨.

• This is relatively harmless.

• When learning about logics, however, this distinction is valuable, because it highlights the two very different methods we can use to distinguish good reasoning from bad.

• Distinguishing ⊢ and ⊨ is also part of understanding Godel's Incompleteness Theorems, which are two of the most important results in logic.

• Whenever we develop a model theory and a proof theory for a specific kind of reasoning, we hope that

  • everything we can prove is valid   (i.e., ⊢  ⟹  ⊨, known as soundness)  and

  • everything that's valid can be proved (⊨  ⟹  ⊢, known as completeness).

• In fact, however, we can only have one or the other for most logics we care about, including type theory and any logic for any part of mathematics.

• Additionally, for all these logics, we can never truly prove soundness.

• It turns out that it's rarely possible to have both soundness and completeness:

  • It's possible for first-order logic—logic for and, or, not, if, all, some, and is.

  • It's not possible for type theory or any mathematics.

• .