Interpretability logic

Key takeaways

Interpretability logic belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Interpretability logic to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Interpretability logic from memory before moving on to harder problems.

Reference excerpt

Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities. Main contributors to the field are Alessandro Berarducci, Petr Hájek, Konstantin Ignatiev, Giorgi Japaridze, Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser, and Domenico Zambella.

Examples

Logic ILM The language of ILM extends that of classical propositional logic by adding the unary modal operator ◻ {\displaystyle \Box } and the binary modal operator ▹ {\displaystyle \triangleright } (as usual, ◊ p {\displaystyle \Diamond p} is defined as ¬ ◻ ¬ p {\displaystyle \neg \Box \neg p} ). The arithmetical interpretation of ◻ p {\displaystyle \Box p} is “ p {\displaystyle p} is provable in Peano arithmetic (PA)”, and p ▹ q {\displaystyle p\triangleright q} is understood as “ P A + q {\displaystyle PA+q} is interpretable in P A + p {\displaystyle PA+p} ”. Axiom schemata:

All classical tautologies

◻ ( p → q ) → ( ◻ p → ◻ q ) {\displaystyle \Box (p\rightarrow q)\rightarrow (\Box p\rightarrow \Box q)}

◻ ( ◻ p → p ) → ◻ p {\displaystyle \Box (\Box p\rightarrow p)\rightarrow \Box p}

◻ ( p → q ) → ( p ▹ q ) {\displaystyle \Box (p\rightarrow q)\rightarrow (p\triangleright q)}

( p ▹ q ) → ( ◊ p → ◊ q ) {\displaystyle (p\triangleright q)\rightarrow (\Diamond p\rightarrow \Diamond q)}

( p ▹ q ) ∧ ( q ▹ r ) → ( p ▹ r ) {\displaystyle (p\triangleright q)\wedge (q\triangleright r)\rightarrow (p\triangleright r)}

( p ▹ r ) ∧ ( q ▹ r ) → ( ( p ∨ q ) ▹ r ) {\displaystyle (p\triangleright r)\wedge (q\triangleright r)\rightarrow ((p\vee q)\triangleright r)}

◊ p ▹ p {\displaystyle \Diamond p\triangleright p}

( p ▹ q ) → ( ( p ∧ ◻ r ) ▹ ( q ∧ ◻ r ) ) {\displaystyle (p\triangleright q)\rightarrow ((p\wedge \Box r)\triangleright (q\wedge \Box r))}

Rules of inference:

“From p {\displaystyle p} and p → q {\displaystyle p\rightarrow q} conclude q {\displaystyle q} ” “From p {\displaystyle p} conclude ◻ p {\displaystyle \Box p} ”. The completeness of ILM with respect to its arithmetical interpretation was independently proven by Alessandro Berarducci and Vladimir Shavrukov.

Logic TOL The language of TOL extends that of classical propositional logic by adding the modal operator ◊ {\displaystyle \Diamond } which is allowed to take any nonempty sequence of arguments. The arithmetical interpretation of ◊ ( p 1 , … , p n ) {\displaystyle \Diamond (p_{1},\ldots ,p_{n})} is “ ( P A + p 1 , … , P A + p n ) {\displaystyle (PA+p_{1},\ldots ,PA+p_{n})} is a tolerant sequence of theories”. Axioms (with p , q {\displaystyle p,q} standing for any formulas, r → , s → {\displaystyle {\vec {r}},{\vec {s}}} for any sequences of formulas, and ◊ ( ) {\displaystyle \Diamond ()} identified with ⊤):

All classical tautologies

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Interpretability logic

Start with the simplest possible case. Write down what Interpretability logic claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Interpretability logic before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Interpretability logic ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Interpretability logic

In research: Interpretability logic appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Interpretability logic in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Interpretability logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Interpretation (philosophy), Provability logic, so understanding it makes those chapters shorter.
In everyday life: Look for Interpretability logic outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Interpretability logic in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Interpretability logic means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Interpretability logic out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Interpretability logic in simple terms?

Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities…

Why does Interpretability logic matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Interpretability logic?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Interpretability logic.