Lesson 5: Entailment is not validity

Why this matters: a language can look logically tidy in its own model while the inference is not valid in general. If you mix up entails and valid_entails, you will overclaim what a grammar proves.

Two questions

Call	Question	Scope
`entails`	Do the premises imply the conclusion here?	The one declared model
`valid_entails`	Do they imply it in every model up to size N?	All non-empty finite models

The same inference, two verdicts

In the level 3 model, only ana is a sun. So this holds:

entails(model, ["exists x (lex:sun(x))"], "lex:sun(ana)")
-> Entailment(holds=True, ...)

But the inference is not valid in general. Another model can put the only sun on a different individual. Bounded search finds it:

conlang valid \
  --premise "exists x (lex:sun(x))" \
  --conclusion "lex:sun(ana)" \
  --max-domain 2

invalid: countermodel at domain size 2
  domain: d0, d1
  constants: ana=d0
  lex:sun: [['d1']]
  premises: [True]

The countermodel makes the premise true and the conclusion false. That is why declared-model entailment and validity are different claims.

Watch the vacuous case

entails holds when a premise is false. That is material implication, not evidence that the conclusion is true:

entails(model, ["lex:moon(ana)"], "lex:sun(ana)")
-> holds=True because moon(ana) is false

Retrieval check

Which call uses only the declared model?

Which verdict can be “unknown”?

Which evidence shows an inference is not valid?

Do it once

Run the conlang valid command above.
Read the countermodel: which individual is ana, and which is the sun?
Now evaluate the same inference in the level 3 model with entails.
Ask: why can both answers be correct?

Expected: the declared model has one sun at ana, so the inference holds there. The countermodel moves the sun away, so the inference is not valid across all models.

Primary source: the feature guide sections on model entailment and validity, plus the glossary entries for Entailment, Validity, and Countermodel.

Ask your teacher: if a formula feels obvious, ask whether it is obvious in the model or valid in every model.

Previous: Meaning is not the model. Reference: Entailment vs validity cheat sheet.