Symbolic LogicPredicate Logic: Translation and Truth

Truth in a Small World

In a world with a few named objects, ∀x Fx is true when every object is F, and ∃x Fx when at least one is.

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Truth tables decided truth in statement logic. In predicate logic the role of a row is played by a world: a domain of objects, with each predicate true of some of them. Here you evaluate quantified sentences (Quantifier Negation) in small worlds you can see.

Key idea

A universal sentence such as $\forall x(Fx \to Gx)$ is true in a world when its inside holds for every object in the domain; one counterexample makes it false. An existential sentence such as $\exists x(Fx \land Gx)$ is true when its inside holds for at least one object, a witness. Evaluate the inside by testing each object in turn.

Why it works

A quantified sentence makes a claim about the whole domain, so its truth depends on which objects exist and what they are like. Change the world and the truth value can change; that is why the same sentence can be true here and false there. A small world lets you check every object by eye, which is exactly what the definitions ask for.
Four shapes. Each sentence shows its value and a witness or counterexample. Edit the world and watch them change.

Worked examples

Worked example: A universal conditional

Is $\forall x(Cx \to Rx)$ true in the world above?
  1. Only circles can break it: $a$ (red) and $d$ (blue).
  2. $d$ is a circle that is not red: a counterexample.

Answer: False

Worked example: A universal that holds

Is $\forall x(Lx \to Cx)$ true?
  1. Only large objects matter: $a$ is the only large one.
  2. $a$ is a circle.
  3. Every other object makes the conditional true by having a false antecedent.

Answer: True

Worked example: An existential

Is $\exists x(Tx \land Lx)$ true?
  1. Look for a large triangle: $c$ is the only triangle, and it is small.

Answer: False

Common mistake

Checking only the objects that make a universal conditional "about something". For $\forall x(Lx \to Cx)$, the small objects pass automatically, since their antecedent is false. And for an existential, one witness is enough; you do not need all objects to pass.

Worked example: Nested connectives

Is $\forall x(Rx \lor Bx)$ true?
  1. Every object is red or blue: $a$ red, $b$ blue, $c$ red, $d$ blue.

Answer: True. (In this world of shapes, every object has one of the two colors.)

One more point about domains: the same sentence can be true in one world and false in another, but in a single world it has exactly one truth value. That is what makes a world work like a row of a truth table, and why the next lesson can use worlds as counterexamples.

  • $\forall$: check every object; stop at the first counterexample.
  • $\exists$: look for one witness; stop when you find it.
  • Inside the quantifier, use the connectives' truth tables as usual.

Worlds also show what a sentence does not say. $\exists x(Sx \land Bx)$ is true here because of $b$, but it would stay true if every square were blue. Changing objects one at a time in the visual is a quick way to feel what each sentence demands.

Practice

Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.

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Sources and further reading

  • forall x: Calgary, an open introduction to formal logic (P. D. Magnus, Tim Button and others; Open Logic Project), chapter on interpretations
  • Patrick J. Hurley and Lori Watson, A Concise Introduction to Logic, 13th ed., section 8.5, finite universe method
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