TD3 - Logic

TD3 - Logic

Logical consequence and implication

Multiple Select

"Murel and Peter will both receive their computer science degree in June"

• "Murel will

  receive her

  Computer

  Science degree

  in June" is a

  logical

  consequence of

  "Murel and

  Peter will both

  receive their

  Computer

  Science degree

  in June"

• "Murel and

  Peter will both

  receive their

  Computer

  Science degree

  in June" is a logical

  consequence of

  "Murel will

  receive her

  Computer

  Science degree

  in June"

• "Murel and

  Peter will both

  receive their

  Computer

  Science degree

  in June"

  logically entails

  "Murel will

  receive her

  Computer

  Science degree

  in June"

• "Murel will

  receive her

  Computer

  Science degree

  in June"

  logically entails "Murel and

  Peter will both

  receive their

  Computer

  Science degree

  in June"

• None of the

  above

​Explanations

We can say that B : "Murel will receive her Computer Science degree

in June" is a logical consequence of A : "Murel and Peter will both receive their Computer Science degree in June" because here, A⇒B is a tautology (will always be true). Then A |= B (A is a logical consequence of B or A logically entails B).

Multiple Choice

"I swim and you play tennis"

• logically entails

  "I don't swim"

• logically entails

  "I swim"

• is a logical

  consequence of

  "I swim"

• is a logical

  consequence of

  "you don't play

  tennis"

• None of the

  above

​Explanations

We can say that B : "I swim" is a logical consequence of A: ""I swim and you play tennis" because here, A⇒B is a tautology (will always be true). Then A |= B (A is a logical consequence of B or A logically entails B).

Multiple Choice

P |= Q is the same as P⇒Q

• True

• False

​Explanations

It's false, P |= Q is the same as P⇒Q if and only if P⇒Q is a tautology (P⇒Q is always true).

Multiple Choice

P ≡ Q does not mean the same as P↔Q

• True

• False

​Explanations

It's true, because P ≡ Q means that P |= Q and Q |= P and it also works the other way round. Moreover, P ≡ Q if and only if P↔Q is a tautology.

Multiple Choice

Φ |= Ψ if and only if Ψ ⇒ Φ is a tautology

• True

• False

• Incorrect. It's: if φ ⊨ ψ, then

  ψ → φ is a

  tautology

• Incorrect. It's: if ψ→ φ is a

  tautology, then

  φ ⊨ ψ

• None of the above

​Explanations

It's false because as we said before, the right answer would be Φ |= Ψ if and only if Φ⇒Ψ is a tautology (Φ⇒Ψ is always true).

Multiple Choice

If A logically entails C and C is logically equivalent to C', then A logically entails C'.

• True

• False

• Impossible to

  know

​Explanations

It's true, because as we said before C ≡ C' means that C |= C' and C' |= C, moreover, A |= C so A |= C'.

​

Multiple Choice

If A logically entails C and A is logically equivalent to A', then A' logically entails C.

• True

• False

• Impossible to

  know

​Explanations

It's true, because as we said before A ≡ A' means that A |= A' and A' |= A, moreover, A |= C so A' |= C.

​

Multiple Choice

Φ |= Ψ if and only if Φ⇒Ψ is a tautology

• True

• False

• Incorrect. It's : if Φ |= Ψ then Φ⇒Ψ is a tautology

• Incorrect. It's : if Φ⇒Ψ is a tautology then Φ |= Ψ

• None of the above

​Explanations

It's true because as we said before, Φ |= Ψ if and only if Φ⇒Ψ is a tautology (Φ⇒Ψ is always true).

Multiple Select

|= Φ means that

• Φ is True

• Φ is False

• Φ is a tautology

• any interpretation is a model for Φ

• Φ is a contingent formula

Multiple Choice

Is ¬P a logical consequence of (P ⇒Q) and ¬Q (Use truth-tables).

• Yes

• No

Multiple Choice

Is (P ⇒R) a logical consequence of (P ⇒Q) and (Q ⇒R) (Use truth-tables).

• Yes

• No

Multiple Choice

(1) (A^C) V (B⇒C)

(2) ¬C V B

Therefore (3) ¬B V C (Use truth-tables)

• It's a valid argument

• It's an invalid argument

• We cannot know