JA5. Logic¶

Task 1¶

Based on the concepts of basic logical laws (rules of inference) for propositions, that you learned this week, check the validity of the arguments. Show all steps and write the logical laws applied at each step (wherever required).

\[A \rightarrow B, A \lor C, C \rightarrow \neg B , D \rightarrow B, \neg C \Lambda D \Rightarrow B \Lambda ¬ C\]

$A \rightarrow B$ // Per Modus Ponens, we can infer $B$.
$A \lor C$ // Per Disjunctive Syllogism, we can infer $C$.
$C \rightarrow \neg B$ // Per Modus Tollens, we can infer $\neg C
$D \rightarrow B$ // we can infer $B$ and $D$.
$\neg C \Lambda D \Rightarrow B \Lambda ¬ C$ // before that both sides evaluates to the same truth value.

From (1), $A$ is true, thus $B$ is true | false.
From (2), $C$ is true | false, thus $A$ is true | false.
From (3), $B$ is true | false, and C is true | false
From (1), (2), (3), A = true, B = true, C = false.
From (4), Because $B$ is true, $D$ is must be true.
From (5) first, $\neg C$ is true, and $B$ is true, thus $B \Lambda ¬ C$ is true.
From (5): second: $\neg C$ is true, and $D$ is true, thus $\neg C \Lambda D$ is true.

Now let’s consider the opposite way (set C to true):

From (1), $A$ is true, thus $B$ is true | false.
From (2), $C$ is true | false, thus $A$ is true | false.
From (3), $B$ is true | false, and C is true | false
From (1), (2), (3), A = false, B = false, C = true.
From (4), Because $B$ is false, $D$ is must be false.
From (5) first, $\neg C$ is false, and $B$ is false, thus $B \Lambda ¬ C$ is false.
From (5): second: $\neg C$ is false, and $D$ is false, thus $\neg C \Lambda D$ is false.

We see that we proved that the compound proposition is valid for both possible cases C, as it seems every value can be determined if we decided to know the value of C.

Task 2¶

Explain quantifiers and its types. Provide examples of propositions involving quantifiers in words and represent them symbolically. Discuss the truth of the predicates you used as examples. (Ensure that you do not take the same examples discussed in the textbooks or the internet).

Also according to (Levin, 2021, p.199), A predicate is a statement that can be true or false depending on some variables. A predicate logic is a branch of logic that is concerned with the relationships between these predicates. The truth values of each predicate -and then the truth value of the compound predicate- can change according to the values of the variables. The predicate logic extends the propositional logic and all the laws of propositional logic are valid in predicate logic.

There are two types of quantifiers: the first one is the universal quantifier $\forall$ which means “for all” and the second one is the existential quantifier $\exists$ which means “there exists”. The universal quantifier is used to express that a predicate is true for all values of the variable, while the existential quantifier is used to express that a predicate is true for at least one value of the variable.

As an example, let’s consider the following predicates:

$P(x)$: “x > 0”, and we can say that $\forall x \in \Z^{+}, P(x)$ is true; where $\Z^{+}$ is the set of all positive integers excluding zero.
$P(x)$: “x^2 + 4x -3 = 0”, after we calculate $\triangle$ we find that $\triangle > 0$ and we can say that $\exists x \in \R, P(x)$ is true; where $\R$ is the set of all real numbers.