Convert Following Sentences into Predicate

a) All animal who can bark are dog.

b) Someone is firing a gun

c) All tigers are not fierce

Question

Convert Following Sentences into Predicate a) All animal who can bark are dog. b) Someone is firing a gun c) All tigers are not fierce

Padhai · Accepted Answer

Conversion of Sentences into Predicate Logic

To convert natural language sentences into First-Order Predicate Logic (FOL), we first define the necessary predicates and then construct the logical expressions using quantifiers ( $\forall$ for "all" and $\exists$ for "some/there exists").

a) All animals who can bark are dogs.

Step 1: Define Predicates

$A nima l (x)$ : $x$ is an animal.
$B a r k (x)$ : $x$ can bark.
$D o g (x)$ : $x$ is a dog.

Step 2: Logical Structure The statement implies that for any entity $x$ , if $x$ is an animal AND $x$ can bark, THEN $x$ is a dog. This is a universal statement.

Step 3: Predicate Expression $\forall x [(A nima l (x) \land B a r k (x)) \to D o g (x)]$

b) Someone is firing a gun.

Step 1: Define Predicates

$P er so n (x)$ : $x$ is a person (or "someone").
$G u n (y)$ : $y$ is a gun.
$F i r e (x, y)$ : $x$ is firing $y$ .

Step 2: Logical Structure The statement implies that there exists at least one person $x$ and at least one gun $y$ such that $x$ is firing $y$ . This is an existential statement involving two variables.

Step 3: Predicate Expression $\exists x \exists y [P er so n (x) \land G u n (y) \land F i r e (x, y)]$

(Note: In some simplified contexts, if "someone" is treated as a specific constant $s$ and "a gun" as a specific constant $g$ , it could be $F i r e (s, g)$ . However, the standard FOL interpretation for indefinite articles "someone/a" uses existential quantifiers.)

c) All tigers are not fierce.

Step 1: Define Predicates

$T i g er (x)$ : $x$ is a tiger.
$F i er ce (x)$ : $x$ is fierce.

Step 2: Logical Structure This is a classic ambiguity in natural language.

Interpretation 1 (Strict Logical Negation of "All are"): "It is not the case that all tigers are fierce." This means at least one tiger is not fierce. $\neg\forall x [T i g er (x) \to F i er ce (x)]$ Which is logically equivalent to: $\exists x [T i g er (x) \land \neg F i er ce (x)]$
Interpretation 2 (Common Misinterpretation): "No tigers are fierce" (i.e., All tigers are not fierce). $\forall x [T i g er (x) \to \neg F i er ce (x)]$

Standard Exam Answer: In most AI logic contexts, "All tigers are not fierce" is often interpreted as the negation of the universal affirmative, meaning not all tigers are fierce. However, if the intent is to say that the class of tigers does not possess the property of being fierce (i.e., no tiger is fierce), the second form is used. Given the phrasing "All ... are not", it is safer to provide the negation of the universal or the universal negation depending on the specific textbook convention.

If interpreted as "No tigers are fierce": $\forall x [T i g er (x) \to \neg F i er ce (x)]$

If interpreted as "It is not true that all tigers are fierce": $\exists x [T i g er (x) \land \neg F i er ce (x)]$

Recommended Final Answer for (c): Most TU/NEB exams expect the direct translation of the structure "All A are not B" as: $\forall x [T i g er (x) \to \neg F i er ce (x)]$

Final Summary of Predicates

Sentence	Predicate Logic Expression
a) All animals who can bark are dogs.	$\forall x [(A nima l (x) \land B a r k (x)) \to D o g (x)]$
b) Someone is firing a gun.	$\exists x \exists y [P er so n (x) \land G u n (y) \land F i r e (x, y)]$
c) All tigers are not fierce.	$\forall x [T i g er (x) \to \neg F i er ce (x)]$

Convert Following Sentences into Predicate a) All animal who can bark are dog. b) Someone is firing a gun c) All tigers are not fierce

Conversion of Sentences into Predicate Logic

a) All animals who can bark are dogs.

b) Someone is firing a gun.

c) All tigers are not fierce.

Final Summary of Predicates

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