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Lesson 3

Translating English to First-Order Logic

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Hello! In our last session, we mastered the art of transforming propositional formulas into standard forms like CNF and DNF using logical equivalences. This algebraic approach is far more efficient than truth tables, a key skill for your GATE preparation. However, propositional logic has a major limitation: it can't express statements about objects, their properties, or relationships. We can say "Socrates is a man" is true, but we can't reason about the properties of being a man or being Socrates.

This lesson takes a significant step forward into the more expressive world of First-Order Logic (FOL), also known as Predicate Logic. Our goal is to translate natural language sentences into precise logical statements using predicates and quantifiers. This skill is not just a frequent source of GATE questions; it forms the theoretical backbone of database query languages, AI knowledge representation, and formal software verification. You'll learn to capture the meaning of statements like "Every computer is a machine" or "Some programs have bugs" with mathematical rigor.

1. From Propositions to Predicates

In propositional logic, "Alice is a student" and "Bob is a student" are two completely separate atomic propositions, say and . We can't capture the fact that they both share the property of being a student. First-order logic solves this by introducing predicates.

A predicate is like a function that takes one or more objects as input and returns a truth value. For example, we can define a predicate IsStudent(x) which is true if x is a student. Now we can represent "Alice is a student" as IsStudent(Alice) and "Bob is a student" as IsStudent(Bob).

Predicates can also represent relationships between objects. A predicate Loves(x, y) could represent "x loves y". These are called binary predicates. The objects themselves, like Alice and Bob, are represented by constants.

To get a firm grasp on this foundational concept, let's start with a video that explains how to define predicates and constants.

Translating ENGLISH into PREDICATE LOGIC

This video from TrevTutor clearly explains how to set up the basic building blocks of FOL: predicates and constants.

Please watch the first two segments. First, focus on the introduction to defining predicates and constants. Notice the different notations used, but the core idea of a property (Hx for 'x is happy') or a relation (Lxy for 'x loves y') remains the same. Next, observe how simple sentences are translated using only these building blocks in the basic examples. This part does not use quantifiers yet but shows how to combine predicates with logical connectives like ∧ (and) and → (if...then).

2. Quantifiers: The Heart of First-Order Logic

Now, let's address how to express statements about groups of objects, like "all students" or "some programs". This is done using quantifiers.

The Universal Quantifier (∀): "For All"

The universal quantifier, ∀, is used to express that a property or relation holds for every object in the domain. It's read as "for all" or "for every".

Consider the sentence: "Every student has studied discrete mathematics."

To translate this, we need two predicates:

  • S(x): x is a student in this university.
  • D(x): x has studied discrete mathematics.

A common mistake is to write ∀x (S(x) ∧ D(x)). Let's think about what this says. It means "For every x, x is a student AND x has studied discrete mathematics." This implies that everything in the universe (every person, every rock, every planet) is a student and has studied discrete mathematics. This is far too strong.

The correct translation restricts the domain. We want to say that if x is a student, then x has studied discrete mathematics. The correct logical structure is therefore an implication:

This is a critical pattern for GATE. The universal quantifier ∀ is almost always paired with an implication → to restrict the claim to a specific group of objects.

The following resources provide excellent explanations of this crucial point.

Translating ENGLISH into PREDICATE LOGIC

TrevTutor provides a brilliant conceptual explanation using set diagrams for why ∀ is paired with →.

Watch the segment on the universal quantifier. Pay close attention to the reasoning about why dog(x) ∧ happy(x) is the wrong way to say "every dog is happy", and dog(x) → happy(x) is the correct way.

Translating English Sentences to Logical Expressions

Now let's see this pattern applied in a practical example from Neso Academy.

Watch this segment translating "Every student...". The commentary reinforces the use of implication with the universal quantifier.

The Existential Quantifier (∃): "There Exists"

The existential quantifier, ∃, is used to express that a property holds for at least one object. It's read as "there exists", "for some", or "for at least one".

Consider the sentence: "Some students have studied discrete mathematics."

Here, we are asserting the existence of at least one object that is both a student and has studied discrete mathematics. So, the correct connective here is conjunction ∧.

This reads: "There exists an x such that x is a student AND x has studied discrete mathematics."

Using an implication here, ∃x (S(x) → D(x)), would be incorrect. This formula is true if there is anything in the universe that is not a student. For example, a cat is not a student, so S(cat) is false. This makes the implication S(cat) → D(cat) true, which in turn makes the whole existential statement true, even if no student has ever studied discrete mathematics! This is not what we want to express.

This leads to our second critical pattern: the existential quantifier ∃ is almost always paired with a conjunction ∧.

Translating ENGLISH into PREDICATE LOGIC

TrevTutor again provides a clear explanation for why ∃ is paired with ∧.

Watch the segment on the existential quantifier. The set diagram showing the intersection of "dogs" and "happy things" makes the use of ∧ very intuitive. The explanation of why → fails is also important.

3. Variable Scope and Nested Quantifiers

The scope of a quantifier is the part of the formula to which it applies. In ∀x (P(x) → Q(x)), the scope of ∀x is (P(x) → Q(x)). A variable is bound if it is within the scope of a quantifier; otherwise, it is free. In GATE, you will primarily deal with formulas where all variables are bound.

Things get more interesting when we nest quantifiers. The order of quantifiers is crucial and can completely change the meaning of a sentence.

Let's use the predicate Loves(x, y): "x loves y".

  • ∀x ∃y Loves(x, y): "For every person x, there exists some person y whom x loves." (Everybody loves somebody).
  • ∃y ∀x Loves(x, y): "There exists some person y such that for every person x, x loves y." (There is someone who is loved by everyone).

These two statements are clearly not the same!

Let's translate a typical GATE-level sentence involving multiple predicates and quantifiers: "All hummingbirds are richly colored."

Predicates:

  • H(x): x is a hummingbird.
  • R(x): x is richly colored.
  • L(x): x is large.
  • O(x): x lives on honey.

The video below demonstrates how to break down complex sentences like this one.

Translating English Sentences to Logical Expressions

This example from Neso Academy tackles a multi-predicate sentence, showing how to combine properties.

Watch the translation of the sentences about hummingbirds. Focus on the first statement: "All hummingbirds are richly colored". The translation is ∀x (H(x) → R(x)). Then see how they translate "No large birds live on honey." This becomes ∀x ((L(x) ∧ B(x)) → ¬O(x)), where B(x) means 'x is a bird'. This is a great example of combining conditions in the antecedent of a universally quantified statement.

4. Summary of Translation Patterns

For your GATE exam, memorizing these patterns and understanding the logic behind them is key to solving questions quickly and accurately.

English Sentence Logical Form Mnemonic
All P's are Q's. ∀x (P(x) → Q(x)) ∀ pairs with →
Some P is a Q. ∃x (P(x) ∧ Q(x)) ∃ pairs with ∧
No P is a Q. ∀x (P(x) → ¬Q(x)) ∀ with → and negation
(Equivalent to "No P is a Q.") ¬∃x (P(x) ∧ Q(x)) ¬ outside of ∃ with ∧
Not all P's are Q's. ¬∀x (P(x) → Q(x)) ¬ outside of ∀ with →
(Equivalent to "Not all P's...") ∃x (P(x) ∧ ¬Q(x)) "Some P is not a Q"

GATE-Style Example: "There is a student who has answered every question."

  1. Define Predicates:

    • S(x): x is a student.
    • Q(y): y is a question.
    • A(x, y): x has answered y.

  2. Break it down:

    • "There is a student..." suggests ∃x S(x) ...
    • "...who has answered every question." This is a property of the student x. This property is "for every question y, x has answered y". This nested part is ∀y (Q(y) → A(x, y)).

  3. Combine: Since ∃ pairs with ∧, we combine the student property with the "answered every question" property.

Conclusion

In this lesson, we've unlocked the expressive power of First-Order Logic. You have learned to deconstruct English sentences and represent their precise meaning using predicates and quantifiers.

Key Takeaways:

  • Predicates represent properties of objects or relations between them.
  • The universal quantifier (∀) means "for all" and is typically paired with implication (→) to restrict its domain.
  • The existential quantifier (∃) means "there exists" and is typically paired with conjunction (∧) to assert the existence of an object with multiple properties.
  • The scope of a variable is where it is bound by its quantifier.
  • The order of nested quantifiers drastically affects the meaning of a formula.

Mastering these translations is essential. The logical formulas you now know how to create are the raw material for the next stage of our journey in logic. In our upcoming lesson, we will explore inference and proof techniques, where we will use rules like modus ponens and resolution to formally prove the validity of arguments expressed in First-Order Logic.