3. The Boolean Connective
Boolean Connectives
Truth-functional Connectives
Truth-functional connectives are essential in forming complex sentences.
Key Questions to Consider
What do connectives do?
What does it mean when we say that connectives are truth-functional?
What determines the truth value of a First-Order Logic (FOL) sentence?
How is double negation different in natural language (e.g., English) and in FOL?
Connectives
Connectives are used to connect simpler claims to create more complex ones.
First studied by the 19th-century English logician George Boole, leading to the term "Boolean connectives."
The first connectives to study are similar to the English words "and," "or," and "not."
In programming, Boolean refers to objects with a binary value of true or false.
Assumptions in FOL Model
A simplifying assumption is made about how complex claims are constructed: - The connectives of FOL are truth-functional.
The truth value of a sentence constructed with connectives relies on the truth values of the component sentences alone.
Example of Connectives
For the statement "a is a cube and a is small": - The truth of the sentence can be assessed once the truth of both components is established.
Many English connectives are not truth-functional; even some usages of "and" differ in meaning.
The truth of complex sentences can be determined by analyzing atomic sentences and their connectives.
Truth tables can be used to describe the meanings of connectives.
Negation Connective
Corresponds to the English word "not".
Example: "I am not happy with my car" vs. "I am unhappy with my car," where "un-" represents negation.
Negation serves to reverse the meaning of a sentence. - In other words, asserting a sentence means declaring it true, while asserting its negation claims it is false.
Negation is the simplest connective, as it operates on a single component sentence.
Symbol for Negation: - Usually read as "not."
Always precedes the sentence being negated (e.g., "not Boring(class)").
Syntax of Negation
Syntax rules define how to build FOL sentences using symbols.
Negation must precede any sentence in FOL format.
A negative sign creates the negation, altering the original claim's meaning.
Question: Identify well-formed FOL sentences (e.g., Home¬(john), ¬Home(john), etc.).
Semantics of Negation
Truth tables clarify the semantics of negation:
P
¬P
T
F
F
T
The truth value for ¬P is based solely on P's truth value:
If P is true (T), then ¬P is false (F).
If P is false (F), then ¬P is true (T).
Negation in English vs. FOL
Double Negation: - Example: "It doesn't make no difference" intensifies the negation, unlike in FOL where double negation cancels itself out.
Another Example: "Clio is not unhappy about leaving" does not equate to Clio being happy; rather, it suggests ambivalence.
Logical concepts are binary, whereas experiential language can have degrees of meaning (e.g., ecstatic vs. happy).
FOL Model of Negation
The FOL model captures the core characteristics of negation effectively, emphasizing that negation doesn't connect but transforms a sentence.
Conjunction
Conjunction in FOL corresponds to the English word "and".
Questions to Consider: - How is conjunction defined?
How can we determine the truth of a conjunctive sentence?
How do we avoid ambiguity in complex FOL formulas?
Conjunction connects two FOL sentences, P and Q, forming P ∧ Q.
Symbol: ∧ (wedge or caret) read as "P and Q".
Example of Conjunction
Two sentences: Home(john) and Home(mary) yield: - Home(john) ∧ Home(mary).
This is true if both John and Mary are at home.
Flexibility of Conjunction in English
English allows for various expressions of conjunction: - "John and Mary are at home" vs. "John is at home, and Mary is at home."
In FOL, only complete sentences can be conjoined, unlike in English which can include noun phrases or predicates.
Ambiguity in Conjunction
Conjunction without explicit markers can lead to misinterpretation in FOL.
Example: - Sentence structure can suggest temporal precedence (e.g., "Max went home, and Clio fell asleep").
In FOL, there are no temporal implications, as shown: WentHome(max) ∧ FellAsleep(clio) is equivalent to FellAsleep(clio) ∧ WentHome(max).
Semantics of Conjunction in FOL
Truth values of conjunction are determined purely by constituent truth values via a truth table:
P
Q
P∧Q
T
T
T
T
F
F
F
T
F
F
F
F
Disjunction is expressed using the word "or" in English.
Example: "John is at home, or Mary is at home" signifies one or both being true.
Symbol in FOL: ∨.
Complex meanings in English are simplified in FOL to join only complete sentences.
Different Meanings of Disjunction
English expresses both inclusive and exclusive disjunctions: - Inclusive: Indicates either or both can be true (e.g., "You could have soup or salad" can mean both).
Exclusive: Indicates only one can be true (e.g., "You can have soup or salad, but not both").
In FOL, the disjunction is always inclusive, with the logic representing the truth across both statements wherever applicable.
Truth Value of Disjunction Example
For the statement "John is at home, or Mary is at home": - True if John is home alone, Mary is home alone, or both are home.
False only if neither John nor Mary is home.
Ambiguity in Disjunction and FOL
The sentence "Home(clio) ∨ Home(max)" can be interpreted in two ways: 1. Conjunctive Claim: Clio is not at home, and Max is at home.
Negation of Both: Neither Clio nor Max is at home.
Parentheses resolve ambiguity of how negation and conjunction relate (e.g., determining which elements of a sentence are being negated).
Summary of Scope of Negation
The scope of negation defines the extent to which negation applies within a sentence.
In FOL: - Example 1: Clio is not at home, but Max is at home, defines the scope of negation as the single atomic formula Home(clio).
Example 2: In the context of a conjunction, negation can apply to the entire conjunction when properly enclosed in parentheses.
Equivalence and Translation
De Morgan’s Laws
De Morgan’s Laws express relationships between negation, conjunction, and disjunction.
The first law states that the negation of a conjunction is equivalent to the disjunction of the negations: - ¬(P ∧ Q) ⟷ ¬P ∨ ¬Q
The second law states that the negation of a disjunction implies both elements are false: - ¬(P ∨ Q) ⟷ ¬P ∧ ¬Q
Translation from English to FOL
Translation seeks to ensure a FOL sentence matches the meaning of a natural language sentence under the same truth conditions.
Consider stylistic choices and how to best express complex sentences without losing meaning or context: e.g., quantifying negation in practice to achieve proper logical representation.
Connotations in Natural Language
Words such as "but," "however," and "moreover" express conjunction in English but carry different implications compared to their FOL representations.
These nuances affect how we interpret complex natural language structures and their FOL equivalents, influencing how they are understood rhetorically.