PHIL222 - Syntax of Propositional Logic L2

Natural Languages:
- Include languages like English, Spanish, and French.
- Enable complex expression of thoughts, emotions, and nuances but often lack a systematic and unambiguous structure.
- Ambiguities inherent in natural language can lead to multiple interpretations, making them less suited for formal logic.
Formal Languages:
- Defined by a strict set of symbols known as an alphabet and explicit formation rules for constructing well-formed expressions.
- Examples include programming languages like Python or Java and algebraic notation used in chess.
- Highly structured, allowing for precise communication of ideas and processes, which is crucial for logical reasoning and mathematical proofs.

Representation:
- Basic propositions are denoted by capital letters, such as A, B, C, and extend through the alphabet to P, Q, R, … , Z.
- In situations requiring more propositions, they can be numerically designated to maintain clarity (e.g., A2, A3, B2, …).
Characteristics:
- Basic propositions do not possess any logically significant internal structure; they form the foundational building blocks of logic.
- Their simplicity aids in the establishment of more complex statements and logical operations.

Purpose: Glossaries are essential for specifying the correspondence between basic propositions and sentence letters for clarity in logical expressions.
Example:
- A: Tickets are available.
- B: Tickets are expensive.
- C: We go to the concert.
Argument Structure:
- P1: If A and not B, then C indicates a conditional relationship where the premise involves both the availability and price of tickets.
- P2: Not C concludes that attending the concert is contingent on the prior analysis of propositions A and B.
- Conclusion: Therefore, not A or B highlights the logic of existential implications within argument forms.

Connective: Negation is represented by the symbol: ¬.
Usage: Positioned directly before the proposition being denied, yielding expressions like ¬A to indicate the opposite of A.
Meaning: Defines conditions such as "it is not the case that," "it is false that," or simply reversing the state of the proposition.

Connective: Conjunction is signified by the symbol: ∧.
Placement: Utilized between two propositions to form compound truth statements (e.g., (A ∧ B)).
Indication: Usually marked by conjunctional phrases like "and," "but," or through the use of commas in lists to represent simultaneous affirmations.

Glossary Examples:
- P: I got paid today.
- H: I am happy.
Translations:
- I got paid today and I am happy can be represented symbolically as (P ∧ H).
- I got paid today but I am not happy translates to (P ∧ ¬H).
- I wasn’t paid today, yet I am happy is denoted as (¬P ∧ H).
- It is not the case that I got paid today but I am unhappy can be represented as ¬(P ∧ ¬H).

Connective: Disjunction is symbolized by the notation: ∨.
Placement: Occurs between two propositions to create an inclusive or alternative statement (e.g., (A ∨ B)).
Indication: Typically indicated by wording such as "or," "either," or "instead of" to denote alternative conditions.

Connective: The conditional connective is expressed with the symbol: →.
Placement: Follows the antecedent and precedes the consequent in propositional expression (e.g., (A → B)).
Components: A represents the antecedent (the initial condition), while B denotes the consequent (the result).
Expressions: Commonly utilizes conditional phrases like "if," "then," or "only if" to establish logical dependency.

Patterns:
- Conditional forms typically follow sentence structures such as "if A then B" or "A only if B," demonstrating dependency.
Complexity: These expressions can vary widely in their construction, demanding a precise understanding of antecedents and consequents.

Connective: Biconditional is denoted with the symbol: ↔.
Meaning: Signifies that propositions A and B are logically equivalent (A ↔ B).
- This implies A is both a sufficient and necessary condition for B, establishing an equivalence.
Expression: Generally articulated in natural language as "if and only if," emphasizing the dual dependency of truth values.

Task: Translate the sentence: "You’re given a discount if and only if you have a club card or a coupon."
Identifying Connectives:
- A: You’re given a discount.
- B: You have a club card.
- C: You have a coupon.
Answer: This translates to (A ↔ (B ∨ C)), illustrating the conditional dependency among the propositions involved.

Introduction: Well-formed formula (wff) variables represented by lowercase Greek letters (α, β, γ,…) facilitate discussion about propositions in a general manner.
Purpose: Wff variables are crucial to enabling logical discourse without needing to specify each individual proposition, thus promoting abstract logical reasoning.
Examples: For instance, in the statement "if α is true, then ¬α is false," α may symbolize either basic propositions or more complex compound propositions.

Symbols include:
- Sentence letters (A, A2, B, B2, …) which represent individual propositions.
- Connectives: ¬, ∧, ∨, →, ↔ that indicate logical operations.
- Punctuation symbols: ( ) used to specify the structure of propositions and the relationships between them.

Wffs definition: Well-formed formulas (wffs) are recursively defined as follows:
- Any basic proposition qualifies as a wff, thereby establishing the baseline for logical expressions.
- If α and β are both wffs, then the expressions ¬α, (α ∧ β), (α ∨ β), (α → β), and (α ↔ β) are also wffs.
- According to this definition, nothing else qualifies as a well-formed formula, maintaining the integrity of logical expressions.

Example: Demonstrating that (¬P ∧ (Q ∨ R)) is a well-formed formula involves systematic justification of each operation and the proper arrangement of components.
Details: Each compound proposition comprises a main connective, which is determined by the last operation performed in its construction sequence.

Not-wffs examples: Invalid constructions include PQ R¬ S¬T, ((U ∧ W)), and ((X → (↔ Y )∨).
Valid wffs include**:
- P (which is valid without any connectives).
- ¬(Q → Q) (with the main connective being the negation).
- ((R ↔ ¬S) ∨ (T ∧ U)) (where ∨ is the main connective).
- (W → (X ∧ (Y ∨ ¬Z))) (with → as the principal connective).
- ((¬A ∧ (¬A ∨ A)) ∧ ¬(¬B → B)) (where ∧ serves as the main connector in the expression).

Importance: Parentheses are critical in determining the main connective of any wff.
Usage: Any negation situated outside parentheses acts as the main connective; in absence of parentheses, the least surrounded two-place connective is identified as the main connective.
Examples: Proper structuring illustrated through examples shows how dividing propositions through parentheses is essential for accurate logical interpretation, as missing parentheses can lead to considerable misinterpretation of meaning.