Foundations of Logic: Sentences and Arguments 🗣️
Logic operates only on sentences that possess a truth value, organized into structured arguments to support conclusions.
The Concept of a Statement
DefinitionIn logic, we make a strict distinction between Statements and Non-Statements. Logic is the study of valid reasoning between statements; therefore, non-statements cannot serve as the building blocks of an argument.
A statement is a sentence that is either True (T) or False (F). If a sentence can be assigned a truth value, it is logically usable.
These are sentences that lack a truth value. They cannot be true or false.
- ❓ Questions: (e.g., "Where is Nizwa city?")
- ✋ Requests/Commands: (e.g., "Please shut the door")
- ❗ Exclamations: (e.g., "Wow!")
- 🤝 Proposals/Suggestions: (e.g., "Let's go to a movie")
Argument Components
StructureAn argument is not just a collection of sentences; it is a specific relationship where certain sentences (premises) are intended to provide support for a claim (conclusion).
🧱 Premises
The sentences that act as the evidence, reasons, or support for the claim.
🏁 Conclusion
The sentence that is claimed to follow from the premises; the central claim.
Premise Indicators:
- Since
- For
- Because
- On account of
Conclusion Indicators:
- Therefore
- Thus
- It follows that
- Consequently
Evaluating Arguments: Deduction and Induction 🧠
Arguments are assessed differently based on whether they claim certainty (deduction) or probability (induction).
The Standard of Certainty
DeductionIn deductive reasoning, premises are intended to provide an absolute guarantee for the truth of the conclusion.
Purely structural. An argument is valid if and only if it is impossible for the premises to be true and the conclusion to be false simultaneously.
Structure + Truth. An argument is sound if it is valid AND all its premises are actually true in the real world.
Example: "All dogs are white. Tom is a dog. Therefore, Tom is white."
Logic works (Valid), but the premise is false (Unsound).
The Standard of Probability
InductionInductive reasoning provides probable, but not certain, support for the conclusion.
Unlike deduction, we use Strength to measure induction. Strength refers to the degree to which the premises make the conclusion likely.
| Feature | Deductive Argument | Inductive Argument |
|---|---|---|
| Goal | Certainty / Necessity | Probability / Likelihood |
| Primary Metric | Validity | Strength |
| Failure State | Invalidity | Weakness |
Logical Properties: Consistency, Truth, and Equivalence 🔄
Logic allows us to determine if sets of ideas can coexist and if different sentences actually mean the same thing.
Consistency in Sets
Set TheoryConsistency is a property of a set of sentences (multiple sentences at once).
✅ Consistent
It is possible for all members of the set to be true at the same time.
❌ Inconsistent
It is impossible for all members to be true (they contradict).
Consistent: {Nizwa is in Oman. Two is even. Today is Monday.}
Inconsistent: {Henry likes ice cream. Ice cream is dairy. Henry likes no dairy.}
Impossible to be false (True by necessity).
Impossible to be true (False by necessity).
Truth depends on empirical facts.
Sentence Relationships
Two sentences are logically equivalent if it is impossible for one to be true while the other is false.
- Any two logically true sentences are equivalent.
- Any two logically false sentences are equivalent.
Sentential Logic (SL): Symbolization and Connectives 🔢
The mechanics of translating English into SL using atomic letters and truth-functional connectives.
The Building Blocks
Atomic Sentences
Simple sentences represented by single letters (A, B, C...). They cannot be broken down further.
Compound Sentences
Formed by combining atomic sentences using logical connectives.
| Connective | Symbol | English Trigger Words | Truth Rule |
|---|---|---|---|
| Negation | ¬ | "not", "it is not the case that" | Flips the truth value. |
| Conjunction | ∧ | "and", "but", "however", "moreover" | True ONLY if both are True. |
| Disjunction | ∨ | "or" | True if at least one is True. |
| Exclusive Or | ⊕ | "either... or... [but not both]" | True if exactly one is True. |
| Material Conditional | ⊃ | "if... then...", "only if", "provided that" | False ONLY if P is T and Q is F. |
| Biconditional | ≡ | "if and only if", "just in case" | True if both have the same value. |
Can be symbolized as:
- ¬P ∧ ¬Q
- ¬(P ∨ Q)
"P unless Q" can be symbolized as:
- P ∨ Q
- ¬Q ⊃ P
- ¬P ⊃ Q
| P | Q | P ∧ Q | P ∨ Q | ¬P | P ⊃ Q | P ≡ Q |
|---|---|---|---|---|---|---|
| T | T | T | T | F | T | T |
| T | F | F | T | F | F | F |
| F | T | F | T | T | T | F |
| F | F | F | F | T | T | T |