Logical Relations and Truth Conditions in Propositional Logic
Logical Relations
Types of Logical Relations
Logical relations describe how statements connect and interact with each other. Some common types include:
- Synonymy: Two statements have the same meaning. (e.g., “He is a bachelor” and “He has never been married”)
- Entailment: The truth of one statement guarantees the truth of another. (e.g., “He killed John” entails “John is dead”)
- Contradiction: Two statements are incompatible and cannot both be true. (e.g., “He went to Brazil” and “He has never been to Brazil”)
- Presupposition: A statement implies a background assumption. (e.g., “The king of England is ill” presupposes “England has a king”)
- Tautology: A statement is always true due to its logical structure. (e.g., “Rich people are rich”)
Logic and Truth
Logic provides tools to represent and analyze the meaning of sentences, helping us determine their truth values and relationships.
Inference Rules
Inference rules are patterns of reasoning that allow us to draw conclusions from premises. Examples include:
- Modus Ponens: If P implies Q and P is true, then Q must be true. (e.g., “If I have money, then I will buy a bike. I have money. Therefore, I will buy a bike.”)
- Modus Tollens: If P implies Q and Q is false, then P must be false. (e.g., “If there is sunlight, then it is daytime. It is not daytime. Therefore, there is no sunlight.”)
- Hypothetical Syllogism: If P implies Q and Q implies R, then P implies R. (e.g., “If I don’t wake up, then I cannot go to work. If I cannot go to work, then I will not get paid. Therefore, if I don’t wake up, then I will not get paid.”)
- Disjunctive Syllogism: If either P or Q is true, and P is false, then Q must be true. (e.g., “Either it will rain or it will snow. It is not raining. Therefore, it will snow.”)
Truth Conditions and Truth Values
Truth Conditions
Truth conditions are the circumstances that must hold for a statement to be true. They determine the truth value (true or false) of a statement.
Example
The statement “It is raining” is true if and only if it is actually raining at the time and place referred to.
Necessary vs. Analytic Truth
Necessary Truth
A necessarily true statement cannot be false under any circumstances. It is true in all possible worlds. (e.g., “All bachelors are unmarried”)
Analytic Truth
An analytically true statement is true by virtue of its meaning and the definitions of the words involved. (e.g., “All triangles have three sides”)
Example
The statement “2 + 2 = 4” is both necessarily and analytically true.
A Priori vs. A Posteriori Knowledge
A Priori Knowledge
Knowledge that is independent of experience. (e.g., knowledge of logical truths or mathematical axioms)
A Posteriori Knowledge
Knowledge that is based on experience or observation. (e.g., knowledge of historical events or scientific facts)
Synthetic vs. Analytic Statements
Synthetic Statements
Statements whose truth depends on the way the world is. (e.g., “The sky is blue”)
Analytic Statements
Statements whose truth depends on the meaning of the words and logical structure. (e.g., “All bachelors are unmarried”)
Presupposition vs. Entailment
Presupposition
A presupposition is an implicit assumption that is taken for granted in a statement. (e.g., “John stopped smoking” presupposes that John used to smoke.)
Entailment
Entailment is a relationship between statements where the truth of one statement guarantees the truth of another. (e.g., “John is a bachelor” entails “John is unmarried.”)
Modus Tollens
Formal Description
If P implies Q and Q is false, then P is false.
Symbolically: ((P → Q) ∧ ¬Q) → ¬P
Example
If it is raining, then the ground is wet. The ground is not wet. Therefore, it is not raining.
Contradiction
Composite Truth Table
A contradiction is a statement that is always false, regardless of the truth values of its components.
Example
“He went to Brazil and he has never been to Brazil” is a contradiction.
Material Implication
Truth Table
| P | Q | P → Q | |—|—|——-| | T | T | T | | T | F | F | | F | T | T | | F | F | T |
Example
“If it rains, then I will go to the movies” is a material implication. It is only false when it rains and I don’t go to the movies.
Biconditional
Analysis
“If and only if you finish your studying, you will play” is a biconditional statement. It means that you will play if and only if you finish your studying. Both conditions must be met for the statement to be true.
Truth Table
| P | Q | P ↔ Q | |—|—|——-| | T | T | T | | T | F | F | | F | T | F | | F | F | T |
Clash between Natural Language and Propositional Logic
Example
“If you go to the party, then I will come with you” may imply a causal relationship in natural language, suggesting that your going to the party is the reason I will go. However, in propositional logic, the statement is simply a material implication and does not necessarily imply causation.
Truth Table
| P | Q | P → Q | |—|—|——-| | T | T | T | | T | F | F | | F | T | T | | F | F | T |
Logical Negation and Conjunction
Examples
- ¬(Q ∧ P): It is not the case that both Q and P are true.
- ¬Q → P: If Q is not true, then P is true.