Question

Which of the following a tautology ? (a) p \(\Lambda(\sim \mathrm{p})\) (b) \(\mathrm{p} \Lambda \mathrm{c}\) (c) \(p \mathrm{~V} \mathrm{t}\) (d) \(\mathrm{p} \Lambda \mathrm{p}\)

Short answer

None of the given propositions (a) p ∧ (¬p), (b) p ∧ c, (c) p ∨ t, and (d) p ∧ p have only "True" values in their truth table result columns, thus none of them is a tautology.

Step-by-step solution

Understanding the notations

We have four propositions to evaluate: (a) p ∧ (¬p) (b) p ∧ c (c) p ∨ t (d) p ∧ p Here, ∧ represents "and", ∨ represents "or", and ¬ represents "not".

Creating truth tables for each proposition

For each proposition, we will create a truth table to determine if it's a tautology. It's a tautology if the result column contains only "True" values. For (a): | p | ¬p | p ∧ (¬p) | |----|----|----------| | T | F | F | | F | T | F | For (b): | p | c | p ∧ c | |---|---|-------| | T | T | T | | T | F | F | | F | T | F | | F | F | F | For (c): | p | t | p ∨ t | |----|----|-------| | T | T | T | | T | F | T | | F | T | T | | F | F | F | For (d): | p | p | p ∧ p | |---|---|-------| | T | T | T | | F | F | F |

Analyzing the truth table results

Now, we check for the result columns in the truth tables: (a) p ∧ (¬p) => F, F (b) p ∧ c => T, F, F, F (c) p ∨ t => T, T, T, F (d) p ∧ p => T, F

Finding the tautology

None of the given propositions have only "True" values in their result column, thus none of them is a tautology.

Key concepts

Truth Tables

In logic, truth tables are a fundamental tool used to determine whether an argument is valid, a statement is true, or in this case, whether an expression is a tautology. A truth table lists all possible combinations of truth values for a given set of propositions and shows the outcome of logical operations based on those values.

Truth tables are constructed by writing out all possible scenarios in which the involved variables can be either true (T) or false (F), and then computing the truth value of the compound propositions for each scenario. The tables often have columns for each individual proposition, followed by columns for each stage of the logical operations, and finally a column showing the results of the entire expression.

For an expression to qualify as a tautology, the result column in its truth table must contain only true (T) outcomes, regardless of the truth values of its components. In the exercise provided, we use truth tables to assess if any of the given propositions is a tautology by looking at the last column of each table.

Logical Conjunction

The concept of logical conjunction refers to the logical 'and' operation, symbolized by the notation \(\wedge\). This operation is binary, meaning it takes two operands. In the context of truth tables, the conjunction of two propositions is true only if both of the propositions are true.

In everyday language, the conjunction works as the word 'and' does. For example, the statement 'It is sunny and warm' is true only when both conditions—sunny and warm—are met. Similarly, in the truth table for the proposition \(p \wedge c\), the only time the result is true (T) is when both \(p\) and \(c\) are true. The presence of even one false (F) makes the conjunction false.

In the exercise's options (a), (b), and (d), logical conjunction is tested. None of these propositions are tautologies because the resulting column contains false (F) values, indicating the compound statement is not always true.

Logical Disjunction

Conversely, the term logical disjunction represents the logical 'or' operation, denoted as \(\vee\). This operation is also binary and results in a true value when at least one of the operands is true. The only case in which a disjunction is false is when both operands are false.

The disjunction acts like the inclusive 'or' in common usage, where 'coffee or tea' means you could have either one or both. For instance, the truth table for the proposition \(p \vee t\) will show true results in all rows except for the one where both \(p\) and \(t\) are false. In the context of the exercise, option (c) explores disjunction. While mostly true, it still fails to be a tautology, as it becomes false when both \(p\) and \(t\) are false.

Logical Negation

Lastly, logical negation is a unary operation that inverts the truth value of a proposition. It is symbolized by \(eg\) or sometimes by a tilde (\sim). Simply put, if a proposition \(p\) is true, then \(eg p\) would be false, and vice versa.

Logical negation plays a crucial role in the understanding of contradictions and tautologies. For something to be a contradiction, the conjunction of a proposition and its negation, \(p \wedge eg p\), must always be false. This principle is showcased in option (a) of the exercise, \(p \wedge (eg p)\), which is not a tautology but rather a contradiction since the proposition and its negation can never both be true at the same time.

The concept of negation is inherent in logical systems and is essential for constructing complex logical expressions and understanding logical equations.