Start Learning Logic 1 | Logical Statements, Negations and Conjunction [dark version]

Logic in mathematics starts with propositions: meaningful declarative statements that have a definite truth value—either true or false. A proposition must be unambiguous enough that, once the relevant definitions are agreed on, its truth can be evaluated. “Mars is a planet” becomes a proposition once “Mars” and “planet” are defined; “Pluto is a planet” is also a proposition under the same agreement, but it evaluates to false. Mathematical examples are often cleaner because definitions are built in: “one plus one equals two” is true once the meanings of the numerals and symbols are fixed, while “the number five is smaller than the number two” is a proposition that evaluates to false.

Not every sentence qualifies. “Good morning” lacks a well-defined truth value, so it is not a proposition. Likewise, a sentence that looks mathematical can fail to be a proposition if it contains an unfixed variable. For instance, “I have a variable X plus 1 and this is equal to 1” does not have a single truth value until X is assigned a specific value; once X is fixed, the statement becomes a proposition. Such variable-dependent constructions are common in mathematics, and they’re flagged as “predicates” (pretty kits in the transcript).

Once propositions are in place, logic builds new propositions using logical operations. The first operation introduced is negation. For a proposition A, the negation is written with a symbol in front of A and read as “not A.” Negation flips the truth value: if A is true, not A is false; if A is false, not A is true. The transcript emphasizes that correct negation is determined by a truth table, which lists all possible truth values for the input and the corresponding output. A key subtlety is that negation must be defined across both possible “worlds,” not just the one you already know. For example, if A is “2 + 2 = 5,” then A is false; but not A must still be false in the hypothetical world where A would be true. The truth table captures both cases.

The next operation is conjunction, combining two propositions A and B using a wedge symbol read as “A and B.” Conjunction is true only when both A and B are true simultaneously. With two inputs, the truth table expands from two rows to four, reflecting all combinations of truth values. The transcript also offers an intuitive circuit analogy: treat A and B as switches, where “true” means the switch is on and “false” means it’s off. The lamp (the output of A and B) turns on only when both switches are on; any case where at least one switch is off yields false. The takeaway is that logical operations are defined precisely through truth tables, ensuring unambiguous foundations for later work in mathematics.