forallx: An Introduction to Formal Logic

This text introduces Sentential Logic (SL), a formal language designed to analyze the logical structure of arguments. By breaking down complex English sentences into atomic components represented by capital letters, SL allows for the systematic evaluation of validity. The core of this approach is the use of five truth-functional connectives—negation, conjunction, disjunction, conditional, and biconditional—which allow for the construction of complex logical statements from simpler ones.

To determine the logical properties of these statements, the author employs truth tables. A truth table is a mechanical, exhaustive procedure that maps every possible combination of truth values (True or False) for the atomic sentences involved. Because SL is truth-functional, the truth value of any complex sentence is entirely determined by the truth values of its atomic parts and the specific rules governing its connectives.

Truth tables provide a rigorous method for classifying sentences and arguments:

• Tautologies: Sentences that are true in every possible scenario.
• Contradictions: Sentences that are false in every possible scenario.
• Contingent Sentences: Sentences that can be either true or false depending on the circumstances.
• Validity: An argument is valid if there is no scenario (row in the truth table) where the premises are true and the conclusion is false.

This mechanical approach ensures that logical evaluation is objective and free from the ambiguities often found in natural language. While SL simplifies complex English expressions, it effectively captures the essential logical structure required to determine whether an argument holds up under formal scrutiny.