Propositional logic

Propositional logic represents statements with letters such as p and q, leaving their internal structure unanalyzed. A letter might stand for a sensor reporting an alarm, but the formal symbol does not describe the sensor or the properties of alarms. Formation rules specify which combinations of letters and connectives count as formulas. This syntactic question is distinct from the semantic question of when a formula is true.

Parentheses identify the scope of connectives. For example, p∧(q∨r) requires p to hold, whereas (p∧q)∨r may hold solely because r does. Making grouping explicit prevents a reader from supplying an unintended interpretation.

References: [1]

A classical valuation consistently assigns either true or false to each statement letter. Negation reverses a truth value. Conjunction requires both components to be true, while inclusive disjunction requires at least one. A biconditional is true when its two components have matching truth values.

The material conditional p→q is false exactly when p is true and q is false. In particular, a false antecedent makes this conditional true, regardless of the consequent. This is a stipulated truth function, not a claim that p causes q. Reversing the arrow generally changes the formula: a conditional and its converse are not interchangeable.

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An argument is valid when no valuation makes every premise true and its conclusion false. Let p mean that an alarm is reported and q that an indicator lights up. Consider premises p→q and p, with conclusion q. If p is false, the second premise fails. If p is true but q false, the first premise fails. The premises can therefore both hold only with q true.

The reverse-looking inference, from p→q and q to p, fails. Assign p false and q true: both premises hold, but the conclusion does not. This countervaluation is enough to establish invalidity; it does not require a story about a particular device.

Validity concerns the relationship between premises and conclusion, not whether the premises accurately describe the world. A tautology holds under every valuation, whereas satisfiability requires only one valuation making a formula true. For example, p∨¬p is a tautology, while p∧¬p is unsatisfiable.

References: [3]

Treating statements as units deliberately discards information. Reasoning about objects, their properties, and expressions such as “every” or “some” calls for richer resources, notably first-order logic. An inference may depend on structure that disappears when whole sentences become independent letters.

Ordinary conditionals also carry meanings not uniformly captured by the material conditional, including counterfactual expectations. Consequently, formalization is a substantive step rather than automatic word replacement. A correct calculation can establish a relation between formulas while failing to settle the intended natural-language question. The appropriate lesson is to check the representation, not to mistake a limitation of this model for the absence of meaningful reasoning.

References: [4]