Logical reasoning and valid arguments

Statements have a truth value

Logical reasoning examines whether conclusions follow from stated information. A proposition, or statement, is a declarative sentence that has a definite truth value in its intended setting: true or false. “Seven is prime” is true, and “Nine is even” is false. A question such as “What time is it?” and a command such as “Close the book” are not propositions. An expression such as x > 3 becomes a statement only after the value or allowed quantification of x is established.

Letters p and q can stand for propositions. Working with letters separates the logical structure from the subject matter. Negation, written ¬p or “not p”, reverses the truth value. If p is true, ¬p is false; if p is false, ¬p is true. For p: “The integer is even”, the negation is “The integer is not even”. The domain matters: an integer is either even or odd, but calling a general real number “not even” does not necessarily make it an odd integer.

Combine statements with and and or

The conjunction p ∧ q means both p and q. It is true only when both statements are true. The disjunction p ∨ q means p or q in the inclusive mathematical sense: at least one is true, and both may be true. Everyday speech sometimes uses an exclusive “or”, meaning one but not both, so read whether exclusivity is explicitly required. In ordinary propositional exercises, an unqualified logical disjunction is inclusive.

A truth table systematically lists possible truth assignments. For two statements, use four rows: (p. In that order, p ∧ q has values T, F, F, F; p ∨ q has values T, T, T, F. Add a separate column for each intermediate expression in a compound proposition. With three statements there are eight input rows. Keeping the input order consistent prevents omitting a case or comparing columns that refer to different cases.

Understand an implication

The conditional means “if p, then q”. The antecedent is p and the consequent is q. It is false precisely when p is true and q is false: that case breaks the promised condition. Its truth-table values in the four-row order above are T, F, T, T. In this formal use, an implication does not claim that p causes q or that p is actually true. It describes the permitted truth-value relationship.

For a mathematical example, let p be “n is divisible by 4” and q be “n is even”, with n an integer. The implication is true because any multiple of 4 is a multiple of 2. This makes p sufficient for q and q necessary for p. An even number need not be divisible by 4: n = 6 demonstrates that. Necessary and sufficient conditions are different unless both directions have been established. A biconditional p ↔ q means both and.

Distinguish converse, inverse and contrapositive

The converse of is q → p. The inverse is ¬p → ¬q. The contrapositive is ¬q → ¬p. The original implication and its contrapositive have identical truth tables, so they are logically equivalent. The converse and inverse are equivalent to each other, but not generally to the original. For the divisibility example, “If n is not even, n is not divisible by 4” is the valid contrapositive. “If n is even, n is divisible by 4” is the false converse.

To negate a conditional, assert the case in which it fails: is equivalent to p ∧ ¬q. It is not generally equivalent to ¬p → ¬q. To negate “If a shape is a square, it has four equal sides”, a counterexample would need to be a square without four equal sides. Merely naming a nonsquare does not refute the original conditional. Write the logical form before changing a complicated verbal statement.

Use equivalent forms and quantify carefully

De Morgan's laws give ¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q. Thus “not both p and q” allows either one to be false, while “neither p nor q” requires both false. These are different claims. The proposition p ∨ ¬p is a tautology, true under every assignment. The proposition p ∧ ¬p is a contradiction, false under every assignment. A proposition true in some assignments and false in others is contingent.

A universal statement refers to every element of a stated domain; an existential statement says at least one element has the property. The negation of “All members passed” is “At least one member did not pass”, not “No member passed”. The negation of “Some member passed” is “No member passed”. In mathematical language, “some” means at least one and does not automatically exclude all. To disprove a universal claim, one valid counterexample within its domain is enough.

Test arguments, not just individual claims

An argument is valid when its conclusion must be true whenever all its premises are true. From and p, infer q; this is often called modus ponens. From and ¬q, infer ¬p; this is modus tollens. By contrast, and q do not establish p. A number being even cannot prove it is divisible by 4. Likewise, and ¬p do not establish ¬q, since a number not divisible by 4 can still be even.

Validity concerns the structure of an inference, while soundness additionally requires true premises. An argument may have a valid form but begin with false factual assumptions. A truth table can test validity by checking every row where all premises are true and looking for a false conclusion. Self-check: negate “Every rectangle is a square”, form the contrapositive of “If x > 4 then x > 2”, and assess the inference “n is even, therefore n is divisible by 4”. The answers are “At least one rectangle is not a square”; “If then x ≤ 4”; and invalid, with n = 6 as a counterexample. State the domain and avoid replacing a logical conclusion with an assumption that merely sounds familiar.