Symbolic logic studies the formal structure of arguments by translating ordinary claims into precise symbols. This cheat sheet covers propositional logic, where whole statements are connected by truth-functional operators, and predicate logic, where statements are analyzed using predicates, variables, and quantifiers. Students need these tools to test validity, build proofs, and avoid relying on ambiguous natural language.
The goal is to make common symbols, rules, and translation patterns easy to check while studying or solving problems.
In propositional logic, the core ideas are truth values, connectives such as ¬, ∧, ∨, →, and ↔, and methods such as truth tables and derivations. In predicate logic, the key additions are predicates, individual constants, variables, domains, and quantifiers such as ∀ and ∃. Validity means that there is no interpretation in which all premises are true and the conclusion is false.
Many proof methods depend on preserving truth through rules such as modus ponens, universal instantiation, and existential generalization.
Key Facts
- Negation reverses truth value: if P is true then ¬P is false, and if P is false then ¬P is true.
- A conjunction P ∧ Q is true only when both P and Q are true.
- An inclusive disjunction P ∨ Q is false only when both P and Q are false.
- A conditional P → Q is false only when P is true and Q is false.
- A biconditional P ↔ Q is true exactly when P and Q have the same truth value.
- Universal quantification ∀x Fx means every object in the domain has property F.
- Existential quantification ∃x Fx means at least one object in the domain has property F.
- A valid argument has this form: if premises P1, P2, ..., Pn are all true, then conclusion C must be true.
Vocabulary
- Proposition
- A proposition is a statement that has a truth value, either true or false.
- Connective
- A connective is a logical operator, such as ¬, ∧, ∨, →, or ↔, that forms compound propositions from simpler ones.
- Predicate
- A predicate is an expression such as Fx or Loves(x, y) that attributes a property or relation to one or more objects.
- Quantifier
- A quantifier is a symbol, usually ∀ or ∃, that states how many objects in a domain satisfy a predicate.
- Domain
- The domain is the set of objects over which variables range in a predicate logic interpretation.
- Validity
- Validity is the property an argument has when it is impossible for the premises to be true and the conclusion false.
Common Mistakes to Avoid
- Treating P → Q as meaning P causes Q is wrong because the material conditional only describes a truth-functional relation between P and Q.
- Forgetting that P ∨ Q is inclusive is wrong because in standard propositional logic P ∨ Q is true when P, Q, or both are true.
- Negating a quantified statement incorrectly is wrong because ¬∀x Fx is equivalent to ∃x ¬Fx, and ¬∃x Fx is equivalent to ∀x ¬Fx.
- Using an existential witness as if it were arbitrary is wrong because a name introduced from ∃x Fx may refer only to some object, not every object.
- Confusing validity with truth is wrong because validity applies to argument form, while truth applies to individual statements under an interpretation.
Practice Questions
- 1 Make a truth table for (P → Q) ∧ P and determine whether Q follows by modus ponens.
- 2 Translate into predicate logic: Every philosopher is a thinker, using Px for x is a philosopher and Tx for x is a thinker.
- 3 Negate the sentence ∀x (Student(x) → Reads(x)) and simplify the negation using quantifier rules.
- 4 Explain why an argument can be valid even if one or more of its premises are actually false.