Using truth tables on large propositions would be very time consuming and difficult. This is why it is better to use logical identities which you can use to prove equivalences. They include:
- Distributive laws
- De Morgan's laws
- Associative laws
- Commutative  laws
- Double-negation law
- Identity laws
- Domination laws
- Idempotent Laws
- Implication laws
- Contrapositive law

Logical Identities (proving logical equivalences)

A chain of logical deductions that leads to a proposition form an original (base) set of axioms.

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an inferential argument for a mathematical statement that uses assumptions to logically show the conclusion is valid

Proofs (mathematics)

Axioms (propositions)

A declarative statement that is either true or false

Propositions (mathematics)

Consists of premises followed by a conclusions.

Syllogism

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Formal Proofs (of propositions)

- Two parallel lines on a plane have a point in common : (FALSE)

- Ann Arbor is the capital of America : (FALSE)

- 2 + 3 + 5 = 10 : (TRUE)

Examples of propositions

Propositions and connectives are used to make compound propositions. One can make deductions and proofs drawing from the structure of propositions

Propositional Logic

Built from multiple propositions using logical operators to connect the propositions

Compound proposition

If the proposition is not a contradiction then it is satisfiable, otherwise it is not.

How can we know if a proposition is satisfiable?

If two compound propositions have the same truth table they are logically equivalent (p $$\equiv$$ q).
- ((p $$\rightarrow$$ q) $$\land$$ (q $$\rightarrow$$ p) $$\equiv$$ p $$\leftrightarrow$$ q is a tautology)

Logically Equivalent

A proposition that is always false (e.g. $$\lnot$$ p $$\land$$ p)

Contradiction (propositions)

A proposition that is always true (e.g. $$\lnot$$ p $$\lor$$ p)

Tautology (propositions)

A predicate is an open declarative statement containing one or more variables. It becomes a proposition when every variable is assigned an object or value from its domain or is bound by a quantifier, so that the resulting statement can be true or false.

Predicate (mathematical logic)

Statements that justifies a conclusion. They are assumptions that something is true.

Premises

Syllogisms that begin with all, none or some.

Categorical Syllogisms

People have a tendency to think a syllogism is valid if it looks like it is true.

Belief Bias

People apply mental models using their imagination to solve deductive reasoning problems. The mental model is the representation of a situation in someones mind and can help determine validity of syllogisms.

Mental model approach

Syllogisms that have a premise of "if... then..."

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