Explicitly defines objects and relations among those objects. For example take the following sentence:
- "Dan is taller than both his mothers, but they’re the same height.”

The objects: 
- Dan
- M1 = Mother 1
- M2 = Mother 2

The relations:
- T(x, y) = "x is taller than y"

The predicate:
- T(Dan, M1) $$\land$$ T(Dan, M2) $$\land$$ $$\lnot$$ T(M1, M2) $$\land$$ $$\lnot$$ T(M2, M1)

Predicate logic

How to refer to someone or everyone (at least one or all)...

Universal quantifier ($$\forall$$)
- "for all", "every", "any"

Existential quantifier ($$\exists$$)
- "there exists", "at least one", "some"

Predicates and quantification

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.

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A declarative statement that is either true or false

Propositions (mathematics)

- 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)

Axioms (propositions)

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

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