Logic: Terminology

1

Natural languages have many kinds of sentences, but only a declarative sentence, i.e. statement, can be made, e.g. a subject of a discussion, attempting to determine its truth value, which is to say, whether the statement is true or false.

How does one go about determining the truth of a statement? One doesn't. One lets oneself become influenced by the ..khhm. Let me reprase that: How did one go about it in 1901? Ah, the life and truth were simple back ..argh. Can truth ever be determined? Why, no, of course. And there's no need to. It's just a label anyway, a point of view as one floats through the universe shifting from red to blue, remembering to avoid such questions.

Classically, known truth value means a sta

A declaration (with a known value) can become a premise in an argument, which is a collection of statements put together in order to show that the truth value of the conclusion (the final statement of an arg), follows (can be derived).

can be assigned a truth value.

Classically, there are only two truth values: true (truth, veracity) and false (falsity, falseness) which are the elements of the Boolean set, 𝔹 = {⟙, ⟘}.

Declarations from a natural language are convertible to a formal language of symbolic logic, where they become logical propositions.

An atomic proposition is a logical formula that cannot be further dissected. There are various kinds of propositions divided by their function and context:

Logical propositions:

• variables: p q r

  • quantifiers

    • universal ∀

    • existential ∃

• constants:

  • truth values, e.g. {true, false}, {⟙, ⟘}

  • equality, '='

• logical connectives:

  • usual: ¬ ∧ ∨ → ⟷

  • extra: ⊕

  • minimal: ↑ (NAND), ↓ (NOR)

  • mini-set: {¬, ∧} or {¬, ∨}

• predicates:

  • property: unary predicate, ∃x. x ∈ ℕ ∧ P(x)

  • relation: polyadic predicate

formula sequent cedent hypothesis antecedent conclusion consequent succedent

protasis