Arguments

An argument has one or more premises and one conclusion.

Denoted by \(P_1, P_2, \ldots \therefore C\).

Statements

Premises and conclusions must be statements: sentences that are true/false.

Validity

An argument is valid if and only if there is no possibility where the premises are true and the conclusion is false.

Soundness

An argument is sound if and only if it is valid and all its premises are true.

Propositional logic

Form

Validity is a matter of form. Any argument of the same form is valid.

Logical connectives

Symbol Name Meaning
\(\neg\) negation not
\(\wedge\) conjunction and
\(\vee\) disjunction or
\(\implies\) conditional if … then …
\(\iff\) biconditional if and only if

Symbolization

Symbolization key

| Letter | Statement | |-|-|

Atomic sentences

Any sentence that doesn’t contain a connective will be symbolized with a letter.

Symbols

Propositional logic contains three kinds of symbols:

  • Atomic sentences
  • Connectives
  • Brackets

Expression

An expression is any combination of symbols.

Sentence

A sentence is an expression defined as follows:

  1. Atomic sentences are sentences.
  2. If \(\phi\) is a sentence, then so is \(\neg \phi\).
  3. If \(\phi\) and \(\psi\) are sentences, then so are
    • \[(\phi \wedge \psi)\]
    • \[(\phi \vee \psi)\]
    • \[(\phi \implies \psi)\]
    • \[(\phi \iff \psi)\]

Main connective

The main connective of a sentence is the last one that is added in building up that sentence.

Arguments

We use \(P \therefore C\) to symbolize an argument.

(TBD)

Predicate Logic

Motivation: there are valid arguments that cannot be expressed in propositional logic.

Many sentences have the form of subject-predicate. Example:

  • \(a\): Alice
  • \(L(x)\): x is a logician
  • \(L(a)\): Alice is a logician

Quantifiers

Name Symbol Meaning
Existential Quantifier \(\exists\), E \(\exists x L(x)\): someone has property L
Universal Quantifier \(\forall\), A \(\forall x L(x)\): everyone has property L

Identity

Name Symbol Meaning
Identity = Two variables are the same

Sentences

An expression is any combination of symbols.

An atomic formula is either in form of \(P(t_1, t_2, \ldots, t_n)\) or \(t_1 = t_2\).

A formula is defined recursively:

  1. Rules from predicate logic
  2. If \(\phi\) is a formula and \(x\) is a variable, then
    • \(\forall x \phi(x)\) is a formula.
    • \(\exists x \phi(x)\) is a formula.

The main logical operator of a formula is the last operator introduced.

The scope of a logical operator is the sub-formula(s) for which it is the main logical operator.

A variable is bound if it occurs within the scope of a quantifier with that variable.

A sentence is any formula with no free variables.

Exam Notes

  • statements sentences that are true or false.
  • expression is any combination of valid symbols.
  • sentence is an expression that follows the rules.
  • A sentence is either tautology (always true), contradiction (always false), or contingent.
  • A set of sentences are equivalent if they have the same truth conditions, consistent if they have a row in which all the sentences are true.
  • provability: \(\phi \vdash \psi\).
  • validity: \(\phi \vDash \psi\).
  • soundness every provable argument is valid, completeness every valid argument is provable.
  • predicate logic break down sentences into subject-predicate.