A.2. Logical forms

Forms for which there is symbolic notation

Symbolic notation English notation or English reading
Negation ¬ φ not φ
Conjunction φ ∧ ψ both φ and ψ (φ and ψ)
Disjunction φ ∨ ψ either φ or ψ (φ or ψ)
The conditional φ → ψ
ψ ← φ 		if φ then ψ
yes ψ if φ 		(φ implies ψ)
(ψ if φ)
Identity τ = υ τ is υ
Predication θτ1…τn θ fits τ1, …, τn

A series of terms τ1, …, τn can be read (series) τ1, …, ən τn (using the expression ən to distinguish this use of and from its use in conjunction and adding series when necessary to avoid ambiguity)

Compound term γτ1…τn γ of τ1, …, τn
γ applied to τ1, …, τn
Predicate abstract [φ]x1…xn what φ says of x1…xn
Functor abstract [τ]x1…xn τ for x1…xn

Universal
quantification

∀x θx forall x θx
everything, x, is such that θx

Restricted
universal

(∀x: ρx) θx forall x st ρx θx
everything, x, such that ρx is such that θx

Existential
quantification

∃x θx forsome x θx
something, x, is such that θx

Restricted
existential

(∃x: ρx) θx forsome x st ρx θx
something, x, such that ρx is such that θx

Definite
description

Ix ρx the x st ρx
the thing, x, such that ρx

Some paraphrases of other forms

Truth-functional compounds
neither φ nor ψ ¬ (φ ∨ ψ)
¬ φ ∧ ¬ ψ
ψ only if φ ¬ ψ ← ¬ φ
ψ unless φ ψ ← ¬ φ
Generalizations

All Cs are such
that ( … they … )

(∀x: x is a C) … x …

No Cs are such
that ( … they … )

(∀x: x is a C) ¬ … x …

Only Cs are such
that ( … they … )

(∀x: ¬ x is a C) ¬ … x …
with: among Bs add to the restriction: x is a B
except Es ¬ x is an E
other than τ ¬ x = τ
Numerical quantifier phrases

At least 1 C is such
that ( … it … )

(∃x: x is a C) … x …

At least 2 Cs are such
that ( … they … )

(∃x: x is a C) (∃y: y is a C ∧ ¬ y = x) ( … x … ∧ … y … )

Exactly 1 C is such
that ( … it … )

(∃x: x is a C) ( … x … ∧ (∀y: y is a C ∧ ¬ y = x) ¬  … y … )
or
(∃x: x is a C) ( … x … ∧ (∀y: y is a C ∧ … y … ) x = y)
Definite descriptions (on Russell’s analysis)

The C is such
that ( … it … )

(∃x: x is a C ∧ (∀y: ¬ y = x) ¬ y is a C) … x …
or
(∃x: x is a C ∧ (∀y: y is a C) x = y) … x …
Glen Helman 15 Jul 2012