| Symbolic notation | English notation or English reading | ||
| Negation | ¬ φ |
not φ
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| Conjunction | φ ∧ ψ |
both φ and ψ
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(φ and ψ)
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| Disjunction | φ ∨ ψ |
either φ or ψ
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(φ or ψ)
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| The conditional |
φ → ψ
ψ ← φ |
if φ then ψ
yes ψ if φ
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(φ implies ψ)
(ψ if φ)
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| Predication | θτ1...τn |
θ fits τ1, ..., ’n τn
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| Identity | τ = υ |
τ is υ
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| Compound term | γτ1...τn |
γ of τ1, ..., ’n τn
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| Predicate abstract | λx φ |
the property of x that φ
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| Complex functor | λx τ |
the function of x yielding τ
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| Universal quantification | ∀x θx |
forall x θx
everything, x, is such that θx |
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| Restricted universal | (∀x: ρx) θx |
forall x st ρx θx
everything, x, such that ρx is such that θx |
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| Existential quantification | ∃x θx |
forsome x θx
something, x, is such that θx |
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| Restricted existential | (∃x: ρx) θx |
forsome x st ρx θx
something, x, such that ρx is such that θx |
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| Definite description | Ix ρx |
the x st ρx
the thing, x, such that ρx |
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| Truth-functional compounds | |||
| neither φ nor ψ |
¬ (φ ∨ ψ)
¬ φ ∧ ¬ ψ |
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| 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) |
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| 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 ... |
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