Symbolization Review

Logical System

8/26/12

[The core of this is from Hugues Leblanc and William A. Wisdom [1972] Deductive Logic p.111 and f.]

Symbolization using the Universal Quantifier (of non relational English)

‘Everything is water’ we understand as meaning ‘Whatever thing you may choose, it is water’ and this becomes

 

(∀x)Wx

Similarly ‘Nothing is water is read as ‘Whatever thing you may choose, it is not water and this becomes

 

(∀x)~Wx

‘Violets are blue’ is read ‘Whatever x you may choose, if it is a violet, then it is blue’ which is

 

(∀x)(Vx⊃Bx)

Everything is either material or spiritual.

 

(∀x)(Mx∨Sx)

Nothing is both round and square.

 

(∀x)~(Rx∧Sx)

All and only friends text.

 

(∀x)(Fx≡Tx)

Everything fell off the shelf.

 

(∀x)(Ox⊃Fx)

All pacifists are Quakers.
Pacifists are all Quakers.
A pacifist is a Quaker.
To be a pacifist is to be a Quaker.
Every (any,each) pacifist is a Quaker.
If anyone’s (someone’s) a pacifist, he’s a Quaker. Only Quakers are pacifists.
None but Quakers are pacifists.

 

(∀x)(Px⊃Qx)

No adults are trustworthy.

 

(∀x)(Ax⊃~Tx)

No babies are unattractive.

 

(∀x)(Bx⊃~Ux)

Old doctors are all conservative.

 

(∀x)((Dx∧Ox)⊃Cx)

Doctors and lawyers are all conservative.

 

(∀x)((Dx∨Lx)⊃Cx) (*note the 'or' *)

People are intelligent if they are witty.

 

(∀x)(Px⊃(Wx⊃Ix))

No intelligent men are lonely if they are young and healthy.

 

(∀x)((Mx∧Ix)⊃((Yx∧Hx)⊃~Lx))

Symbolization using the Universal Quantifier (of English with relations)

Peter hates everything.

 

(∀x)Hpx

Everything is a delusion caused by dehydration.

 

(∀x)(Dx∧Cxd)

Raewyn gets everything she wants.

 

(∀x)(Wrx⊃Grx)

Nothing that Raewyn saw was new.

 

(∀x)(Srx⊃~Nx)

Symbolization using the Existential Quantifier (of non relational English)

‘Something fell off the shelf’ is understood as ‘There is at least one thing, x, such that x fell off the shelf’.

 

(∃x)Fx

‘Some teachers are infallible’ is understood as ‘There is at least one thing, x, such that x is a teacher and x is infallible’.

 

(∃x)(Tx∧Ix)

Some teachers are not infallible.

 

(∃x)(Tx∧~Ix)

Something is a centaur.
There are centaurs.
There is a centaur.
Centaurs exist.

 

(∃x)Cx

Some tigers are mean if they are hungry.

 

(∃x)(Tx∧(Hx⊃Mx))

Some young folk are neither rich nor unhappy.

 

(∃x)((Yx∧Fx)∧~(Rx∨~Hx))

Symbolization using the Existential Quantifier (of English with relations)

Peter doesn’t understand something.

 

(∃x)~Upx

Somebody lost something.

 

(∃x)(∃y)(Px∧Lxy) or (∃x)(Px∧(∃y)(Lxy))

Symbolization using both Quantifiers (of English with relations)

‘Everything has a cause’ becomes ‘Whatever thing you may choose, something (or other) is the cause of it’.

 

(∀x)(∃y)Cyx

‘Something is the cause of everything’ becomes ‘There is a something (the same thing) such that whatever thing you may choose, that something is the cause of it’.

 

(∃y)(∀x)Cyx

Everything is the cause of everything.

 

(∀y)(∀x)Cyx