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