# 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