**Table of Contents**

- Predicate Logic
- How it Works
- Translations into Predicate Logic
- Proof that the Astral Twin Argument against Astrology is Valid
- Rules of Inference for Predicate Logic L′
- Counterexamples in Predicate Logic

##### Predicate Logic

*Deductive Logic*is the formal theory of deductive reasoning.- Developed in 1879 by Gotlob Frege, the founder of modern formal logic,
*Predicate Logic*is the foundation of Deductive Logic. - Predicate Logic consists of:
- a symbolic language
- rules of inference
- a semantics.

##### How it Works

- To determine whether the following argument-form is valid:
- All F’s are H’s.
- All G’s are non-H’s.
- Therefore, nothing is both an F and a G.

- First, translate the argument-form into the language of Predicate Logic
- (x)(Fx
**→**Hx)- Every F is and H

- (x)(Gx
**→**~Hx)- Every G is a non-H

- Therefore, ~(Ex)(Fx & Gx)
- Nothing is both an F and a G

- (x)(Fx
- Then, either
- Prove the translated argument-form valid by deriving the conclusion from the premises using the rules of inference of Predicate Logic
- or

- Show the argument-form invalid by putting forth a counterexample.

- Prove the translated argument-form valid by deriving the conclusion from the premises using the rules of inference of Predicate Logic

##### Translations into Predicate Logic

- Black swans exist. (B: x is black, S: x is a swan)
- (Ex)(Bx & Sx)
- There exists an x such that x is B and x is S

- (Ex)(Bx & Sx)
- An oral contract isn’t worth the paper it is written on. (C: x is a contract, O: x is oral, W: x is worth the paper it is written on)
- (x)( (Ox & Cx)
**→**~Wx)- For any x, if x is O and x is C then x isn’t W

- (x)( (Ox & Cx)
- Nothing easy is worthwhile. (E: x is easy, W: x is worthwhile)
- ~(Ex)(Ex & Wx)
- It’s not the case that there exists an x such that x is E and x is W

- ~(Ex)(Ex & Wx)
- Opposites attract. (O: x and y are opposites, A: x and y attract)
- (x)(y)(Oxy
**→**Axy)- For any x and y if x and y are opposites then x and y attract.

- (x)(y)(Oxy
- There is a guard on duty at all times (though not necessarily the same guard). (G: x is a guard, T: x is a time, D: x is on duty at y)
- (x)(Tx
**→**(Ey)(Gy & Dyx))- For any x, if x is a time then there’s a y such that y is a guard and y is on duty at x.

- (x)(Tx
- The same guard is on duty at all times (poor guy) (G: x is a guard, T: x is a time, D: x is on duty at y)
- (Ey) (Gy & (x)(Tx
**→**Dyx))- There exists a y such that y is a guard and for any x if x is a time then y is on duty at x)

- (Ey) (Gy & (x)(Tx
- No two people have the same fingerprints. (Domain: people, F: x has the same fingerprints as y, S: x is the same person as y))
- ~(Ex)(Ey)(Fxy & ~Sxy)
- There’s no x and y such that x and y have the same fingerprints and x is not the same person as y

- ~(Ex)(Ey)(Fxy & ~Sxy)

##### Proof that the Astral Twin Argument against Astrology is Valid

- Astral Twin Argument against Astrology
- People born within minutes of each other in the same hospital have the same astrological chart.
- But some people born within minutes of each other in the same hospital don’t have the same personality traits.
- Therefore, not everyone with the same astrological chart has the same personality traits.

- Argument in Symbolic Notation
- (x)(y)((Bxy & Hxy)
**→**Axy)- For any x and y, if x and y are born minutes apart and x and y are born in the same hospital, then x and y have the same astrological chart.

- (Ex)(Ey)(Bxy & Hxy & ~Pxy)
- There are x and y such that x and y are born minutes apart and x and y are born in the same hospital and it’s false that x and y have the same personality traits

- Therefore, ~(x)(y)((Axy
**→**Pxy)- It’s false that, for any x and y, if x and y have the same astrological chart then x and y have the same personality traits.

- (x)(y)((Bxy & Hxy)

- Derivation of ~(x)(y)((Axy
**→**Pxy) on line13 from- (x)(y)((Bxy & Hxy)
**→**Axy) on line 1 - (Ex)(Ey)(Bxy & Hxy & ~Pxy) one line 2

- (x)(y)((Bxy & Hxy)

- {1} (x)(y)((Bxy & Hxy)
**→**Axy)- Premise

- {2} (Ex)(Ey)(Bxy & Hxy & ~Pxy)
- Premise

- {3} (Ey)(Bay & Hay & ~Pay)
- Premise

- {4} Bab & Hab & ~Pab
- Premise

- {1} (y)((Bay & Hay)
**→**Aay)- Universal Specification on #1

- {1} (Bab & Hab)
**→**Aab- Universal Specification on #6

- {1,4} ~(Aab
**→**Pab)- Tautological Inference on #4 and #6

- {1,4} (Ey)~(y)(Aay
**→**Pay)- Existential Generalization on #7

- {1,4} ~(y)((Aay
**→**Pay)- Quantifier Exchange on #8

- {1,4} (Ex)~(y)((Axy
**→**Pxy)- Existential Generalization on #9

- {1,4} ~(x)(y)((Axy
**→**Pxy)- Quantifier Exchange on #10

- {1,3} ~(x)(y)((Axy
**→**Pxy)- Existential Specification on #3, #4, #11

- {1,2} ~(x)(y)((Axy
**→**Pxy)- Existential Specification on #2, #3, #12

**Rules of Inference for Predicate Logic L′**

**From Benson Mates’ ****Elementary Logic**

*Premise Introduction*(P): Any sentence may be entered on any line of a derivation.- As the sole premise-number of the line take the line-number.

*Tautological Inference*(T): Any sentence may be entered on a line if it is a tautological consequence of a set of sentences that appear on earlier lines.- As premise-numbers of the new line take all those of the earlier lines.

*Conditionalization*(C): The sentence φ → ψ may be entered on a line if ψ appears on an earlier line.- As premise-numbers of the new line take all those of the earlier line, with the exception of any that is the line number of a line on which φ appears.

*Universal Specification*(US): The sentence φα/β may be entered on a line if (α)φ appears on an earlier line.- As premise-numbers of the new line take all those of the earlier line.
- Note: β is any constant term.

*Existential Generalization*(EG): The sentence (∃α)φ may be entered on a line if φα/β appears on an earlier line.- As premise-numbers of the new line take all those of the earlier line.
- Note: β is any constant term.

*Universal Generalization*(UG): The sentence (α)φ may be entered on a line if φα/β appears on an earlier line and β occurs neither in φ nor in any premise of that earlier line.- As premise-numbers of the new line take all those of the earlier line.
- Note: β is any individual constant.

*Existential Specification*(ES): Suppose that (∃α)ψ appears on a line i of a derivation, that ψα/β appears (as a premise) on a later line j, and that φ appears on a still later line k; and suppose further that the constant β occurs neither in φ, ψ nor in any premise of line k other that Ψα/β; then φ may be entered on a new line.- As premise-numbers of the new line take all those of lines i and k, except the number j.
- Note: β is any individual constant.

*Quantifier Exchange*(Q): The sentence -(α)-φ may be entered on a line if (∃α)φ appears on an earlier line and vice versa. Likewise for the following pairs of sentences: (α)φ and -(∃α)-φ; -(α)φ and (∃α)-φ; -(∃α)φ and (α)-φ.- As premise-numbers of the new line take all those of the earlier line.

*Identity Part A:*α=α may be entered on any line of a derivation with no premise-numbers.- Note: α is any constant term.

*Identity Part B:*φ may be entered on a line if α=β and ψ appear on earlier lines (where φ results from ψ by replacing one or more occurrences of α in ψ with β).- As premise-numbers of the new line take all those of the earlier lines.
- Note: α and β are any constant terms.

**Counterexamples in Predicate Logic**

A *counterexample to an argument-form* is an interpretation of the predicate letters under which the premises are true and the conclusion false, showing the argument-form invalid.

###### Counterexample 1

- Invalid Argument-form
- ~(x)(Fx
**→**Gx)- It’s not the case that all F’s are G’s

- ~(x)(Gx
**→**Hx)- It’s not the case that all G’s are H’s

- Therefore, ~(x)(Fx
**→**Hx)- It’s not the case that all F’s are H’s

- ~(x)(Fx
- Counterexample
- F: x is an integer > 10
- G: x is an integer < 7
- H: x is an integer > 5

- So
- Not every integer greater than 10 is less than 7.
- True

- Not every integer less than 7 is greater than 5.
- True

- Not every integer greater than 10 is greater than 5
- False

- Not every integer greater than 10 is less than 7.

###### Counterexample 2

- Invalid Argument-form
- (x)(Ey)Rxy
- For every entity, at least one entity R’s it.

- (x)(y)(Rxy
**→**Ryx)- For any entities x and y, if x R’s y then y R’s x.

- Therefore, (x)Rxx
- Every entity R’s itself

- (x)(Ey)Rxy
- Counterexample
- R: x and y are integers next to each other so that either x-y=1 or y-x=1

- So
- For every integer, at least one integer is next to it.
- True

- For any integers x and y, if x and y are next to each other, then y and x are next to each other.
- True

- Every integer is next to itself.
- False

- For every integer, at least one integer is next to it.