# Predicate Logic, a Sketch

##### 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
• 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.
##### 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
• 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
• 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
• 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.
• 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.
• 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)
• 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
##### 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.
• 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
1. {1} (x)(y)((Bxy & Hxy)  Axy)
• Premise
2. {2} (Ex)(Ey)(Bxy & Hxy & ~Pxy)
• Premise
3. {3} (Ey)(Bay & Hay & ~Pay)
• Premise
4. {4} Bab & Hab & ~Pab
• Premise
5. {1} (y)((Bay & Hay)  Aay)
• Universal Specification on #1
6. {1}  (Bab & Hab)  Aab
• Universal Specification on #6
7. {1,4} ~(Aab  Pab)
• Tautological Inference on #4 and #6
8. {1,4} (Ey)~(y)(Aay  Pay)
• Existential Generalization on #7
9. {1,4} ~(y)((Aay  Pay)
• Quantifier Exchange on #8
10. {1,4} (Ex)~(y)((Axy  Pxy)
• Existential Generalization on #9
11. {1,4} ~(x)(y)((Axy  Pxy)
• Quantifier Exchange on #10
12. {1,3} ~(x)(y)((Axy  Pxy)
• Existential Specification on #3, #4, #11
13. {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
• 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
###### 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
• 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