Predicate Logic Splitting the Atom

What Propositional Logic can’t do It can’t explain the validity of the Socrates Argument Because from the perspective of Propositional Logic its premises and conclusion have no internal structure.

The Socrates Argument All men are mortal Socrates is a man _______________________ Therefore, Socrates is mortal Translates as: H S __ M

The Problem The problem is that the validity of this argument comes from the internal structure of these sentences--which Propositional Logic cannot “see”: All men are mortal Socrates is a man _______________________ Therefore, Socrates is mortal

Solution We need to split the atom! To display the internal structure of those “atomic” sentences by adding new categories of vocabulary items To give a semantic account of these new vocabulary items and To introduce rules for operating with them.

Predicate Logic Vocabulary Individual Constants: lower case letters of the alphabet (a, b, c,…, u, v, w) Predicates: upper case letters (A, B, C,…, X, Y, Z) Connectives:  ,  , • ,  , and  Variables: x, y and z Quantifiers: Existential ( [variable]), e.g. (x), (y)… Universal ( [variable]) or just ([variable] ), e.g. (x), (y), (x), (y)…

Singular statements Make assertions about persons, places, things or times Examples: Socrates is a man. Athens is in Greece. Thomas Aquinas preferred Aristotle to Plato To display the internal structure of such sentences we need two new categories of vocabulary items: Individual constants (“names”) Predicates

Individual Constants & Predicates Socrates is a man. The name of a person—which we’ll express by an individual constant A predicate—assigns a property (being-a-man) to the person named

More Vocabulary So, to the vocabulary of Propositional Logic we add: Individual Constants: lower case letters of the alphabet (a, b, c,…, u, v, w) Predicates: upper case letters (A, B, C,…, X, Y, Z) These represent predicates like “__ is human,” “__ the teacher of __,” “__ is between __ and __,” etc. Note: Predicates express properties which a single individual may have and relations which may hold on more than one individual

Predicates 1-place predicates assign properties to individuals: __ is a man __ is red 2-place predicates assign relations to pairs of individuals __ is the teacher of __ __ is to the north of __ 3-place predicates assign relations to triples of individuals __ preferred __ to __ And so on…

Translation Socrates is a man. Hs Socrates is mortal. Ms Athens is in Greece Iag Thomas preferred Aristotle to Plato. Ptap We always put the predicate first followed by the names of the things to which it applies.

Predicates designate sets! Individual constants name individuals—persons, places, things, times, etc. We resolve to understand properties and relations as sets—of individuals or ordered n-tuples of individuals (pairs, triples, quadruples, etc.) Fido { brown things } is brown

Ordered Sets We use curly-brackets to designate sets so, e.g. the set of odd numbers between 1 and 10 is {3, 5, 7, 9} Order doesn’t matter for sets so, e.g. {3, 5, 7, 9} = {3, 9, 5, 7} But sometimes order does matter: to signal that we use pointy-brackets to designate ordered n-tuples— ordered pairs, triples, quadruples, quintuples, etc. <Adam, Eve> ≠ <Eve, Adam> <1, 2, 3> ≠ <3, 2, 1>

ε Having a property Fido { brown things } An individual’s having a property is understood as its being a member of a set. “Fido is brown” says that Fido is a member of the set of brown things. ε Fido { brown things }

Translation Singular sentences

What Singular Sentences Say Singular sentences are about set membership. Ascribing a property to an individual is saying that it’s a member of the set of individuals that have that property, e.g. “Descartes is a philosopher” says that Descartes is a member of {Plato, Aristotle, Locke, Berkely, Hume, Quine…} Saying that 2 or more individuals stand in a relation is saying that some ordered n-tuple is a member of a set so, e.g. “San Diego is north of Chula Vista” says that <San Diego,Chula Vista> is a member of {<LA,San Diego>, <Philadelphia,Baltimore>…}

Singular Sentences All men are mortal Socrates is a man _______________________ Therefore, Socrates is mortal We can now understand what 2 and 3 say: 2 says that Socrates ε {humans} 3 says that Socrates ε {mortals}

Ducati is brown. Bd

Tweety bopped Sylvester Bts

General sentences

General Sentences All men are mortal Socrates is a man _______________________ Therefore, Socrates is mortal But we still don’t have an account of what 1 says Because it doesn’t ascribe a property or relation to any individual or individuals.

Venn Diagrams Set theoretical concepts, like union and intersection can be represented visually by Venn Diagrams Venn Diagrams represent sets as circles and Represent the emptiness of a set by shading out the region represented by it and Represent the non-emptiness of a set by putting an “X” in it.

Emptiness Shading in a region says that the set it represents is empty. A This says that there are no As.

There are no unicorns. unicorns  (x) Ux

This says that at least one thing is an A. Non-emptiness ‘X’ in a region says that the set it the region represents is non-empty, i.e. there’s something there A This says that at least one thing is an A.

There are horses. X horses (x) Hx

Set Complement The complement of a set is everything in the universe that’s not in the set. A The shaded area is the complement of A—the set of things that are not A.

Regions of Logical Space Things that are neither A nor B Things that are B but not A Things that are A but not B Things that are A & B A B

This is the intersection of sets A and B Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. This is the intersection of sets A and B A B

Intersection & Conjunction This region represents the set of things that are both A and B. X A B

Set Union Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. This is the union of sets A and B A B

Union and Disjunction A B If we say something is in the union of A and B we’re saying that it’s either in A or in B. A B

Categorical Sentences All S are P No S are P (x)(Sx  Px) (x)  (Sx  Px)   Some S are P Some S are not P (x)(Sx  Px) (x)(Sx   Px)

No horses are carnivores (x)(  (Hx • Cx)

Some horses are brown X horses brown things (x)(Hx • Bx)

This says that something is in both A but not B. Set Complement This says that something is in both A but not B. X A B

Some horses are not brown X horses brown things (x)(Hx • Bx)

All horses are mammals horses horses mammals (x)(Hx  Mx)

All horses are mammals There are no non-mammalian horses horses ∩ non-mammals = { } (the empty set)

General Sentences All men are mortal Socrates is a man _______________________ Therefore, Socrates is mortal We can’t treat “all men” as the name of an individual as, e.g. the sum or collection of all humans Consider: “All men weigh under 1000 pounds.” We want to ascribe properties to every man individually.

The Socrates Argument is Valid! All men are mortal Socrates is a man _______________________ Therefore, Socrates is mortal 1 says that there are no men that don’t belong to the set of mortals 2 says that Socrates ε {men} 3 says that, therefore, Socrates ε {mortals}

All men are mortals men mortals immortal men: nothing here! men mortals There’s nothing in the set of men outside of the set of mortals; the set of immortal men is empty.

Socrates is a man. men mortals This is the only place in the men circle Socrates can go since we’ve already said that there are no immortal men.

Therefore, Socrates is a mortal. men mortals So Socrates automatically ends up in the mortals circle: the argument is, therefore, valid!

Arguments Involving Relations All dogs love their people Bo is Obama’s Dog ____________________ Bo loves Obama If we treat the predicates in these 3 sentences as one- place predicates, we can’t explain the validity of this argument! It looks like we’re ascribing 3 different properties that don’t have anything to do with one another: people-loving, being-Obama’s-dog and Obama-loving

Relations We understand relations as sets of ordered n-tuples, e.g. being to the north of is {<Los Angeles, San Diego>, <Philadelphia, Baltimore>, <San Diego, Chula Vista>…} being the quotient of one number divided by another: x into y is {<2,2,1>, <2,4,2>, <5,45,9>…}

Singular sentences about relations Standing in a relation is also understood in terms of set membership. It’s a matter of being a member of an ordered n-tuple which is a member of a set. Michelle is Barak’s wife. ε {<Eve, Adam>, <Hillary, Bill>…}

Validity of the Dog Argument We can consider being the dog of and loves as relations and designate them by the 2- place predicates “D”and “L” Now the 3 sentences have something in common--they don’t involve 3 different properties; they involve 2 relations: being the dog of and loves So we can link them to show validity All dogs love their people Bo is Obama’s Dog ____________________ Bo loves Obama

Bo is Obama’s dog: Dbo

For all x, y, if x is the dog of y then x loves y (x)(y)(Dxy  Lxy) For all x, y, if x is the dog of y then x loves y All dogs love their people

Validity of the Dog Argument All dogs love their people Bo is Obama’s Dog ____________________ Bo loves Obama (x)(y)(Dxy  Lxy) Dbo ______________ Lbo 1 says that for any x and y, if x is the dog of y then x loves y. 2 says that Bo is the dog of Obama. So it follows that Bo loves Obama!

All dogs love their people __ is the dog of __ __ loves __ (x)(y)(Dxy  Lxy)

Bo is Obama’s Dog <Bo,Obama> __ is the dog of __ __ loves __ Dbo

Bo loves Obama <Bo,Obama> __ is the dog of __ __ loves __ Lbo

Scope

There are dogs and there are cats. (x)Dx  (x)Cx : There exists an x such that x is a dog and there exists an x such that x is a cat

There are dog-cats (x)(Dx  Cx): There exists an x such that x is a dog-and-a-cat

Overlapping quantifiers

Everybody loves somebody (x)(y)Lxy: for all x, there exists a y such that x loves y

Everybody is loved by somebody (x)(y)Lyx: for all x, there exists a y such that y loves x

There’s somebody everybody loves (x)(y)Lyx: There exists an x such that for all y, y loves x

There’s somebody who loves everybody (x)(y)Lxy: there exists an x such that for all y, x loves y

The End