Predicate Logic Rules

Predicate Logic: instantiation and generalization rules

All of the rules of implication and the rules of replacement we learned in propositional logic are applicable in predicate logic.  However, we cannot apply them to any line without first getting rid of the quantifiers.  So, we need some additional rules, implicational rules, that allow us to get rid of the quantifiers.  We also need rules that allow us to add quantifiers.  So, the following four rules are new implication rules, and so must be applied to whole lines not parts of lines.

Instantiation Rules
UI
(x)Fx
Fy

Use this version of the rule if you anticipate using UG later in the proof (the ‘y’ in ‘Fy’ is a variable and not a constant)

(x)Fx
Fa

Use this version of the rule if there is an individual with F (the ‘a’ in ‘Fa’ is a constant and not a variable—it names something specific in the universe of discourse)

EI
(∃x)Fx
Fa

(Beware: the existential name must be new and not occur in the conclusion)

Generalization Rules
UG
Fy
(x)Fx                           

EG
Fa
(∃x)Fx

Fy
(∃x)Fx

  
Examples:
All humans are mortal.  Socrates is a human.  Hence, Socrates is mortal. 

In symbols (the à means the same thing as the horseshoe):
         (x)(Hx à Mx)
         Hs.                   /Ms

We cannot use MP at this stage to get the conclusion.  We need Hs à Ms first and then we can use MP.  Premise 1 is not a conditional statement but a universal.  But we are permitted to remove the universal quantifier because premise 1 tells us that for every x, if x is a human, then x is mortal.  So we can infer:

3. Hs à Ms.    1, UI
4. Ms.              2, 3, MP

EG application
1. (x)Rx
2. Ry                1, UI
3. (∃x)Rx          2, EG

Everything has property R.  So, something has property R
NB: predicate logic assumes that there is at least one thing that exists.  There are logics that deny this assumption—e.g. free logics.

Errors: UI and EI must be applied to entire lines, never parts of a line. 
UI can only be applied to a line that is a universal statement.  So, the following is incorrect

         ~(x)Gx
         ~Gs

This would permit the following invalid argument:

Not everything is Greek.  Hence, Socrates is not Greek. 
The premise can be true while the conclusion is false.  The problem is that the first premise is not a universal statement. It is an existential statement that says that something exists without having G (∃x)~Gx


Here is another example that presents two mistakes
Examples:
(x)Ex à (y)Dy
:. Es à (y)Dy   incorrect use of UI

If everything is an even number, then everything is divisible by two.  So, if 6 is an even number, then everything is divisible by 2. 
The premise is true but the conclusion is false.
The first mistake is that the first premise is not a universal statement. 
The second mistake is that UI is applied to only part of the line.

Another error to avoid is illustrated with the following:
(x)(Ex à Dx)
Es à Dn

Let the first premise be ‘all even numbers are divisible by 2.’ The conclusion says that ‘if six is even then nine is divisible by 2.’ 
The problem here is that we failed to replace the occurrences of x in Ex à Dx with the same constant. 

Errors to avoid with EI
Never existentially instantiate the same name (constant) for multiple lines

1. (∃x)(Sx)
2. (∃x)(Rx)     /(∃x)(Sx & Rx)
3. Sa                            1. EI
4. Ra                            2. EI, incorrect
5. Sa & Ra                   3. 3,4, conj
6. (∃x)(Sx & Rx)         5. EG

Let 1 say that something is a square; Let 2 say that something is round; 3 says that a is a square; 4 says that a is round; 5 says that a is a square and round. YIKES!!!!


Errors to avoid with UG:
Never okay to use UG on the basis of a statement about a single individual

1. Os
2. :. (x)Ox

Socrates is old.  So, everything is old.

Errors to avoid with EG:
Never okay to apply EG to a part of a line

Rules of thumb:
1. apply EI before applying UI
2. Use UI to instantiate free variables or individual constants and apply the rules of propositional logic.



Comments

Post a Comment

Popular posts from this blog

Relations and Multiple Quantifiers in Predicate Logic

Thus far we have been using only one-place predicates. These are predicates that only require one name (put differently, these are properties that only require one object) to complete them. For example, _____ is red = Rx _____ is heavy = Hx _____ is a wife = Wx ( ∃ x)Rx is true if and only if there is at least one thing in the universe that is red. ( ∃ x)Hx is true iff there is at least one thing in the universe that is heavy, and so on. (x)Rx is true iff everything in the universe is red, and so on.  But, as we all know, there are predicates that are more than one place. In other words, there are predicates that require more than one name to complete them (put differently, there are properties-- better-called relations--that require more than one object). For example ____ is heavier than ____ = Hxy ____ is taller than ____ =  Txy ____ loves ____ = Lxy ____ is between ____, _____ = Bxyz (∃ x) (∃y )Hxy is true iff there is something ...
Read more

Predicates in search of nouns

According to standard categorical logic, statements of the form All S are P must have some sort of noun-phrase in the predicate.  Thus ‘All dogs are mammals’ is fine, but ‘All dogs are four-legged’ is not.  Why is this a requirement? Standard categorical logic is based on Aristotelian logic and inherits some of Aristotle’s assumptions. So one reason for the noun-requirement is that given Aristotelian metaphysics predicates without nouns are incomplete. To understand why, we first need to keep in mind that  predicates (parts of language) correspond to  properties (parts of the extra-linguistic—that is, non-linguistic—world).  Predicates without nouns are incomplete because, according to Aristotle,  properties  must inhere in objects in order to exist.  There are no free-floating properties—properties without some subject that they attach to.  Thus, statements like ‘All dogs are four-legged’ while grammatic...
Read more

Meta-Logical Themes: Soundness of Propositional Logic

We have been studying propositional logic (PL) for some time now. PL is about whole or complete propositions and their relationships to each other in terms of truth-preservation. That is, PL investigates the relationships between sets of true propositions to other propositions. How can we be sure that if we are given a set of true propositions (called the premises) that we will through a series of steps wind up with another true proposition and not a false one (i.e. how can we be sure that the relationship between the premise set and the conclusion is, in fact, a valid one). Put differently, we want to make sure that the logic we are using will never take us from truths to falsity. We have been using our rules to show that if we start with a set of (assumed to be) true propositions we can derive another proposition that must be true if the premises are. But we have only studied a finite number of arguments. There is an infinite number of arguments. So, how can we be sure that there i...
Read more