1. 1st-order Predicate Logic (FOL)
Now a “real logic”; Think and concentrate !
First-Order Predicate Logic

2. Simple arguments, where propositional logic does not suffice
All monkeys like bananas.
Judy is a monkey.
 Judy likes bananas.
From the viewpoint of Propositional Logic (PL) the above are simple (atomic) sentences:
p, q, r, and p, q does not entail r.
All students are clever
Charles is not clever
 Charles is not a student
What are the valid schemata of these arguments?
First-Order Predicate Logic

3. Logical form (scheme) of an argument
The schemata of the arguments above remind valid schemata of PL:
p  q, p |= q (modus ponens) or
p  q, q |= p (modus tollens)
But, in PL we cannot refine the analyses of simple sentences. Let us reformulate them:
Every individual, if it is a Monkey, then it likes Bananas
Judy is an individual with the property of being a Monkey
 Judy is an individual that likes Bananas
x [M(x)  B(x)], M(J) |= B(J), where x is an individual variable, M, B are predicate symbols, J is a functional symbol
It is again a schema: For M, B, J we can substitute other properties and individual, respectively. For instance, man for M, mortal for B and Charles for J.
M, B, J are here only symbols which stand for properties and individuals
First-Order Predicate Logic

4. Formal language of FOL (first-order predicate logic) Alphabet
Logical symbols
individual variables: x, y, z, ...
Symbols for truth-connectives: , , , , 
Symbols for quantifiers: , 
Special symbols
Predicate: Pn, Qn, ... n – arity = number of arguments
Functional: fn, gn, hn, „ --
Auxiliary symbols: (, ), [, ], {, }, ...
First-Order Predicate Logic

5. Formal language of FOL Grammar
terms:
each symbol for a variable x, y, ... is a term
if t1,…,tn (n  0) are terms and if f is an n-ary functional symbol, then the expression f(t1,…,tn) is a term; If n = 0, then we talk about individual constant (denoted a, b, c, …)
only expressions due to i. and ii. are terms
First-Order Predicate Logic

6. Formal language of FOL Grammar
atomic formulas:
If P is an n-ary predicate symbol and if t1,…,tn are terms, then P(t1,…,tn) is an atomic formula
(composed) formulas:
each atomic formula is a formula
if A is a formula, then A is a formula
if A and B are formulas, then (A  B), (A  B), (A  B), (A  B) are formulas
if x is a variable and A a formula, then x A and x A are formulas
First-Order Predicate Logic

7. Formal language of FOL 1st-order
We can quantify only over individual variables
We cannot quantify over properties or functions
Example: Leibniz’s definition of identity:
If two individuals have all the properties identical, then it is one and the same individual
P [ P(x) = P(y)]  (x = y) here we need a 2nd-order language, because we quantify over properties
First-Order Predicate Logic

8. Example: the language of arithmetic
We need special functional symbols:
0-ary symbol: 0 (the constant zero) –
constant is a 0-ary functional symbol
unary symbol: s (the successor function)
binary symbols: + and  (functions of adding and multiplying)
Examples of terms (using infix notation for + and ):
0, s(x), s(s(x)), (x + y)  s(s(0)), etc.
Formulas are, e.g.: (= is here a special predicate symbol):
s(0) = (0  x) + s(0), x (y = x  z), x [(x = y)  y (x = s(y))]
First-Order Predicate Logic

9. Transforming natural language into the language of FOL
“all”, “every”, “none”, “nobody”, “any”, ...  
“somebody”, “something”, “some”, “there is”, ...  
A sentence often needs to be reformulated (in an equivalent way)
No student is retired (For any student it holds that he is not retired):
x [S(x)  R(x)]
But: Not all students are retired (It is not true that any student is retired):
x [S(x)  R(x)]  x [S(x)  R(x)]
First-Order Predicate Logic

10. Transforming natural language into the language of FOL
An auxiliary rule:  + ,  +  (almost always)
x [P(x)  Q(x)]  x [P(x)  Q(x)]
It is not true that all P’s are Q’s  Some P’s are not Q’s
x [P(x)  Q(x)]  x [P(x)  Q(x)]
It is not true that some P’s are Q’s  No P is a Q
de Morgan laws in FOL
First-Order Predicate Logic

11. Transforming natural language into the language of FOL
The lift is used only by employees
x [L(x)  E(x)]
All employees use the lift
x [E(x)  L(x)]
Mary likes only the winners:
Hence, for all individuals it holds that if Mary likes him then he must be a winner:
x [L(m, x)  W(x)],
“to like” is a binary relation, not a property !!!
First-Order Predicate Logic

12. Transforming natural language into the language of FOL
Everybody loves somebody sometimes
x y t L(x, y, t)
Everybody loves somebody sometimes but Hitler doesn’t like anybody
x y t L(x, y, t)  z L’(h, z)
Everybody loves nobody – ambiguous
Nobody loves anybody – ambiguous;
Everybody dislikes anybody: x y L’(x, y)  x y L’(x, y)
First-Order Predicate Logic

13. First-Order Predicate Logic
Free, bound variables
x y P(x, y, t)  x Q(y, x)
bound, free free, bound
Formula with clear variables: each variable has only free occurrences, or only bound occurrences; each quantifier “has its own variables”.
For instance, the above formula does not have clear variables: x in the second conjunct is another variable than the x in the first conjunct, similarly for y. Clear formula:
x y P(x, y, t)  z Q(u, z)
First-Order Predicate Logic

14. Substitution of terms for variables
Ax/t  arises from A by a correct (i.e., collisionless) substitution of a term t for the variable x.
There are two rules for a correct substitution:
We can substitute a term t only for free occurrences of a variable x in a formula A, and we have to substitute for all the free occurrences.
No individual variable that occurrs in the term t can become bound in A (in such a case the term t is not substitutable for x in the formula A).
First-Order Predicate Logic

15. First-Order Predicate Logic
Substitution, example
A(x): P(x)  y Q(x, y), term t = f(y)
After executing the substitution A(x/f(y)), we obtain:
P(f(y))  y Q(f(y), y).
The term f(y) is not substitutable for x in A
We’d change the sense of the formula
First-Order Predicate Logic

16. First-Order Predicate Logic
Semantics of FOL !!!
P(x)  y Q(x, y) – is this formula true?
A non-reasonable question;
For, we do not know what the symbols P, Q mean, what they stand for. They are only symbols which can stand for any predicate (property).
P(x)  P(x) – is this formula true?
YES, it is; and it is always so, in all the circumstances. It is necessarily true.
First-Order Predicate Logic

17. First-Order Predicate Logic
Semantics of FOL !!!
x P(x, f(x)) we have to specify first,
x P(x , f(x)) how to understand these formulas:
What do they talk about; we have to choose the universe of discourse: any non-empty set U  
What does the symbol P denote; it is binary, with two arguments; it has to denote a binary relation R  U  U
What does the symbol f denote; it is an unary, one-argument symbol; it has to denote a function F  U  U, denoted F: U  U
First-Order Predicate Logic

18. First-Order Predicate Logic
Semantics of FOL !!!
A: x P(x, f(x)) we have to specify
B: x P(x , f(x)) how to understand these formulas:
Let U = N (the set of natural numbers)
let P denote the relation < (i.e., the set of pairs, where the first element is strictly less than the second one: {0,1, 0,2, …,1,2, …})
Let f denote the function second power x2, i.e., the set of pairs where the second element is the power of the first one: {0,0, 1,1, 2,4, …,5,25, …}
Now we can evaluate the truth values of the formulas A, B
First-Order Predicate Logic

19. First-Order Predicate Logic
Semantics of FOL !!!
A: x P(x, f(x))
B: x P(x , f(x)) We evaluate “from the inside”:
First evaluate the term f(x). Each term denotes an element of the universe. Which one? It depends on the valuation e of the variable x. Let e(x) = 0, then f(x) = x2 = 0.
Let e(x) = 1, then f(x) = x2 = 1, Let e(x) = 2, then f(x) = x2 = 4, etc.
Now by evaluating P(x , f(x)) we have to obtain a truth value: e(x) = 0, 0 is not < 0 False e(x) = 1, 1 is not < 1 False,
e(x) = 2, 2 is < True.
First-Order Predicate Logic

20. First-Order Predicate Logic
Semantics of FOL !!!
A: x P(x, f(x))
B: x P(x , f(x))
The formula P(x , f(x)) is in the given interpretation True for some valuations of the variable x, and False for other valuations.
The meaning of x (x): the formula is true for all (some) valuations of x
Formula A: False in our interpretation I: |I A
Formula B: True in the interpretation I: |=I B
First-Order Predicate Logic

21. Model of a formula, interpretation
A: x P(x, f(x))
B: x P(x , f(x))
We have found an interpretation I in which the formula B is true. The Interpretation structure N, <, x2 satisfies the formula B; it is a model of the formula B.
How to adjust the interpretation in order it were a model of the formula A? There are infinitely many possibilities, infinitely many models.
For instance: N, <, x+1, {N/{0,1}, <, x2, N, , x2, …
All the models of the formula A are also models of the formula B (“what holds for all, it holds also for some”)
First-Order Predicate Logic

22. Model of a formula, interpretation
C: x P(x, f(y)) what are the models of this formula (with a free variable y)?
Let us again
choose a Universe U = N
to the symbol P assign a relation: 
to the symbol f assign a function: x2
Is the structure IS = N, , power a model of the formula C? In order it were so, the formula C would have to be true in IS for all the valuations of the variable y. Hence the formula P(x, f(y)) would have to be true for all valuations of x and y.
But it is not so, for instance, if e(x) = 5, e(y) = 2, then 5 is not  22
First-Order Predicate Logic

23. Model of a formula, interpretation
C: x P(x, f(y)) what are the models of this formula (with a free variable y)?
The structure N, , x2 is not a model of formula C.
A (trivial) model is, e.g., N, N  N, x2. The whole Cartesian product N  N, i.e. the set of all the pairs of natural numbers, is also a relation over N.
Or, the structure N, , F, where F is the function, mapping N  N, such that F associates all the natural numbers with the number 0.
First-Order Predicate Logic