Logic - Uni Salzburg

Logic

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In the beginning

Aristotle +/- 350 B.C.

Organon

19 syllogisms

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In the beginning

Aristotle +/- 350 B.C.

Organon

19 syllogisms

Logic = study of correct reasoning

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Barbara syllogism

All K’s are L’sAll L’s are M’s

———————All K’s are M’s

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Barbara syllogismonly later called so,in the Middle Ages



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from the two premises

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one can always conclude the

conclusion

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conclusion

independent of what the parameters K,L,M are

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conclusion

independent of what the parameters K,L,M are

Logic (Logos, Greek for word, understanding, reason) deals with general reasoning laws in the form of formulas with parameters.

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Propositions

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Propositions

Def. A proposition (Aussage) is a grammatically correct sentence that is either true or false.

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Propositions


logic deals with patterns! what matters are not particular

propositions but the way in which (abstract) propositions

are combined and what follows from them

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Propositions


logic deals with patterns! what matters are not particular

propositions but the way in which (abstract) propositions

are combined and what follows from them

Connectives

∧ for “and”∨ for “or”¬ for “not”⇒ for “if .. then” or “implies”

⇔ for “if and only if”

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Abstract propositions

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Definition

Basis Propositional variables are abstract propositions.

Step (Case 1) If P is an abstract proposition, then so is (¬P).

Step (Case 2) If P and Q are abstract propositions, then so are (P ∧ Q), (P ∨ Q), (P ⇒ Q), and (P ⇔ Q).

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a recursive/inductivedefinition

Definition

Basis Propositional variables are abstract propositions.

Step (Case 1) If P is an abstract proposition, then so is (¬P).

Step (Case 2) If P and Q are abstract propositions, then so are (P ∧ Q), (P ∨ Q), (P ⇒ Q), and (P ⇔ Q).

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…and their structure

⇒

∧¬

a b c

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the tree of((a ∧ b) ⇒ (¬c))

⇒

∧¬

a b c

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⇒

∧¬

a b c

building

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⇒

∧¬

a b c

building

decomposing

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tree representation(no need of parenthesis)


⇒

∧¬

a b c

building

decomposing

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Dropping parenthesis

¬∧ ∨⇒⇔

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priority schema(top binds the most)

¬∧ ∨⇒⇔

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increasing priority

¬∧ ∨⇒⇔

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increasing priority

decreasing priority¬∧ ∨⇒⇔

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Example: ((a ∧ b) ⇒ (¬c))

becomes a ∧ b ⇒ ¬cpriority schema

(top binds the most)

increasing priority

decreasing priority¬∧ ∨⇒⇔

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Truth tables

Conjunction

P Q P∧Q

0 0 0

0 1 0

1 0 0

1 1 1

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Truth tables

Conjunction

P Q P∧Q

0 0 0

0 1 0

1 0 0

1 1 1

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Truth tables

Conjunction

P Q P∧Q

0 0 0

0 1 0

1 0 0

1 1 1 only true when both P and Q are true

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Truth tables

Disjunction

P Q P∨Q

0 0 0

0 1 1

1 0 1

1 1 1

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Truth tables

Disjunction

P Q P∨Q

0 0 0

0 1 1

1 0 1

1 1 1

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Truth tables

Disjunction

P Q P∨Q

0 0 0

0 1 1

1 0 1

1 1 1

true when either P or Q or both are

true

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Truth tables

Negation

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Truth tables

Negationunary connective

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Truth tables

Negation

P ¬P

0 1

1 0

unary connective

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Truth tables

Negation

P ¬P

0 1

1 0

unary connective

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Truth tables

Negation

P ¬P

0 1

1 0

true when P is false

unary connective

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Truth tables

Implication

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Truth tables

Implicationneeds more attention

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Truth tables

Implication

P Q P ⇒ Q

0 0 1

0 1 1

1 0 0

1 1 1

needs more attention

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Truth tables

Implication

P Q P ⇒ Q

0 0 1

0 1 1

1 0 0

1 1 1


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Truth tables

Implication

P Q P ⇒ Q

0 0 1

0 1 1

1 0 0

1 1 1

only false when P is true and Q is false


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Truth tables

Bi-implication

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Truth tables

Bi-implication

P⇔Q

is (P ⇒ Q)∧(Q ⇒ P)

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Truth tables

Bi-implication

P Q P ⇒ Q Q ⇒ P P ⇔ Q

0 0 1 1 1

0 1 1 0 0

1 0 0 1 0

1 1 1 1 1

P⇔Q

is (P ⇒ Q)∧(Q ⇒ P)

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Truth tables

Bi-implication

P Q P ⇒ Q Q ⇒ P P ⇔ Q

0 0 1 1 1

0 1 1 0 0

1 0 0 1 0

1 1 1 1 1

P⇔Q

is (P ⇒ Q)∧(Q ⇒ P)

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Truth tables

Bi-implication

P Q P ⇒ Q Q ⇒ P P ⇔ Q

0 0 1 1 1

0 1 1 0 0

1 0 0 1 0

1 1 1 1 1

P⇔Q

is (P ⇒ Q)∧(Q ⇒ P)

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Truth tables

Bi-implication

P Q P ⇒ Q Q ⇒ P P ⇔ Q

0 0 1 1 1

0 1 1 0 0

1 0 0 1 0

1 1 1 1 1

P⇔Q

is (P ⇒ Q)∧(Q ⇒ P)

true when P and Q have the same truth

value

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Truth-functionsDef. A truth-function or Boolean function is a function f: {0,1}n ⟶ {0,1}

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Property: Every abstract proposition P(a1,..,an) induces a truth-function.

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by its inductive structure, using the

truth tables

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truth tables

{Notation in the book… a, b(0,0) ⟼ 0

(0,1) ⟼ 1

(1,0) ⟼ 0

(1,1) ⟼ 1 13




truth tables

P(a,b): (a ∧ b) ∨ b{Notation in the book… a, b(0,0) ⟼ 0

(0,1) ⟼ 1

(1,0) ⟼ 0

(1,1) ⟼ 1 13




truth tables

a1, .. an are the variables in P (and more) ordered in a sequence

P(a,b): (a ∧ b) ∨ b{Notation in the book… a, b(0,0) ⟼ 0

(0,1) ⟼ 1

(1,0) ⟼ 0

(1,1) ⟼ 1 13

Truth-functions

Property: Every abstract proposition P(a1,..,an) with ordered and specified variables induces a truth-function.

a1, .. an are the variables in P (and more) ordered in a sequence

P(a,b,c):

induces

(a ∧ b) ∨ b

The sequence of specified variables matters!

Note: a, b, c(0,0,0) ⟼ 0

(0,0,1) ⟼ 0

(0,1,0) ⟼ 1

(0,1,1) ⟼ 1

(1,0,0) ⟼ 0

(1,0,1) ⟼ 0

(1,1,0) ⟼ 1

(1,1,1) ⟼ 1

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Equivalence of propositions

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Definition: Two abstract propositions P and Q are equivalent, notation P = Q, iff they induce the same truth-functionval

on any sequence containing their common variables

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Property: The relation = is an equivalence on the set of all abstract propositions

val



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Property: The relation = is an equivalence on the set of all abstract propositions

val

i.e., for all abstract propositions P, Q, R, (1) P = P; (2) if P = Q, then Q = P; and

(3) if P = Q and Q = R, then P = Rval val val

valvalval



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Exampleb b c cAre the following equivalent? and

0 0

0 1

1 0

1 1

b b c ccb b c

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0 0 1

0 1 1

1 0 0

1 1 0

b b c ccb b c

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0 0 1 1

0 1 1 0

1 0 0 1

1 1 0 0

b b c ccb b c

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0 0 1 1 0

0 1 1 0 0

1 0 0 1 0

1 1 0 0 0

b b c ccb b c

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0 0 1 1 0 0

0 1 1 0 0 0

1 0 0 1 0 0

1 1 0 0 0 0

b b c ccb b c

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0 0 1 1 0 0

0 1 1 0 0 0

1 0 0 1 0 0

1 1 0 0 0 0

b b c ccb b c

Their truth values are the same, so they are equivalentb b

valc c

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Tautologies and contradictions

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Def. An abstract proposition P is a tautology iff its truth-function is constant 1.

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Def. An abstract proposition P is a contradiction iff its truth-function is constant 0.

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Def. An abstract proposition P is a contingency iff it is neither a tautology nor a contradiction.22




Def. An abstract proposition P is a contingency iff it is neither a tautology nor a contradiction.

all tautologies areequivalent

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all contradictions are equivalent

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all contradictions are equivalentbut not all

contingencies!

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a recursive/inductivedefinition

Definition

Basis (Case 1) T and F are abstract propositions.Basis (Case 2) Propositional variables are abstract propositions.

Step (Case 1) If P is an abstract proposition, then so is (¬P).Step (Case 2) If P and Q are abstract propositions, then so are (P ∧ Q), (P ∨ Q), (P ⇒ Q), and (P ⇔ Q).

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Propositional LogicStandard Equivalences

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Commutativity and Associativity

P Qval

Q P

P Qval

Q P

P Qval

Q P

Commutativity

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P Qval

Q P

P Qval

Q P

P Qval

Q P

Commutativity

P Qval

Q P

P Q P Q Q P0 1 1 0

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P Qval

Q P

P Qval

Q P

P Qval

Q P

Commutativity

P Q Rval

P Q R

P Q Rval

P Q R

P Q Rval

P Q R

Associativity

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P Qval

Q P

P Qval

Q P

P Qval

Q P

Commutativity

P Q Rval

P Q R

P Q Rval

P Q R

P Q Rval

P Q R

Associativity

P Q Rval

P Q R

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P Qval

Q P

P Qval

Q P

P Qval

Q P

Commutativity

P Q Rval

P Q R

P Q Rval

P Q R

P Q Rval

P Q R

Associativity

P Q Rval

P Q R

P Q R P Q R P Q R

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P Qval

Q P

P Qval

Q P

P Qval

Q P

Commutativity

P Q Rval

P Q R

P Q Rval

P Q R

P Q Rval

P Q R

Associativity

P Q Rval

P Q R

P Q R P Q R P Q R0 1 0

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P Qval

Q P

P Qval

Q P

P Qval

Q P

Commutativity

P Q Rval

P Q R

P Q Rval

P Q R

P Q Rval

P Q R

Associativity

P Q Rval

P Q R

P Q R P Q R P Q R0 1 0 0 1

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Idempotence and Double Negation

Idempotence

P Pval

P

P Pval

P

P Pval

P

P Pval

P

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Idempotence and Double Negation

Idempotence

P Pval

P

P Pval

P

P Pval

P

P Pval

P

Double negation

Pval

P

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T and F

Inversion

Tval

F

Fval

T

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T and F

Inversion

Tval

F

Fval

T

Negation

Pval

P F

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T and F

Inversion

Tval

F

Fval

T

Negation

Pval

P F

Contradiction

P Pval

F

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T and F

Inversion

Tval

F

Fval

T

Negation

Pval

P F

Contradiction

P Pval

F

Excluded Middle

P Pval

T37

T and F

Inversion

Tval

F

Fval

T

Negation

Pval

P F

Contradiction

P Pval

F

Excluded Middle

P Pval

T

T/F - elimination

P Tval

P Fval

P Tval

P Fval

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T and F

Inversion

Tval

F

Fval

T

Negation

Pval

P F

Contradiction

P Pval

F

Excluded Middle

P Pval

T

T/F - elimination

P Tval

P

P Fval

F

P Tval

T

P Fval

P

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Distributivity, De Morgan

Distributivity

P Q Rval

P Q P R

P Q Rval

P Q P R

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Distributivity, De Morgan

Distributivity

P Q Rval

P Q P R

P Q Rval

P Q P R

De Morgan

P Qval

P Q

P Qval

P Q

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Implication and Contraposition

Implication

P Qval

P Q

P Qval

P Q

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Implication

P Qval

P Q

P Qval

P Q

Contraposition

P Qval

Q P

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Implication

P Qval

P Q

P Qval

P Q

Contraposition

P Qval

Q P P Qval

P Q

common mistake!

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Bi-implication and Self-equivalence

Bi-implication

P Qval

P Q Q P

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Bi-implication

Self-equivalence

P Qval

P Q Q P

P Pval

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Bi-implication

Self-equivalence

P Qval

P Q Q P

P Pval

T

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Calculating with equivalent propositions

(the use of standard equivalences)

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Recall…


Property: The relation = is an equivalence on the set of all abstract propositions.

val


i.e., for all abstract propositions P, Q, R, (1) P = P; (2) if P = Q, then Q = P; and

(3) if P = Q and Q = R, then P = Rval val val

valvalval

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Substitution

Simple

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Substitution

Simple

EVERY occurrence of P is substituted!

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Substitution

Simple Sequential

EVERY occurrence of P is substituted!

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Substitution

Simple Sequential

Simultaneous EVERY occurrence of P is substituted!

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Substitution

Simple Sequential


meta rule

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The rule of Leibnitz

Leibnitz

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Leibnitz

single occurrence is

replaced!

51


Leibnitz


replaced!formula that has

as a sub formula

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Leibnitz


replaced!

meta rule

formula that has as a sub formula

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Strengthening and

weakening

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We hadDefinition: Two abstract propositions P and Q are equivalent, notation P = Q, iff (1) Always when P has truth value 1, also Q has truth value 1, and (2) Always when Q has truth value 1, also P has truth value 1.

val

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val

if we relax this, we get

strengthening

53


val

if we relax this, we get

strengthening

in an equivalentformulation

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Strengthening

Definition: The abstract proposition P is stronger than Q, notation P = Q, iff (1) Always when P has truth value 1, also Q has truth value 1, and (2) Always when Q has truth value 1, also P has truth value 1.

val

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Strengthening

Definition: The abstract proposition P is stronger than Q, notation P = Q, iff (1) Always when P has truth value 1, also Q has truth value 1, and (2) Always when Q has truth value 1, also P has truth value 1.

val

Q is weaker than P

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Strengthening

Definition: The abstract proposition P is stronger than Q, notation P = Q, iff always when P has truth value 1, also Q has truth value 1.

val

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Strengthening


val

always when P is true, Q is also true

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Strengthening


val

Q is weaker than P

always when P is true, Q is also true

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Properties

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Properties

Lemma E1: iff is a tautology.

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Properties


Lemma EW1: iff and .

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Properties

Lemma W2: Pval

|ù P



56

Properties

Lemma W2: Pval

|ù P

Lemma W3: If and then Pval

|ù Q Qval

|ù R Pval

|ù R



56

Properties

Lemma W2: Pval

|ù P

Lemma W3: If and then Pval

|ù Q Qval

|ù R Pval

|ù R



Lemma W4: iff is a tautology.

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Standard Weakenings

and-or-weakening

Extremes

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Calculating with weakenings(the use of standard weakenings)

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Substitution

Sequential

Simultaneous

Simple

�r⇠{P sval

|ù r⇠{P s

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Substitution

Sequential

Simultaneous

just holds

Simple

�r⇠{P sval

|ù r⇠{P s

59

Substitution

Sequential


just holds

Simple

�r⇠{P sval

|ù r⇠{P s

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Leibnitz

formula that has as a sub formula

does not hold for weakening!

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Leibnitzdoes not hold for weakening!

Monotonicity


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Logic - Uni Salzburg