propositional logic 05-31-2016 - university of maryland · outline 1 syntax 2 semantics truth...

Propositional Logic

Jason Filippou

CMSC250 @ UMCP

05-31-2016

Jason Filippou (CMSC250 @ UMCP) Propositional Logic 05-31-2016 1 / 38

Outline

1 Syntax

2 SemanticsTruth TablesSimplifying expressions

3 InferenceValid reasoningBasic rules of inference


Propositional Logic: Overview

Propositional logic is the most basic kind of Logic we will examine,and arguably the most basic kind of Logic there is.

It uses symbols that evaluate to either True or False,combinations of those symbols (which we call compoundstatements), as well as a set of equivalences and inferencerules.

Its simplicity allows it to be implemented in computer hardware!


Propositional Logic: Overview

We will study Propositional (and “Predicate” logic) in three(unbalanced) steps:

Syntax.Semantics.Inference (or “Proof theory”).


Syntax

Syntax


Syntax

Syntax

Syntax in Propositional Logic is very easy to grasp.

Components:The (self-explanatory) constant symbols True and False.A pre-defined vocabulary of propositional symbols which we usuallydenote P. Those “map” to either True or False.

Often-used symbols: p, q, r . . .

The negation operator ∼, applied on propositional symbols in P.Examples: ∼p (“not” p), ∼∼p (“not not p”).

The binary operators of conjunction (∧) and disjunction (∨).Examples: p ∧ q, p ∨ ∼q, q ∧ q.

The left and right parentheses ((,)), used to group terms forprioritization of execution or readability.

Examples: (p), (((((. . . (p) . . . ))))), (p ∧ q) ∨ z, p ∧ (q ∨ z).

The binary connectives of implication (“if-then”) (⇒), bi-conditional(“if and only if”, commonly abbrv. iff)(⇔) and logical equivalence:≡.

Examples: p⇒ r, p⇔ (q ∧ ∼r), p ∧ p ≡ p, (p ∧ q) ∨ (p ∧ ∼q) ≡ p.


Syntax

Recap

Syntax for Propositional Logic consists of:{True, False,P,∼,∧,∨, (, ),⇒,⇔,≡}.So what do all of these symbols mean?


Semantics

Semantics


Semantics

Constants / Propositional Symbols

True and False should be self-explanatory, intuitive symbols.

Without agreement on what they mean, we can go no further.Think about them like the notions of a point and a line inEuclidean Geometry.


Semantics

Propositional Symbols and Interpretation

Think of a Propositional Symbol like a binary variable withdomain True, False.

Anything that can be either true or false in our world can bemodelled by such a symbol.

E.g the symbol rain is True if it’s raining today, False otherwise.

Probabilities?


Semantics Truth Tables

Truth Tables



Negation Operator

Beginning from the definitions of our truth assignments forconstants and propositional symbols, we can assign truth to everycompound statement we can build with our syntax.

Basic instrument for doing this: Truth Tables.

E.g negation operator truth table:

p ∼p

False True

True False



Conjunction / Disjunction

What would the truth table for conjunction and disjunction be?

p q p ∧ q p ∨ q

F F F F

F T F T

T F F T

T T T T



Binary connectives

Implication:

p q p⇒ q

F F T

F T T

T F F

T T T

Bi-conditional:

p q p⇔ q

F F T

F T F

T F F

T T T



Natural language examples

Let’s convert the following natural language statements topropositional logic:

1 It’s rainy and gloomy.

2 I will pass 250 if I study.

3 I will pass 250 only if I study.

4 THOU SHALT NOT PASS.

5 All work and no play makes Jack a dull boy.


Semantics Simplifying expressions

Simplifying expressions



Take 3

Do the truth tables for ∼(p ∧ q) and ∼p ∨ ∼q.

What do you observe?



De Morgan’s Laws

For every p, q ∈ P, we have:∼(p ∧ q) ≡ ∼p ∨ ∼q∼(p ∨ q) ≡ ∼p ∧ ∼q

Fundamental result first observed by Augustus De Morgan.



Other logical Equivalences

Convince yourselves about the following:∼p ∨ q ≡ p⇒ qp⇒ q ≡ ∼q ⇒ ∼p (contrapositive)



Tautologies / Contradictions

Tautology: A logical statement that is always True , regardlessof the truth values of the variables in it.

Common notation (also used in Epp): t.

E.g: p ∨ ∼p, p ∨ T

Contradiction: A logical statement that is always False ,regardless of the truth values of the variables in it.

Common notation (also used in Epp): c.

E.g: p ∧ ∼p, p ∧ F



Logical Equivalence cheat sheet

For (possibly compound) statements p, q, r, tautological statement tand contradicting statement c:

Commutativity p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ pAssociativity of binary op-erators

(p ∧ q) ∧ r ≡ p ∧ (q ∧ r) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)

Distributivity of binary op-erators

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

Identity laws p ∧ t ≡ p p ∨ c ≡ pNegation laws p ∨ ∼p ≡ t p ∧ ∼p ≡ cDouble negation ∼(∼p) ≡ pIdempotence p ∧ p ≡ p p ∨ p ≡ pDe Morgan’s axioms ∼(p ∧ q) ≡ ∼p ∨ ∼q ∼(p ∨ q) ≡ ∼p ∧ ∼qUniversal bound laws p ∨ t ≡ t p ∧ c ≡ cAbsorption laws p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ pNegations of contradictions/ tautologies

∼c ≡ t ∼t ≡ c

Those will be posted on our website as a reference.



Practice

Using the equivalences we just established, simplify the followingexpressions:

p ∧ (∼p ∨ q) ∨ (∼(∼(z ∨ ∼q)))(p ∧ r) ∨ ((p ∨ s) ∧ (p ∨ a))


Inference

Inference


Inference Valid reasoning

Valid reasoning



The role of inference

We’ve looked at syntax, or the vocabulary of propositional logic.

Semantics helped us combine the members of the vocabulary intosentences (compound statements) and the notion of equivalencehelped us find equivalent statements, as well as simplifyunnecessarily long sentences.

We haven’t talked about constructing new knowledge!

That’s where inference, (or proof theory in the context of logic)comes to play.



Valid reasoning

All reasoning has to be valid.

Intuitively: the knowledge we infer has to obey the constraints ofthe world defined by the stuff we already know.

Formal definition later.

Examples:

All men are mortal. Socrates is a man. Therefore, Socrates ismortal.All men are mortal. Socrates is mortal. Therefore, Socrates is aman.All men are mortal. Socrates is not mortal. Therefore, Socrates isnot a man.



Complete reasoning

The notion of “complete” reasoning is one that we won’t examinemuch, if at all, in 250.

Intuitively, if we have a rule (or a set of rules) that can produce allof the knowledge that logically follows from the stuff that wealready know, we have a complete reasoning system.



Premises and conclusions

All reasoning systems consist of rules.

All rules consist of premises and conclusions.

We will write rules in the following manner:

Premise 1

Premise 2

. . .

P remise n

∴ Conclusion

Some authors prefer the form:

Premise 1, Premise 2, . . . , Premise n

Conclusion



Definition of validity

Split rule to premises and conclusions

Critical rows: The rows of a truth table where all premises areTrue .

The rule is valid if the conclusion is also True for all critical rows.



Definition of validity

Valid rule

Premise 1

Premise 2

Premise n

Figure 1: A pictorial representation of valid reasoning.


Inference Basic rules of inference

Basic rules of inference



Modus Ponens

The cornerstone of deductive reasoning.

Modus Ponensp

p⇒ q∴ q

Theorem (Validity of Modus Ponens)

Modus Ponens is a valid rule of reasoning.

Proof.

p q p⇒ q

F F TF T TT F FT T T



Modus Tollens

Modus Tollensp⇒ q∼q

∴ ∼p

Proof?



Other valid rules of inference

The following are mentioned on Epp (but there exist many more).

Disjunctive addition

p∴ p ∨ q

Conjunctive simplification

p ∧ qp, q

Disjunctive syllogism

p ∨ q∼q∴ p

Hypothetical syllogism

p⇒ qq ⇒ r∴ p⇒ r

Prove their validity as an exercise!



Valid inference rules cheat sheet

Modus Ponens Modus Tol-lens

Disjunctiveaddition

Conjunctiveaddition

p

p⇒ q

∴ q

∼q

p⇒ q

∴ ∼p

p

∴ p ∨ q

p, q

∴ p ∧ q

ConjunctiveSimplification

Disjunctivesyllogism

HypotheticalSyllogism

p ∧ q

∴ p, q

p ∨ q

∼p

∴ q

p⇒ q

q ⇒ r

∴ p⇒ r

Note that disjunctive syllogism is symmetric, i.e if ∼q is the premise, p is theconclusion.



Take 5

Are the following inference rules valid?

Rule 1 Rule 2 Rule 3

p ∨ q

p⇒ r

q ⇒ r

∴ r

p⇒ q

q

∴ p

p⇒ q

∼p

∴ ∼q

YES: Division Into Cases NO: Converse Error NO: Inverse Error



Difference with language

Background knowledge oftentimes blurs the distinctionbetween valid and invalid arguments.

Consider the following arguments:

If my pet ostrich could do 100meters in under 10 seconds, it

could participate in theOlympics.

If these tracks are Bigfoot’s,Bigfoot exists.

My pet ostrich can do 100 me-ters in under 10 seconds.

Bigfoot exists.

∴ My pet ostrich can partici-pate in the Olympics.

∴ These tracks are Bigfoot’s.



Proof by contradiction

A very popular proof methodology, which we will be using a lot, isproof by contradiction.

Intuitively, we want to prove something, so we assume that itdoesn’t hold (i.e its converse holds), and we arrive at acontradiction.

Formally, the following rule is sound:

Proof by contradiction∼p⇒ c∴ p

Very important to convince yourselves that the rule is sound!


propositional logic 05-31-2016 - university of maryland · outline 1 syntax 2 semantics truth...

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