[PPT] - Propositional Logic Jason Filippou CMSC250 @ UMCP 05-31-2016 PowerPoint Presentation

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Propositional Logic

Jason Filippou

CMSC250 @ UMCP

05-31-2016

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Outline

1 Syntax 2 Semantics

Truth Tables Simplifying expressions

3 Inference

Valid reasoning Basic rules of inference

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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 compound statements), as well as a set of equivalences and inference rules. Its simplicity allows it to be implemented in computer hardware!

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Propositional Logic: Overview

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

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

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Syntax

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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 usually denote 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 for prioritization 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.

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Syntax

Recap

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

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Semantics

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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 in Euclidean Geometry.

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Semantics

Propositional Symbols and Interpretation

Think of a Propositional Symbol like a binary variable with domain True, False. Anything that can be either true or false in our world can be modelled by such a symbol. E.g the symbol rain is True if it’s raining today, False otherwise. Probabilities?

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Semantics Truth Tables

Truth Tables

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Semantics Truth Tables

Negation Operator

Beginning from the definitions of our truth assignments for constants and propositional symbols, we can assign truth to every compound 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

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Semantics Truth Tables

Conjunction / Disjunction

What would the truth table for conjunction and disjunction be?

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Semantics Truth Tables

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

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Semantics Truth Tables

Binary connectives

Implication:

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Semantics Truth Tables

Binary connectives

Implication: p q p ⇒ q F F T F T T T F F T T T

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Semantics Truth Tables

Binary connectives

Implication: p q p ⇒ q F F T F T T T F F T T T Bi-conditional:

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Semantics Truth Tables

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

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Semantics Truth Tables

Natural language examples

Let’s convert the following natural language statements to propositional 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. Jason Filippou (CMSC250 @ UMCP) Propositional Logic 05-31-2016 15 / 38

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Semantics Simplifying expressions

Simplifying expressions

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Semantics Simplifying expressions

Take 3

Do the truth tables for ∼(p ∧ q) and ∼p ∨ ∼q. What do you observe?

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Semantics Simplifying expressions

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.

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Semantics Simplifying expressions

Other logical Equivalences

Convince yourselves about the following:

∼p ∨ q ≡ p ⇒ q

p ⇒ q ≡ ∼q ⇒ ∼p (contrapositive)

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Semantics Simplifying expressions

Tautologies / Contradictions

Tautology: A logical statement that is always True , regardless

f 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

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Semantics Simplifying expressions

Logical Equivalence cheat sheet

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

Commutativity p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p Associativity 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 ≡ p Negation laws p ∨ ∼p ≡ t p ∧ ∼p ≡ c Double negation

∼(∼p) ≡ p

Idempotence p ∧ p ≡ p p ∨ p ≡ p De Morgan’s axioms

∼(p ∧ q) ≡ ∼p ∨ ∼q ∼(p ∨ q) ≡ ∼p ∧ ∼q

Universal bound laws p ∨ t ≡ t p ∧ c ≡ c Absorption laws p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p Negations of contradictions / tautologies

∼c ≡ t ∼t ≡ c

Those will be posted on our website as a reference.

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Semantics Simplifying expressions

Practice

Using the equivalences we just established, simplify the following expressions:

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

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Inference

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Inference Valid reasoning

Valid reasoning

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Inference 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 into sentences (compound statements) and the notion of equivalence helped us find equivalent statements, as well as simplify unnecessarily 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.

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Inference Valid reasoning

Valid reasoning

All reasoning has to be valid. Intuitively: the knowledge we infer has to obey the constraints of the world defined by the stuff we already know. Formal definition later. Examples:

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

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Inference Valid reasoning

Complete reasoning

The notion of “complete” reasoning is one that we won’t examine much, if at all, in 250. Intuitively, if we have a rule (or a set of rules) that can produce all

f the knowledge that logically follows from the stuff that we

already know, we have a complete reasoning system.

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Inference Valid reasoning

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 . . . Premise n ∴ Conclusion Some authors prefer the form: Premise 1, Premise 2, . . . , Premise n Conclusion

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Inference Valid reasoning

Definition of validity

Split rule to premises and conclusions Critical rows: The rows of a truth table where all premises are True . The rule is valid if the conclusion is also True for all critical rows.

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Inference Valid reasoning

Definition of validity

Valid rule Premise 1 Premise 2 Premise n

Figure 1: A pictorial representation of valid reasoning.

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Inference Basic rules of inference

Basic rules of inference

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Inference Basic rules of inference

Modus Ponens

The cornerstone of deductive reasoning. Modus Ponens p p ⇒ q ∴ q

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Inference Basic rules of inference

Modus Ponens

The cornerstone of deductive reasoning. Modus Ponens p p ⇒ q ∴ q Theorem (Validity of Modus Ponens) Modus Ponens is a valid rule of reasoning.

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Inference Basic rules of inference

Modus Ponens

The cornerstone of deductive reasoning. Modus Ponens p p ⇒ q ∴ q Theorem (Validity of Modus Ponens) Modus Ponens is a valid rule of reasoning. Proof. p q p ⇒ q F F T F T T T F F T T T

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Inference Basic rules of inference

Modus Tollens

Modus Tollens p ⇒ q

∼q

∴

∼p

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Inference Basic rules of inference

Modus Tollens

Modus Tollens p ⇒ q

∼q

∴

∼p

Proof?

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Inference Basic rules of inference

Other valid rules of inference

The following are mentioned on Epp (but there exist many more). Disjunctive addition p ∴ p ∨ q Conjunctive simplification p ∧ q p, q Disjunctive syllogism p ∨ q

∼q

∴ p Hypothetical syllogism p ⇒ q q ⇒ r ∴ p ⇒ r Prove their validity as an exercise!

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Inference Basic rules of inference

Valid inference rules cheat sheet

Modus Ponens Modus Tol- lens Disjunctive addition Conjunctive addition p p ⇒ q ∴ q

∼q

p ⇒ q ∴ ∼p p ∴ p ∨ q p, q ∴ p ∧ q Conjunctive Simplification Disjunctive syllogism Hypothetical Syllogism 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 the conclusion.

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Inference Basic rules of inference

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

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Inference Basic rules of inference

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

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Inference Basic rules of inference

Difference with language

Background knowledge oftentimes blurs the distinction between valid and invalid arguments. Consider the following arguments: If my pet ostrich could do 100 meters in under 10 seconds, it could participate in the Olympics. 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.

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Inference Basic rules of inference

Proof by contradiction

A very popular proof methodology, which we will be using a lot, is proof by contradiction. Intuitively, we want to prove something, so we assume that it doesn’t hold (i.e its converse holds), and we arrive at a contradiction. Formally, the following rule is sound: Proof by contradiction

∼p ⇒ c

∴ p Very important to convince yourselves that the rule is sound!

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