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Propositional Logic
Jason Filippou
CMSC250 @ UMCP
05-31-2016
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Outline
1 Syntax
2 SemanticsTruth TablesSimplifying expressions
3 InferenceValid reasoningBasic 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 compoundstatements), as well as a set of equivalences and inferencerules.
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
Syntax
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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.
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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
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 inEuclidean Geometry.
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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?
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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 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
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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
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:
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
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
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
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 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.
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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⇒ qp⇒ q ≡ ∼q ⇒ ∼p (contrapositive)
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Semantics Simplifying expressions
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
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Semantics Simplifying expressions
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.
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Semantics Simplifying expressions
Practice
Using the equivalences we just established, simplify the followingexpressions:
p ∧ (∼p ∨ q) ∨ (∼(∼(z ∨ ∼q)))(p ∧ r) ∨ ((p ∨ s) ∧ (p ∨ a))
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Inference
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 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.
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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 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.
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Inference Valid reasoning
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.
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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
. . .
P remise 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 areTrue .
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 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
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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
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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
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Inference Basic rules of inference
Modus Tollens
Modus Tollensp⇒ q∼q
∴ ∼p
Proof?
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Inference Basic rules of inference
Modus Tollens
Modus Tollensp⇒ 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 ∧ qp, q
Disjunctive syllogism
p ∨ q∼q∴ p
Hypothetical syllogism
p⇒ qq ⇒ 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
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.
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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
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
Jason Filippou (CMSC250 @ UMCP) Propositional Logic 05-31-2016 36 / 38
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Inference Basic rules of inference
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.
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Inference Basic rules of inference
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!
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