Symbolic logic
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Transcript Symbolic logic
Logical Form: general rules
◦ All logical comparisons must be done with
complete statements
◦ A statement is an expression that is true or false
but not both
If p or q then r
If I arrive early or I work hard then I will be promoted
◦ Tautologies and Contradictions
A Tautology (t) is a statement that is always true
A Contradiction (c) is a statement that is always false
The use of symbols
◦ ~ denotes negation (Not)
If p = true, ~p = false
◦ ^ denotes conjunction (And)
p^q = true iff (if and only if) p = true and q = true
◦ v denotes disjunction (Or)
p vq = true iff p = true or q = true or p^q = true
◦ XOR: exclusive or
P XOR q = (p vq) ^ ~(p^q), “p or q but not both”
◦ Order of operations
~ is first, ^ and v are co-equal
P^q v r is ambiguous, so parenthesis need to be used: (p^q)
vr
~p^q = (~p) ^ q
Inequalities
◦ x ≤ a means x < a or x = a: (x < a) v (x = a)
Same for x ≥ a
◦ a ≤ x ≤ b means (a ≤ x) ^ (x ≤ b)
◦ a (NOT)> x = a ≤ x
Same for opposite
◦ a (NOT) ≤ x = a > x
Same for opposite
Truth Tables
◦ Every expression has a truth table
◦ This table represents all the possible evaluations of
the expression
◦ To build a truth table, construct a table with cells
corresponding to every possible value of the
variables and the resulting value of the expression
Logical equivalence
◦ Two statement forms are logically equivalent iff
their truth tables are entirely the same
Ex: p^q = q^p
P = ~(~p)
Showing non-equivalence
◦ Two methods:
Use truth tables: this takes a long time
Use an example statement like “0 < 1”
The following are known as axioms. Use these to
simplify logical forms easily
◦ Commutative Laws: p^q = q^p , pvq = qvp
◦ Associative Laws: (p^q)^r = p^(q^r), (pvq)vr = pv(qvr)
◦ Distributive Laws: p^(qvr) = (p^q)v(p^r)
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p v(q^r) = (pvq)^(pvr)
Identity Laws: p^t = p, pvc = p
Negation Laws: pv~p = t, p^~p = c
Double Negative Law: ~(~p) = p
Idempotent Laws: p^p = p, pvp = p
Universal Bound Laws: pvt = t, p^c = c
De Morgan’s Laws: ~(p^q) = ~pv~q, ~(pvq) = ~p^~q
Absorption Laws: p√(p^q) = p, p^(pvq) = p
Negations of t and c: ~t = c, ~c = t
If Structures
◦ Statement form: “if p then q”
Noted: p→q, p is Hypothesis, q is conclusion
Truth Values: p→q is false iff p = true and q = false
In statement forms, “→” is evaluated last
Division Into Cases: Show pvq→r=(p→r)^(q→r)
◦ Build truth table and evaluate each term separately
◦ Then fill in each side of the equation and compare the
values
An If statement can be translated into an Or
◦ p→q = ~pvq
◦ People often use this equivalence in everyday language.
◦ By De Morgan’s Law
~(p →q) = p^~q
Caution: The negation of an If does not start with “if”
The Contrapositive of an If
◦ The contrapositive of p →q is ~q →~p
A contrapositive is always logically equivalent to the
original statement, so it can be used to solve
equations
A contrapositive is both the converse and the inverse
of a statement
The Converse and Inverse
◦ The Converse of p →q is q →p
◦ The Inverse of p →q is ~p →~q
Neither is logically equivalent to the original statement
If tomorrow is Easter then tomorrow is Sunday
If tomorrow is Sunday then tomorrow is Easter?
Only If
◦ “p only if q” means that p may occur only if q occurs
Equivalent to: ~q →~p
Equivalent to: p →q
This does not mean “p if q”, which says that if q is true, p
must be true
An argument is a sequence of statements and
an argument form is a sequence of statement
forms.
◦ A basic argument is: p→q
p
:q
_ All statements except the final one are the premises
_ The final is the conclusion
_ This is read: “If p then q; p occurs, therefore q
follows
_ The argument is valid iff the conclusion is true
when all of the premises are true
Testing an argument for validity
◦ Identify the premises and conclusion
◦ Construct a truth table showing the possible truth
values for each statement and statement form
◦ If a situation exists in which all of the premises are
true but the conclusion is false, the argument form
is invalid
To simplify, fill in all rows where all premises are true
Modus Ponens: A famous argument form
◦ p→q: p:: q
◦ If p occurs then q occurs: p occurs:: therefore q
occurs
Modus Tollens
◦ p →q: ~q:: ~p
◦ If q doesn’t occur, p can’t occur
◦ A rule of inference is an argument form that is
valid.
There are infinitely many of them
Modus Ponens and Tollens are rules of inference
Generalization
Specialization
Elimination
Transitivity
Contradiction Rule:
◦ p::pvq and q::pvq
◦ p occurs, therefore either p or q occurred
◦ Used to classify events into larger groups
◦ p^q::p and p^q::q
◦ Both p and q occur, therefore p occurred
◦ Used to put events into smaller groups
◦ Pvq: ~q::p and pvq:~p::q
◦ P or Q can occur: Q doesn’t:: p must
◦ you can choose one by ruling the other out
◦ p →q:q →r::p →r
◦ If p then q: if q then r:: therefore if p then r
◦ ~p →c::p
◦ If the negation of p leads to a contradiction, p must be true.
Proof by Division Into Cases
◦ pvq: p →r:q →r:: r
◦ p or q will occur: if p then r: if q then r:: r occurs
◦ You may only know one thing or another. You must
simply show that in either case, the result is the
same
The Biconditional (iff)
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This is: “p if, and only if q”
Denoted: p↔q and is coequal with →
p iff q = (p→q) ^ (q→p)
If p has the same truth value as q, p↔q is true
An error in reasoning that results in an invalid
argument
Three kinds
Using ambiguous premises (Statements that are not
T/F)
Begging the Question: assuming the conclusion without
deriving it from the premises
Jumping to a Conclusion: verifying the conclusion
without adequate grounds
Converse Error:
◦ p →q: q:: p – FALSE
◦ If p then q: q occurs:: p must occur – FALSE
Inverse Error
◦ p →q: ~p:: ~q - FALSE
◦ If p then q: p doesn’t occur:: q can’t occur - FALSE