PROPOSITIONAL LOGIC

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PROPOSITIONAL LOGIC
Connectives and formulas
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Negation

• Its not raining.

Its not true that its raining.
p Its raining
its not true that p
p
The proposition p is called a negation of
proposition p.
The sub-proposition p of the negation p is the
negand of the negation.
The negation p is true when its negand is false
and false otherwise.
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Negation
Keeping your promises is not always good.
Its not true that keeping your promises is
always good.
a
Its not true that a
Keeping your promises is always not good.
Keeping your promises is never good.
Its not true that keeping your promises is ever
good.
e
Its not true that e
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Conjunction

• The battery is flat and the starter is dead.

p The battery is flat q The starter is dead
p and q
p q
The proposition pq is called a conjunction of
propositions p and q.
The sub-propositions p and q of the conjunction
pq are called conjuncts of the proposition pq.
The conjunction pq is true when both its
conjuncts are true and false otherwise.
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Conjunction
Dennett and Searle are right.
Dennett is right and Searle is right
ds
Dennett and Searle are friends.
Dennett is a friend and Searle is a friend
???
Dennett is right, but Searle is not.
Dennett is right and Searle is not right.
ds
You make one false move and I shoot.
I shoot and you make one false move.
???
Peter is a clever boy.
Peter is clever and Peter is a boy
cb
Steven is a perfect stranger.
Steven is perfect and Steven is stranger
???
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Disjunction

• Its raining or its snowing

p Its raining q Its snowing
p or q
p v q
The proposition pvq is called a disjunction of
propositions p and q.
The sub-propositions p and q of the disjunction
pvq are called disjuncts of the proposition pvq.
The disjunction pvq is true when at least one of
its disjuncts is true and false otherwise.
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Disjunction
Inclusive disjunction
In Ireland, its raining or its sunny.
Its raining or its sunny (or both).
r v s
Exclusive disjunction
In Slovakia, its raining or its sunny.
Its raining or its sunny (but not both).
(r v s) (r s)
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Conditional

• If everything is determined, then people are not
  free.

p Everything is determined q People are not
free
If p, then q
p ? q
The proposition p?q is called a conditional.
The sub-proposition p of the conditional p?q is
called an antecendent, and the sub-proposition q
is called a consequent of the proposition p?q .
The conditional p?q is false when its antecendent
is true and its consequent is false and true
otherwise.
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Conditional
p I play golf q Its sunny
q?p

• I play golf if its sunny.

I play golf only if its sunny.
p?q
q?p
I play golf provided its sunny.
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Biconditional

• I play golf if its sunny. (If its sunny, then I
  play golf)
• I play golf only if its sunny. (If I play golf,
  then its sunny)

I play golf if and only if its sunny
p I play golf q Its sunny
p if and only if q
p ? q
The proposition p?q is called a biconditional of
propositions p and q.
The sub-propositions p and q of the biconditional
p?q are called its left-hand and right-hand
expressions respectively.
The biconditional p?q is true when its
expressions have the same truth value and false
otherwise.
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Connectives - summary
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Language of forms, formulas
Proposition to formula

• Find atomic sub-propositions, ie.
  sub-propositions which do not contain another
  sub-propositions
• Create a dictionary, ie. assign letters to the
  atomic propositions
• Re-formulate the sentence (expressing the
  proposition) using atomic sub-propositions and
  connectives
• Use parentheses Enclose every sub-proposition
  (apart from the atomic ones) in parentheses
• Replace the atomic propositions with
  appropriate letters, and connectives with their
  symbols

A proposition which does not contain a proper
sub-proposition (and is unrelated to other
sub-propositions) is called an atomic proposition.
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Language of forms, formulas
Either Crumm is guilty, or both he and Moriarty
are guilty.
Dictionary c Crumm is guilty m
Moriarty is guilty
Crumm is guilty, or Crumm is guilty and Moriarty
is guilty.
(Crumm is guilty or (Crumm is guilty and Moriarty
is guilty))
(c v (c m))
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Language of forms, formulas
m Moriarty is guilty c Crumm is guilty
Either Moriarty and Crumm are both guilty, or
Crumm is innocent.
((m c) v c)
If Crumm is guilty, then so is Moriarty but
surely they arent both guilty.
((c ? m) (m c))
Crumm is guilty if Moriarty is, unless theyre
both innocent.
((m c) ? (m ? c)) ((m ? c) v (m c))
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Language of forms, formulas
Atomic formulas (for atomic propositions)
p, q, a, b, c, s1, s2, s18, p25, ...
Every atomic formula is a formula. If A is a
formula, so is A. If A and B are formulas, so
are (A B), (A v B), (A ? B) and (A ?
B). Nothing else is a formula.
p, p, (pq), (qp), (s?(pq)), ((pq)?(qp)),
... well formed formulas (wff)
p, (pq), (pqr), (pq?qp), ... NOT well
formed formulas
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Language of forms, formulas
A formula which is not atomic is called
complex. The main connective of a complex formula
is the connective which was added last in its
construction.
((p q) ? r) ((p v q) ? r) (((p ? q) (q ?
r)) v (p ? q))
To make our lives easier, we will drop the
(outermost) parentheses around the main
connective, unless it is a negation.
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Argument forms (cont.)
Modus Ponens
Modus Ponens
If p, then q p Therefore, q
p ? q p Therefore, q
Modus Tollens
Modus Tollens
If p, then q not q Therefore, not p
p ? q q Therefore, p
Disjunction Introduction
Disjunction Introduction
p ? r q ? r Therefore, (p v q) ? r
If p, then r If q, then r Therefore, if p or q,
then r
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Argument forms (cont.)
The mind is the brain. The mind is not the brain.
Therefore, both Dennett and Searle are right.
a The mind is the brain d Dennett is right
s Searle is right
a a Therefore, d s