Valid Arguments in Propositional Logic
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An argument in propositional logic is a sequence of propositions. All
but the final proposition in the argument are called premises and the
final proposition is called the conclusion.
Example:
Example:
“If you have a current password, then you can log onto the network.”
“You have a current password.”
Therefore,
“You can log onto the network.”
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An argument form
in propositional logic is a sequence of compound
propositions involving propositional variables.
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An argument form is valid if no matter which particular propositions
are substituted for the propositional variables in its premises, the
conclusion is true if the premises are all true.
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The conclusion is true if the premises are all true.
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Either team A or Team B will win the match
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Team B lost
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Therefore Team A won
----------------------------------------------
The general form of this argument is:
Either P or Q
Not P
Therefore Q
(P
∨
Q)
∧
┐P→Q , or
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We can always use a truth table to show that an argument form is
valid. We do this by showing that whenever the premises are true,
the conclusion must also be true.
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Step 1
: Construct truth table for the premises and conclusion.
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Step 2
: Find the rows in which all the premises are true(critical rows).
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Step 3
: Check conclusion of all critical rows.
a)
If in each critical rows the conclusion is
true
then the argument is
valid
.
b)
If there is a row in which conclusion is
false
then the argument form
is
invalid
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Example : Determine whether the following argument is valid or invalid
(Hint: use informal method)
Solution:
put all premises true: (p → q) = T
q → (p → r) = T
p = T
.........................................
Since p = T and (p → q) = T, then q = T
Now, since q = T and q → (p → r) = T then Q := (p → r) = T
We have p = T and (p → r) = T then r must be true. Therefore r = T and
the argument is
valid
.
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An argument in propositional logic consists of premises leading to a conclusion. Valid arguments are those where the truth of the premises implies the truth of the conclusion. To determine validity, you can construct a truth table to check if the conclusion always holds when all premises are true. This systematic approach helps in ensuring logical consistency.
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Presentation Transcript
Valid Arguments in Propositional Logic Valid Arguments in Propositional Logic 1 L Al-zaid Math1101
Argument Argument An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. Premises Conclusion Example: If you have a current password, then you can log onto the network. You have a current password. Therefore, You can log onto the network. ? ? ? ---------------- ? 2 L Al-zaid Math1101
An argument is An argument is valid conclusion is true. conclusion is true. valid if the truth of all its premises implies that the if the truth of all its premises implies that the An argument form in propositional logic is a sequence of compound propositions involving propositional variables. An argument form is valid if no matter which particular propositions are substituted for the propositional variables in its premises, the conclusion is true if the premises are all true. The conclusion is true if the premises are all true. 3 L Al-zaid Math1101
Example: Example: Either team A or Team B will win the match Team B lost Therefore Team A won ---------------------------------------------- The general form of this argument is: Either P or Q Not P Therefore Q (P Q) P Q , or 4 L Al-zaid Math1101
Rules of Inference for Propositional Logic Rules of Inference for Propositional Logic We can always use a truth table to show that an argument form is valid. We do this by showing that whenever the premises are true, the conclusion must also be true. 5 L Al-zaid Math1101
Checking the validity of an Argument form Checking the validity of an Argument form Step 1: Construct truth table for the premises and conclusion. Step 2: Find the rows in which all the premises are true(critical rows). Step 3: Check conclusion of all critical rows. a) If in each critical rows the conclusion is true then the argument is valid. b) If there is a row in which conclusion is false then the argument form is invalid. 6 L Al-zaid Math1101
Example: Example: Determine whether the following argument is valid or invalid ? ? ? P q q Formal T T F T F ? ? ? ? T F F F T F T T T F F F T T T Solution put all premises true: p q = T p = T, then p = F Now, put q = T. So, we still have p q = T, but q = T, Then q = F. The conclusion is False whereas all premises are true, Therefor, the argument is invalid. Informal 7 L Al-zaid Math1101
Example : Determine whether the following argument is valid or invalid (Hint: use informal method) ? ? ? ? ? ? ? Solution: put all premises true: (p q) = T q (p r) = T p = T ......................................... Since p = T and (p q) = T, then q = T Now, since q = T and q (p r) = T then Q := (p r) = T We have p = T and (p r) = T then r must be true. Therefore r = T and the argument is valid. 8 L Al-zaid Math1101
Exercise: Exercise: Determine whether the following argument is valid or invalid: ? ? ? (? ?) ? ? ? ? 9 L Al-zaid Math1101
