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Propositions and connectives: proposition true false …myhome.sunyocc.edu/~matthewg/251/CN1-1 Propositions... · Propositions and connectives: A proposition is a sentence that is

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Page 1: Propositions and connectives: proposition true false …myhome.sunyocc.edu/~matthewg/251/CN1-1 Propositions... · Propositions and connectives: A proposition is a sentence that is

Propositions and connectives:A proposition is a sentence that is either true or false but not both.Lower case letters, such as p, q, r are used as symbols to represent simple propositions.

p: “Pi is the ratio of circumference of a circle to its diameter” is a true proposition.q: “Pi equals 22/7” is a false proposition. r: “It is raining” may be true or false, according to the time and place.s: “Logic is important” is at best an ambiguous statement.

The manipulation of simple propositions to form compound propositions is called propositionalcalculus. Special logic symbols are used to denote the five basic operations:

negation, not: ¬conjunction, and: vdisjunction, (inclusive or): w [simply read “or”]

conditional implication, implies: 6 [also read “only if”, or “If ..., then ... .”]

biconditional, if and only if: : [also written “iff”]

Using p: “The parade is proceeding.” and q: “The number pi is rational.” and r: “It is raining.”,we can form the following compound statements:

¬ q : “The number pi is not rational.”p v r : “The parade is proceeding and it is raining.” q w r: “The number pi is rational or it is raining.”

p 6 r : “The parade is proceeding only if it is raining.”

p : q : “The parade is proceeding if and only if the number pi is rational.”And we can translate compound statements into symbolic form:

“If the parade is proceeding, then it is not raining.” : p 6 ¬ r“Either it is raining or the parade is proceeding.” : r w p“It is raining, but the number pi is not rational.” : r v ¬ q

Note that the compound propositions do not need to make “sense”, but they do have truth values

which can be computed according to the tables given in the next section.

There are several variants that involve the conditional implication. An implication is equivalent to its contrapositive, but is not equivalent to its converse.

Given the implication p 6 q, the contrapositive is ¬q 6 ¬p, and the converse is q 6 p.If s: “You are studying all of this unit.” and l: “You are learning logic.” what is s 6 l ?

The implication “If you are studying all of this unit, you are learning basic logic.” has thecontrapositive: “If you’re not learning basic logic, then you’re not studying all of this unit.” And the converse is: “If you are learning basic logic, then you are studying all of this unit.”

Other variant wordings of s 6 l are:“Studying all of this unit implies you are learning basic logic.”“Studying all of this unit is sufficient for learning basic logic.”“You will learn basic logic, if you study all of this unit.”“Learning basic logic necessarily follows from studying all of this unit.”

Confusion of these different forms with the converse is the source of much illogical reasoning.

© 2005 Prof. George Matthews