Notes on Introduction to Logic -- Phil A101

William Jamison - Instructor

Lecture on Propositional Logic:

Propositional Logic: Statements

            You may think of the statements discussed here as compound statements that include A, E, I, O statements joined together using logical operators, however, we will not be concerned with those atomic statements in this section. Here the focus will be on the logical operators and how they function.

Compound Propositions and Logical Operators

Truth-Functional Operators


Propositional Abbreviations and Schemas

            These are tools we can use to simplify our examination of complex arguments.



            “And” and other words we use to mean both conjoined statements must be true for the compound statement to be true. The symbol in most texts is the dot, and is called the “dot operator” but for html ease of use I will “+” on these pages.


Truth Tables

            If you are interested in Wittgenstein, who invented the truth table method, and what he thought the truth tables showed us about how our language works, visit the IEP for a wonderful summary of his life and work and pay special attention to the description of the Tractatus Logico-Philosophicus.  You can also visit Wittgenstein quotes for some thought provoking ideas on what language can do.



            “Not” or other words we use to mean the statement (or compound statement) following the symbol “~” is not the case.



            “Or” or other words we use to mean the disjuntion of two statements requires the truth of either one of the disjuncted statements or both in order for the compound statement to be true. If we want to translate a statement to reflect a speaker’s intention to mean only one or the other, but not both statements disjuncted in the compound statement, then we do this by symbolizing the compound statement using the schema  [(p v q) + ~(p + q)]. I will use the symbol “v” for disjunction.


Material Implication

            While it seems that a logical system of symbolic notation can be made using only conjunction, disjunction, and negation, a system that has only those symbols becomes too complex to make our work as simple as is efficient. So most contemporary systems use other symbols that represent combinations of those three. These extra symbols can be shown to be equivalent to combinations of +, v, and ~ by means of truth tables.

            Material Implication is one of these extra symbols. We use the horse shoe in the text, but most machine fonts use an arrow to the right of some kind. I will use “>” on these pages.


The compound statement (p > q) is equivalent to (~p v q).




Material Equivalence


            A second symbol we use to be more efficient is represented in most texts by three lines that look like = with an extra line on top. For ease of use on these pages I will just use “=”. It means that the statement to the right of  = has exactly the same truth value as the statement to the left. So, for example, the equivalence from above can be represented as:


            (p > q) = (~p v q)


The operator “=” can be seen to be equivalent to a combination of +, v, and ~ in the following examples:



p = p (Which is the Law of Identity – a thing is what it is)

            ~(p = ~p) (Which is the Law of Non-Contradiction – it is not the case that a thing is not identical to itself.



Propositions with More Than One Logical Operator


Truth Table Construction


Logically Equivalent Statements

Logical Equivalence and Material Equivalence




Contingent Statements





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