So, I have given an introduction to proofs and logic symbols, but I have not properly explained logic. Please note, I am using my Discrete Mathematics textbook [1] for this post. Also, this is a “crash course” in logic. There is much more to be learned, so make sure you view the proper resources for more information.
Wolfram Mathworld [2] says that logic is
The formal mathematical study of the methods, structure, and validity of mathematical deduction and proof
From page 1 of [1] we see that the study of logic should begin with
..the basic building blocks of logic – propositions…
This book[1] tells that,
A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.
EXAMPLES of Propositions:
- 4+8=12
- 4+8=13
- Water is
- Water is
All of the above examples are propositions, they are either true or false, but not both. We see that examples 1 and 3 are true and examples 2 and 4 are false. However, as illustrated in the following examples, not everything is a proposition.
EXAMPLES of non-propositions:
- What is your name?
- x+2=15
- x+y=12
- Be smart.
Examples 1 and 4 are not declarative sentences and are thus not propositions. Furthermore, examples 2 and 3 are neither true nor false and thus are not propositions. (These examples are based off examples 1 & 2 in section 1.1 of [1].)
We usually denote propositions by using variables, much like we use variables to represent numbers. The most commonly used variables for propositions are “p, q, r, s,…” We assign a truth value to each proposition. If a proposition is true, we assign it “T” for true and if it is false, we assign it “F” for false.
Page 2 of [1] tells us,
The area of logic that deals with propositions is called the propositional calculus or propositional logic. It was first developed systematically by the Greek philosopher Aristotle more than 2300 years ago.
I am going to leave this post here with more to come regarding logic in a later post.
[1] Rosen, K. H. (2007). Discrete Mathematics and its Applications (3rd Ed.). New York: McGraw Hill.
[2] http://mathworld.wolfram.com/Logic.html
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