Posted 21 August 2009
This chapter covers:
¨ The major types of logical constructions used in mathematical assertions.
¨ The methods of deduction that may be used with each type of logical construction.
The assertion “P and Q” is a logical construction. It claims that both assertions are true. The assertion “P or Q” claims that at least one of them is true. You can determine the truth of these two compound assertions provided you know the truth of P and of Q. You don’t need to know anything else about P and Q except their truth value to determine the meaning of “P and Q” and “P or Q”.
The truth values for a logical construction are given unambiguously by truth tables, one for each connective such as “and” and “or”. For example, from the tables shown here you can see immediately that if P is true and Q is false, then “P and Q” is false but “P or Q” is true.
The truth tables control the meaning. The meaning and connotation of the English words describing the connectives (“and” and “or” in the example above) do not affect the truth of the construction. They usually suggest the truth but can be misleading. This is one aspect of the translation problem.
you know that the assertion “P and Q”
is true, then you know just from the form of the assertion
that P is true. For example, “
Each part of the discussion of math reasoning on this website describes a type of logical construction and gives the methods of deduction appropriate for that type of logical construction.
The translation problem is the task of extracting the mathematical and logical structure hidden in mathematical prose. The discussions of logical constructions and methods of deductions in this part of the website describe the difficulties with the translation problem.
¨ Word problems: To “translate a word problem into math” is to find the equation(s) (or other symbolic assertions) that contain the information in a word problem, with the intent of solving them. Finding the equations may be easy or hard and solving them may independently easy or hard.
Mathematical logic (or proof theory) is a branch of mathematics that uses to model mathematical statements and proofs. Doing this requires:
¨ Giving a strict mathematical definition of theorem and proof.
Mathematical logic is quite technical but very powerful; it has put mathematical reasoning on a sound scientific basis and it has enabled logicians to prove certain impossibility theorems such as The description of proofs given on this website is informal and does not include all the technicalities necessary to make logic a part of math.
In mathematical logic, assertions, connectives, quantifiers and rules of deduction are typically represented using symbols, and the symbolic system developed this way is studied as a mathematical object.
Assertions containing a variable x might be represented as P(x) and Q(x). Then expressions built up out of these may be represented in one of these ways (this list is not exhaustive):
means “for all x, P(x)” (universal quantifier).
means “there is an x for which P(x) is true” (existential quantifier).
means that P(x) is false (also written as or ).
means “P(x) or Q(x)” (also written P(x) + Q(x)).
means “P(x) and Q(x)” (also written P(x)Q(x) or P(x) & Q(x)).
means “If P(x), then Q(x)” (also written or ). See conditional assertions.
These websites provide more complete descriptions of this symbolism and of the theory behind it.
The mathematical structures used in mathematical logic that correspond to a mathematician’s proof are often called formal proofs. However, the phrase “formal proof” can also describe proofs written in narrative form in mathematical English.
More about quantifiers
H.-D. Ebbinghaus, J. Flum and W.
Thomas, Mathematical Logic. Springer-Verlag
D. van Dalen, Logic and Structure (
¨ Mathworld on first order logic
¨ Suber on symbolic logic