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Formal logic

Formal logic is a set of rules for making deductions that seem self evident. Syllogisms like the following occur in every day conversation.

All humans are mortal.Mathematical logic formalizes such deductions with rules precise enough to program a computer to decide if an argument is valid.

Socrates is a human.

Therefore Socrates is mortal.

This is facilitated by representing objects and relationships symbolically. For example we might use for the set of humans, for the set of mortal creatures and for Socrates. We use the symbolic expression `' to indicate that object is a member of set . Thus we represent `Socrates is a human' with . We use the `quantifier' to indicate that all objects satisfy some condition. For example all men are mortal can be written as . This reads that every that has the property of being human must also have the property of being mortal. Then we restate the syllogism as follows.

Logic assumes something cannot be both true and not true. It looks only at the truth value of a proposition. It involves simple relationships between these truth values. These can be represented by truth tables as shown in Table 3.1. The only logical operations required are the three in this figure. Others such as implication represented by `' can be constructed from these three. is the same as . implies requires that either both and are true or is false.

Determining the truth of a logical expression that contains no
quantifiers (like ) is a straightforward
application of simple rules. One can use a truth table to evaluate
each subexpression starting with those at the root of the
expression tree as shown in Table 3.2. If a logical expression
contains quantifiers than we need to evaluate a logical
relationship over a range of values to determine the truth of the
expression. If the range is infinite then there is no general way
to evaluate the expression. We can use induction^{3.2}to prove that some statements hold
for all integers but for that we need to go beyond logic to
mathematics.

Completed
second draft of this book

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**Next:** Formal mathematics **Up:** Mathematical
structure **Previous:** Logically determined unsolvable
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