Introduction to our Logically Precise Language

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Our logically precis language will have the following elements

• Sentences (which are formed of:)
  • Terms
  • Predicates
• Connectives
• Quantifiers
• Functions

Reasoning tools
We will also add to our language tools for doing proofs. These include:

• Proof methods
  • Direct derivation
  • Indirect derivation
  • Conditional derivation
  • Universal derivation
• Inference rules

Method of introduction
To introduce elements to our language, we will follow the same pattern:

• Syntax (how can we combine this symbol with other symbols)
• Semantics (what does this symbol mean)
• Inference rules (what follows from the proper use of this symbol?)

Sentences

• An unambiguous sentence is one that is either true or false, never both, never neither. As logicians doing first order logic, this is the only property that we care about for setences!
• A sentence alone we represent with upper case letters from P through Z. Think of this as just a shorthand, so we do not have to write out a whole sentence in English or some other language.
• The syntax for a sentence is that it can stand alone. That is, you may at any time assert P or Q or R. They might be false, but you have written down a grammatical string in our language. A sentence alone is called an atomic sentence.
• The semantics is that any atomic sentence is either true or false, never both, never neither. If we write down an atomic sentence, we are assumed to have asserted it, and thus to have claimed or supposed that it is true.
• An obvious inference rule then is Repitition, or R. This is a trivial rule but sometimes useful for bookkeeping. Repition says that if P is true, then I can write assert it as often as I like without asserting something false. Thus, if I am willing to write "P", I should feel free to write it again as often as I like.

"...not..."

• We also want to be able to combine sentences together. We need tools to allow us to do that. We call these connectives.
• The first, simplest connective corresponds to our English "...not..."
• We will write "~" for "... not ..."
• The syntax for "~" is that you can put it in front of any sentence and that produces a sentence. Thus, if P is a sentence, then so is ~P.
• The semantics for "~" is that it changes the truth value of whatever it precedes. If P is true, then ~P is false. If P is false, then ~P is true.
• Given the semantics of "~", we can observe that if ~~P is true, then P is true, and if P is true, then ~~P is true. Thus, we introduce the rule Double Negation or DN.
• "~" also translates such phrases as "it is not the case that...."

"if...then..."

• A powerful logical notion is captured in our everyday English phrase "if ... then ...." We will symbolize this using "-->".
• Syntax: if "-->" has a sentence on each side, it forms a new sentence. That is, for any sentences P and Q, P --> Q is a sentence.
• Semantics: these are given by the truth table for "-->", which shows us that the condition is true in all cases but where the antecendent (the first sentence) is true and the consequent (the second sentence) is false.
• Inference rules: modus ponens (MP), modus tollens (MT).
• "(P-->Q)" also translates such English phrases as
  • If P, Q
  • Q if P
  • Provided that P, Q
  • Q provided that P
  • P only if Q
  • In case P, Q
  • Q in case P
• We call the first part of a conditional the "antecedent" and the second part the "consequent."

Truth functions
These connectives are called "truth functional connectives" because they are defined in terms of their truth function: they take in 1 or more truth values as an input and give out only one truth value as an output. The truth tables define their semantics or meaning. You must memorize the truth tables.