The central logical notions of statements and arguments are introduced. The notion of the standard form of an argument is presented, as are the criteria adopted for judging the quality of an argument. 

• Explain what a statement is, and discuss how statements are related to sentences.

• Determine whether or not a sentence of English expresses a statement, and if so, to identify the statement expressed.

• Explain what an argument is, and determine whether or not a given passage constitutes an argument.

• Identify the premises and conclusion of an argument, and represent the argument in standard form.

• List the criteria an argument must meet in order to be considered a good argument, and explain why each criterion is necessary.

• Determine whether premises support the conclusion jointly or independently.

• Determine whether a given argument is a bad or a good argument, and why this is the case.

The syntax of the language of sentential logic is introduced. Translation to and from sentential formulae and syntactic analysis of expressions is the focus.

• Discern the logical structure of English sentences.

• Symbolize English sentences as formulae of sentential logic.

• Explain the grammar of the logical language of sentential logic.

• Construct and identify formulae of sentential logic.

• Construct and use parse trees.

The semantics of the language of sentential logic are discussed. The notions of truth-values and truth-functions are introduced, and a number of semantics tools for the evaluation of both formulae and arguments are presented.

• Explain what a truth-value assignment is. 

• Give the truth-conditions for the logical connectives.

• Determine the truth-value of a formula relative to a given truth-value assignment.

• Construct a truth-table for a given formula or argument.

• Use truth-tables to analyze arguments and formulae.

• Explain what tautological, contingent, and contradictory formulae are.

• Find a counterexample to an invalid argument, using a truth-table or truth-tree.

The notion of a derivation is introduced, and the inference rules for the binary connectives are presented.

• Explain the structure of derivations. 

• Apply and identify applications of rules of inference within a derivation.

• Establish the validity of rules of inference.

The inference rules relating to negation are introduced.

• Apply and identify applications of the inference rules for negation.

• Explain the structure of indirect rules of inference.

• Find contradictions to use in applications of indirect rules.

Strategies for completing derivations efficiently and easily are discussed, and a number of derived rules of inference are presented.

• Approach proof construction problems in a strategic fashion.

• Apply and identify applications of derived rules.

• Provide explanations of and explain some significant properties of the logical connectives.

A number of metamathematical notions are introduced and a new connective is added to the language.

• Show that two formulae are logically equivalent just in case their biconditional is a tautology.

• Explain how a biconditional can be considered logically equivalent to a formula in disjunctive normal form.

• Find a disjunctive normal form equivalent to any formula.

• Explain the connection between truth-tables and Boolean circuits.

The first step in the transition from sentential to predicate logic is taken, by introducing the notions of predicates and singular terms.

• Identify and distinguish between predicates and singular terms.

• Analyze the internal logical structure of English sentences.

• Construct atomic formulae from predicates and singular terms according to syntactic rules.

• Semantically interpret and evaluate formulae constructed using predicates and singular terms.

The transition from sentential to predicate logic is completed by introducing the notions of quantifiers and variables.

• Analyze the internal logical structure of English sentences involving quantity terms.

• Construct arbitrary predicate formulae according to the syntactic rules.

• Semantically interpret and evaluate predicate formulae.

• The derivation system for sentential logic is expanded to cover predicate arguments, by adding introduction and elimination rules for the quantifiers.

• Learning Objectives:

• Apply and identify applications of the inference rules for the quantifiers.

The strategies for efficient derivation completion are reconsidered in the predicate context, and a number of derived rules for predicate logic are presented. A normal form for predicate logic, called prenex normal form, is also introduced.

• Extend the strategic considerations for the construction of proofs to predicate logic.

• Develop, apply, and identify applications of derived rules.

• Convert an arbitrary formula into an equivalent formula in prenex normal form.

The notion of identity is introduced and discussed, and additions are made to the syntax, semantics, and derivation system in order to incorporate this notion.

• Translate English sentences involving identity into the extended language of predicate logic.

• Apply and identify applications of the inference rules for identity.

• Give the Russellian analysis of sentences involving definite descriptions.

The notion of functions is introduced and discussed, and additions are made to the syntax and semantics in order to incorporate this notion.

• Use function symbols in translating both English sentences and mathematical statements.

• Use syntactic and semantic techniques with function symbols.