This concise text treats logic as a tool, “generated so that half the work involved in thinking is done for you by somebody else (the rules and laws of the logic).” Gabbay explains in a clear and careful manner how formal features of, and formal relations between, ordinary declarative sentences are captured by the systems of propositional and predicate logic.

# Logic With Added Reasoning

1. Arguments and Validity

- Validity and Arguments
- A little argument
- A theory of validity
- Examples of valid and invalid arguments
- Valid arguments
- Invalid arguments
- Validity and the structure of an argument
- Validity is independent of meaning
- Determining validity by changing to an easier meaning
- Other forms of argument
- Modus Ponens and friends
- More valid arguments
- Conclusions
- Exercises

2. Truth Functionality

- Embedded sentences
- Basic and simple sentences found in larger sentences
- Truth values
- Propositions
- Truth values of complex sentences
- Truth functional sentences
- Non-truth functional sentences
- Different words, same truth function
- Same word, different truth functions
- A problem for truth functionality
- Exercises

3. Formalisation of Truth Functions

- Formalising truth functions
- Formalising and, not and or
- Formalising sentences
- Truth Tables
- How to read truth tables
- Negation
- Conjunction
- Disjunction
- Material equivalence
- The formalisation of material equivalence
- Material equivalence in sentences
- Exercises

4. Truth Tables and Tautologies

- Tautologies and how to find them
- Truth tables with more than two letters
- How to write the exclusive “or”
- Exercises

5. Material Implication and Validity

- Material implication in theory
- The truth conditions of material implication
- The truth table for material implication
- Counterfactuals, a problem for the truth tables
- Material implication does not imply any causality
- Only if
- Material implication in practice
- More interesting tautologies
- DeMorgan’s Laws
- Some jolly big truth tables

- Truth tables for analysing arguments
- Exercises

6. The Tableaux Method

- Indirect Proof: a preliminary
- The way of the tableau for tautologies
- The rules of the tableau
- Exercises
- The way of the tableau for validity
- Exercises

7. Propositional Logic: The Interesting Bits

- What is so interesting?
- Only three truth functions are necessary
- Only two truth functions are necessary
- Only one truth function is necessary
- Sheffer stroke
- Another sufficient truth function
- The big problem with propositional logic
- Natural language is more than a few sentences
- Exercises

8. Where Sheffer Can Put His Stroke

- Everyone will have a stroke, eventually
- A shelf shuffling game
- An interesting property of truth tables
- Back to the shelf game
- What’s the connection?
- Sheffer stroke
- What we have just been doing

9. Syllogisms and Venn Diagrams

- The Syllogism
- The Greek’s theory
- The modern theory
- Formalising syllogisms
- Venn diagrams
- What to do with your Venn diagrams
- Some intuition for Venn diagrams
- Venn diagrams for validity
- Three circle Venn diagrams
- Venn and Aristotle
- Problems with Venn diagrams
- Complicated arguments can be difficult to draw
- Venn diagrams cannot do truth functions
- Exercises

10. Predicate Logic: On Natural Language

- Names and Predicates
- Formalisation of sentences
- Exercises
- Quantifiers and Variables
- An important and helpful convention
- Sentences with multiple Quantifiers
- Syllogisms in predicate logic
- Buckets of eggs
- Exercises

11. The Tableaux and Identity

- A BIG warning
- The extra rules
- Choice of variables
- Exercises
- Arguments in predicate logic
- The standard arguments
- Arguments with multiple qualifiers
- Exercises
- Identity
- Identity and more new rules
- Identity for sentences of quantity
- There is at least
- There are at most
- There are exactly
- Exercises
- Will it ever end?

A. The complete tableaux rules

B. Famous Truth Tables

C. A brief summary of Classical Logic

References

Index