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Full Description
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.
Contents
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
