Logic with Added Reasoning

Michael Gabbay teaches in the department of Philosophy at Kings College, the University of London.

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