Andreas Flache

Manu Muñoz-Herrera

Introduction to formal logicTutorial Week 3 - Application of Theories

Block A 2012/2013

http://manumunozh.wix.com/apptheories

Summary Syllogistic Logic

D: Drug dealers in the U.S. P: People who protested in Eastern Germany S: Students in the course Application of Theories

Language

Capital letters are used for categories

Small letters are used for individuals

d: Tom the drug dealer c: Charlie the protester s: Silvia the student

5 words

To formulate a correct wff, you need only five words:

all no some is not

8 Wff’s (woofs)

There are only eight (8) forms of wffs:

all A is B no A is B some A is B some A is not B x is B x is not B x is y x is not y

More notation

a ≠ AA ≠ A’ ≠ A’’

Use of capital letters

All syllogistic wff’s have the verb is

Wff’s beginning with a word (not a letter) use two capital letters

Wff’s beginning with a letter (not a word) begin with a small

some C is D

g is C

If a wff begins with a letter, the second letter can be either capital or small

(g is D) or (g is d)

The Star Test

The Star Test (1)

A syllogism is a vertical sequence of one or more wff’s in which each letter occurs twice and the letters form a chain.

Each wff has at least one letter in common with the wff just below it (if there is one)

no P is Bsome C is B----------------some C is not P

The Star Test (2)

Distributed: An instance of a letter is distributed in a wff if it occurs just all or anywhere after no or not.

To test validity, use the star test. Star premise letters that are distributed and conclusion letters that are not distributed.

The syllogism is valid if and only if every capital letter is starred exactly once and there is exactly one star on the right hand side.

all A is Bsome C is A----------------some C is B

Valid or not?

The Star Test (3)

all A* is Bsome C is A----------------some C* is B*

A, B and C have only one *Only one star on the right hand side (B*)The argument is valid

Translation

Different ways to say: all A is B

Every (each, any) A is B Whoever is A is B A’s are B’s Those who are A are B If a person is A, then he or she is B If your are A, then you are B Only B’s are A’s None but B’s are A’s No one is A unless he or she is B No one is A without being B A thing is not A unless it is B It is false that some A is not B

Different ways to say: no A is B

A’s are not B’s Ever (each, any) A is non-B Whoever is A is not B If a person is A, then he or she is not B If you are A, then you are not B No one that is A is B There is not a single A that is B Not any A is B It is false that there is an A that is B It is false that some A is B

Different ways to say: some A is B

A’s are sometimes B’s One or more A’s are B’s There are A’s that are B’s It’s false that no A is B

Different ways to say: some A is not B

One or more A’s are not B’s There are A’s that are not B’s Not all A’s are B’s It’s false that all A is B

Contradictions: If the one is false, the other one is true

no A is B is contradictory to some A is B.

all A is B is contradictory to some A is not B

Deriving Conclusions

Steps to derive conclusions

Step 1: Translate the premises and starStep 2: Figure out the letters in the conclusions (the two letters that occur just once in the premises)Step 3: Figure out the form of the conclusionStep 4: Add the conclusion and do the star test

Form of the conclusion

If all premises are universal (all, no), then use (all A is B) or (no A is B) Otherwise, use (some A is B) or (some A is not B)

Use (x is A) or (x is not A)

Use (x is y) or (x is not y)

If both conclusion letters are capital

If just one conclusion letter is small

If both conclusion letters are small

Use no or not if there are any negative premises

Idiomatic arguments

Hence, thus, so, therefore...It must be, it can’t be...This proves (or shows) that...

These often indicate premises

These often indicate conclusions

Because, for, since, after all...I assume that, as we know...For these reasons...