Formal System

A site about formal logic, literature, philosophy and simulations. And formal systems!

Sentential Logic – 7. Strategic Assumptions — December 18, 2012

Sentential Logic – 7. Strategic Assumptions

A strategic assumption allows you to draw a conclusion based on the existence or no-existence of a single interpretation.

–          Tautology

Strategic assumption: show that the statement is not a tautology, so assume that the statement is false.

Outcomes:

a)      If you find an interpretation under this assumption: the statement is not a tautology; it is either a contradiction or a contingent statement.

b)      If you don’t find an interpretation under this assumption, it means you have disproved (proved against) the assumption, so the statement is a tautology.

 

–          Contradiction

Strategic assumption: show that the statement is not a contradiction, so assume the statement is true.

Outcomes:

a)      If you find an interpretation under this assumption: the statement is not a contradiction; it is either a tautology or a contingent statement.

b)      If you don’t find an interpretation under this assumption, it means you have disproved (proved against) the assumption, so the statement is a contradiction.

 

–          Contingent statement

Use the two assumption tests for tautology and contradiction. If you find an interpretation under the two assumptions, it means the statement is not a tautology nor is it a contradiction, so the statement must be a contingent statement.

 

–          Semantic (in)equivalence

Strategic assumption: connect the statements with a <>operator and assume that the new statement is false.

Outcomes:

a)      If you find an interpretation under this assumption: the statements are semantically inequivalent.

b)      If you don’t find an interpretation under this assumption, it means you have disproved (proved against) the assumption, so the statements are semantically equivalent.

 

–          (In)consistency

 

Strategic assumption: try to show that the set of statements is consistent so assume all the statements are true.

 

Outcomes:

 

a)      If you find an interpretation under this assumption: the set of statements is consistent.

b)      If you don’t find an interpretation under this assumption, the set of statements is inconsistent.

 

 

–          (In)validity

 

Strategic assumption: try to show that the argument is invalid so assume all the premises are true and the conclusion is false.

 

Outcomes:

 

c)       If you find an interpretation under this assumption, the argument is invalid.

d)      If you don’t find an interpretation under this assumption, the argument is valid.

Note: I will update this post in the near future with some examples.

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Sentential Logic – 6. Logical Conditions —

Sentential Logic – 6. Logical Conditions

About truth tables:

Number of rows = Constant1 (2) x Constant2 (2)…

Example: 4 rows = Constant1 (2) x Constant2 (2) x Constant3 (2) x Constant 4 (2)

Note: every row of a truth table is an interpretation.

Reminder: an interpretation is a specific set of truth values for a set of constants.

Looking at single statements

Working with truth tables: The three categories of SL statements

–          Tautologies: a statement that is always true under every interpretation. Example: Pv-P. Regardless of the truth value of P, at least a part of the statement is true, so the statement will always be true.

–          Contradictions: a statement that is always false under every interpretation. Example: P^-P. Regardless of the truth value of P, at least a part of the statement is false, so the statement will always be false.

–          Contingent statements: a statement that is true under at least one interpretation and false under at least one interpretation. Example: P>Q. The statement is true when P is true and Q is true but it is false when P is true and Q is false.

Summary:

If a statement evaluates as:

a)      False in every row, it is a contradiction.

b)      True in every row, it is a tautology.

c)       True in at least one row and false in at least one row, it is a contingent statement.

Looking at pairs of statements

Semantic equivalence: it refers to when two statements have the same truth value under all interpretations.

Looking at two or more statements

Reminder: argument => premise(statement1,statement2)>conclusion(statement3)

Consistency: when a set of statements is consistent, at least one interpretation makes them all true.

Inconsistency: when a set of statements is inconsistent, no interpretation makes them all true.

Validity: when an argument is valid, no interpretation exists under which all of the premises are true and the conclusion is false.

Invalidity: when an argument is invalid, at least one interpretation exists under which all of the premises are true and the conclusion is false.

Summary of logical conditions for Truth Table tests

For 1 sentence

Tautology: statement is true on every row.

Contradiction: statement is false on every row.

Contingent statement: statement is true on at least one row and false on at least one row.

For 2 sentences

Semantic equivalence: both statements have the same truth value on every row.

Semantic inequivalence: both statements have different truth values in at least one row.

For 2 sentences or more

Consistency: all statements are true at least on one row.

Inconsistency: all statements are not true on any row.

Validity: in every row where all the premises are true, the conclusion is also true.

Invalidity: all the premises are true and the conclusion is false in at least one row.

Logical Tapestry: Network of logical conditions

Tautology and contradiction

A tautology can be turned into a contradiction (or vice versa) by negating the whole statement with a –operator.

 

Tautology and semantic equivalence

Connecting any two semantically equivalent statements with a <>operator turns the resulting statement into a tautology.

Note: connecting any two semantically inequivalent statements with a <>operator turns the resulting statement into either a contradiction or a contingent statement.

About learning, adaptation, machines, evolution and cybernetics — May 28, 2012

About learning, adaptation, machines, evolution and cybernetics

I was reading one of the supplementary chapters of Wiener’s “Cybernetics: or Control and Communication in the Animal and the Machine” and I came across a very interesting chapter about learning. But first I will put things in context.

Norbert Wiener was a mathematician and the founder of the field of cybernetics. Cybernetics deals with 3 tasks:

Task 1- How (efficiently) systems process information

Task 2- How (efficiently) systems react to information

Task 3- How (efficiently) systems change to improve Task 1 and Task 2

The 3 tasks can be summarised under the notion of “feedback”, which is a closed chain of cause and effect where a system ‘feeds back’ on itself. ( I will expand on this on another article).

The chapter of the book was “On learning and self-reproducing machines”. According to Wiener, the two characteristics of living systems are the power to learn and the power to self-reproduce.

Wiener states that reproduction (or phylogenetic learning) and learning (or onto-genetic learning) can be understood as similar processes by looking at both as adaptation mechanisms. Phylogenetic learning can be seen as the adjustment of a species in order to adapt to the requirements of an environment, while onto-genetic learning is the adjustment of a member of a species in order to adapt to the requirements of an environment. Phylogenetic learning spans through the lifespans of the individuals of a species, as opposite to onto-genetic learning that occurs during an individual’s lifetime.

Birds, for example, have a greater capacity for phylogenetic learning that for onto-genetic learning. It translates into birds executing seemingly complex behaviours such as flying or nest-building without any significant amount of instruction from the mother.

On the other end, mammals and especially, humans, are so prone to onto-genetic learning over phylogenetic learning to the extent that phylogenetic learning itself is devoted to ensure good onto-genetic learning capabilities. It translates into a new level of flexibility that not only allows smooth adaptation to the environment but also allows to adapt the environment to the individual as well. Regarding artificial intelligence and learning,  this means that the more flexible the system can get the better the results, thus, I consider original approaches to achieve a flexible structure in a system crucial for the sake of significant progress in the field.