Classnotes
8/24/00
There are six main logical properties and relations, defined for us on page 24. They are DEDUCTIVE VALIDITY, LOGICAL CONSISTENCY, LOGICAL EQUIVALENCE, LOGICAL TRUTH, LOGICAL FALSEHOOD AND LOGICAL INDETERMINACY.
The definitions of these terms indicate that each expresses a general truth relation or property of sentences or sets of sentences. Each definition appeals to the concept of possibility, but this is unacceptably vague. The formal study of logic attempts to provide a mechanism for evaluating sentences and sets of sentences for the logical properties and relations by giving strict sense to the concept of logical possibility.
Formal systems:
A formal system consists necessarily in two components, and a third may be added.
(1) A formal language consisting of
(b) grammar (a set of rules for well-formed expressions.)
(2) Transformation rules (a set of rules for manipulating well-formed expressions of the language.)
(3) A semantic scheme (a protocol for interpreting or assigning meanings to the expressions of the language.)
Logic and formal systems:
Logicians employ formal systems to provide models for evaluating the logical properties and relations. Formal logical systems are intended to represent only those features of natural language that are logically relevant (i.e. relevant to the evaluation of the six properties and relations.) A logical system is any formal system that can serve this purpose.
In Philosophy 160 (Formal logic), you were introduced to two such logical systems: the sentential logic and the predicate logic. The components of the two systems were informally introduced and you learned the techniques involved in using them for logical evaluation.
Sentential logic:
Sentential logic is a system that takes sentences as the basic logical units, and is concerned with the consequences of the way sentences are compounded together by certain kinds of connective expression.
The language SL:
The formal language of the system of sentential logic is called SL. This is not an abbreviation for ‘sentential logic’. It is a name for the first component of the system. The vocabulary of SL includes symbols that are intended to model the logical role of individual sentences and symbols that are intended to model a special class of sentential connectives.
A SENTENTIAL CONNECTIVE is any expression used to form compound sentences from other sentences.
A sentential connective is used TRUTH-FUNCTIONALLY if and only if the compound sentence it generates has a truth-value that is dependent on (is a function of) the truth-values of the component sentence(s) from which the compound is generated.
SL employs capital letters to model sentences.
SL employs five "truth-functional connectives". These model the logical role of many natural language sentential connectives.
~ (called ‘tilde’) models the logical role of every natural language expression with the same function as ‘NOT’.
& (called ‘ampersand’) models the logical role of every natural language expression with the same function as ‘AND’.
Ú (called ‘wedge’) models the logical role of every natural language expression with the same function as ‘OR’.
É (called ‘horseshoe’) models the logical role of every natural language expression with the same function as ‘IF…THEN…’.
º (called ‘triple bar’) models the logical role of every natural language expression with the same function as ‘IF AND ONLY IF’.
Mathematical logicians and computer scientists are acquainted with a
slightly different set of symbols for these connectives:
Ø Ù Ú ® «
08/29/00
Formal syntax of SL.
The vocabulary and rules for well-formed sentences of SL are found on pp. 62-3.
We are employing metalanguage (a mixture of English and some special symbols) to talk about SL. We will refer to expressions of SL by the use of inverted commas (single quotation marks.) E.g. we will use this expression
‘(A & B)’
to make reference to (i.e. talk about) the SL expression
(A & B)
We also need metalinguistic variables. We will use
P Q R
as variables for SL sentences.
A protocol for showing that a given expression is an SL sentence:
STEP 1: Establish that all symbols are SL symbols.
STEP 2: Identify all atomic sentences. They are SL sentences by rule 1.
STEP 3: Identify all sentential components of one connective. State by what rules and which previous steps they are to be regarded as SL sentences.
STEP 4: Continue this procedure step by step for larger components until the whole sentence in question is reached.
Semantic scheme for SL:
The semantic scheme is given by stipulating two components:
1. The characteristic truth tables for the SL connectives (pp. 67-8).
2. The concept of a Truth Value Assignment (TVA, p.68).
These components together give us a basis for assigning truth values to all sentences of SL.
A truth table is a table exhibiting the truth value of an SL sentence on any given TVA.
09/05/00
The TF properties and relations:
The TF properties and relations are formal counterparts of the informal logical properties and relations defined in the glossary on page 24.
They are tabulated in the glossary on pp. 100-1.
A new logical concept - entailment:
A set of SL sentences can be said to TRUTH-FUNCTIONALLY ENTAIL an SL sentence. This means that there is no TVA on which the members of the former are all true while the latter is false.
Abbreviated like this:
G |-- P
On page 100 there are twelve elementary metatheory propositions whose
proofs depend on the definitions of the TF properties and relations and
on the details of the semantic scheme. They are excellent exercises.
09/07/00
Review of SD.
09/12/00
The SD properties (see glossary page 210).
Exercise set 15 on page 200 contains three important metatheory propositions
whose proofs depend on the definitions of the SD and TF properties.
Again, they are excellent exercises.
09/14/00
FIRST TEST
09/19/00
The main object of this part of the course:
Show that the SD-properties and relations match up with the TF-properties and relations. I.e. show that what we are deriving in SD is appropriate for our intentions given the scheme of interpretation we have chosen.
We will prove that every SD derivation corresponds to a TF entailment and vice-versa. We can put this as follows:
A method of metatheoretical proof:
Our metatheory proofs will be somewhat informal. We will employ a method of proof that is based upon a principle related to those of mathematical induction.
The weak principle of mathematical induction: If 1 has a given property F, and for any positive integer n, if n has F, then n + 1 has F, then all positive integers have F.
An argument schema based on this principle
1 has F
If n has F, then n +
1 has F
Every positive integer has
F
The strong principle of mathematical induction: If when any positive integer up to and including n has some property F, it follows that n + 1 has F, then all positive integers have F.
An argument schema based on this principle
If every m less than or
equal to n has F, then n + 1 has F
Every positive integer has
F
A hybrid principle for our metatheory proofs:
The principle of proof by mathematical induction: If the first
member of some sequence has a property F, and, for any n, if every member
up to and including the nth member has F then the n+1th
member has F, then every member of the sequence has F.
Steps in a proof based on the principle of mathematical induction:
To show that every member of some set of items {I1…In} has a given property F.
STEP ONE: order the items (singly or by groups) according to some relevant feature.
[Step one is called the ordering.]
STEP TWO: show that the first item (or the items in the first group) has F.
[Step two is called the basis step and is the proof of the basis clause.]
STEP THREE: show that if every item up to and including the kth item (or the items in the kth group) has F, then the k+1th item (items in the k+1th group) has F.
[Step three is called the inductive step. It is proved by a form of conditional proof. Assume the antecedent of what is to be proved – this is called the inductive hypothesis – and derive the consequent.]
STEP FOUR: every item has the property.
[Step four is the conclusion.]
09/21/00
Some metatheorems of SL:
(See p.222 for what this means.)
M6.2 SD is sound for sentential logic.
M6.3 SD is complete for sentential logic.
(See p.229 for what these mean.)
M6.4 SL is truth-functionally compact. (See p.245.)
One other thing we’ll want to prove.
M6.5* SD is consistent relative to negation.
This means that there is no SL sentence P such that Ø |-- P and Ø |-- ~P.
[* You won’t find this one in the textbook.]
M6.5 is a purely syntactic result.
M6.2 and M6.3 are results about the relationship between
syntactic and semantic concepts.