Lecture 9: Formal proofs in the Fitch system for Natural Deduction

Syntax and semantics

What is the meaning of an expression (philosophy of language)?

Reference:

• The things we talk about. • How it relates to other expressions.

  • ‘cat’ refers to the set of cats. • ‘cat’ is used in the sentence ‘Cats are nice’.

Use:

• Analogously, what is the meaning of ‘ A ^ B ’ ?

Semantic view:

• Symbolization key, truth table

Syntactic view:

• Relation to ¬(¬A _ ¬B), etc.

These distinctions are crucial for thinking about and dealing with formal languages

Motivation and terminology

What we frequently want is not just a ‘static’ characterization of valid inference,

• but a way of constructing a valid inference step-by-step

• by building a path from some given premises to a conclusion.

This is a proof or deduction of the conclusion from the premises

What is a proof?

Definition

• A proof is a finite sequence of sentences, such that

each element in this sequence is either a premise,

or obtained from earlier sentences by the application of an inference rule

Inference rules

Inference rules typically refer to one or two (or more) sentences

and to their main connectives.

For example, two simple rules:

1. From ‘ A^ B’ we can infer ‘ A’. Sometimes written as: A^ B

A

2. From ‘ A! B’ and ‘ A’, we can infer ‘ B’. Also written as: A! B A

B

These rules of inference can be regarded as mini-proofs:

• They correspond to valid inferences — can be justified semantically using truth tables!

• In other words, each application of an inference rule just is a valid inference.

• A succession of such mini-proofs (i. e., applications of inference rules)

constitutes a a proof or deduction

Symbol for deduction

If there is a proof of the conclusion ‘ Q ’ from the premises ‘ P1 ’, . . . , ‘ Pn ’, we write:

P1 , ... , Pn ` Q

This mirrors our use of the entailment sign ‘ |= ’, but it conceptually very different!

In the case where we have a proof of ‘ Q ’

without calling on any premises (we’ll see this later), we write:

` Q.

Warning: Make sure never to confuse ‘ ✏ ’ with ‘ ` ’ !

• Entailment is a semantic notion, while provability is a syntactic notion

Soundness of proof-systems

What we ultimately are looking for,

is a syntactic notion of deduction that corresponds to our semantic notion of entailment.

As a first step, we have, formally:

If P1 , ... , Pn ` Q , then P1 , ... , Pn ✏ Q.

If we have this, we’ll say that the proof-system is sound.

Warning: This notion is relative to our definition of ‘ ` ’ and ‘ ✏ ’ !

Warning: This is a different notion than the soundness of an argument !!!

(Mathematicians like to use ambiguous terminology to confuse non-mathematicians!)

The soundness of our proof system is crucial:

otherwise we could have proofs that are invalid

The Fitch System of Natural Deduction

Conventions for parentheses

First, some bracketing conventions:

• For ease of writing, we will relax our bracketing rules as long as we remain unambiguous.

• For example, we will accept A _ (B ^ C ) ,

this sentence is uniquely readable: it is clearly equivalent to (A _ (B ^ C )) .

• But, we will not accept A _ B ^ C (this sentence is not uniquely readable

Methodological reflections on inference rules

Ideally, we would like our inference rules to be:

• Simple: cite just a few lines as justification

• Obvious: the rules themselves should be as clear and simple as possible,

based on simple valid inferences

• Schematic: can be described just by forms (syntax) of sentences involved

• Few in number: as not to strain our memory too much

Aside: In fact, early proof systems, such as

Frege’s Begriffsschrift or Hilbert’s axiomatic systems

had just one rule of inference: Modus ponens.

But, there are trade-offs. For example:

1. Reducing the number of rules might have a negative impact on the obviousness.

2. Having fewer rules often also implies longer chains of reasoning

(this is why we will introduce ‘derived rules’ later on.)

Systems of Natural Deduction

The type of proof system we use for TFL is called Natural Deduction.

• It was developed independently in the 1930s by

Stanisław Jaśkowski (1906–1965) and Gerhard Gentzen (1909–1945).

• The particular version we use is known as the Fitch System, named after

the American logician Frederic Fitch (September 9, 1908 – September 18, 1987).

For each connective Natural Deduction systems have characteristically two kinds of rules:

• introduction (‘I’ or ‘intro’) rules

• and elimination (‘E’ or ‘elim’) rules.

As their names suggest,

an introduction rule an elimination rule introduces a new connective eliminates a connective

into a sentence. from a sentence.

E. g., ^I introduces ‘^’ to E. g., ^E eliminates ‘^’ from

(A^ B) (A^ B)

using A and B. yielding either A or B.

Our next task is to explain the introduction and elimination rules for each of our connectives.

Rules for ‘ ^ ’

Eliminating ‘ ^ ’

What can we infer, if we know P ^ Q ?

Clearly, if you know that ‘Sylvester is a dog and Snoopy is a cat’,

you also know that ‘Sylvester is a dog’ and you know that ‘Snoopy is a cat’.

Note, this exactly the kind of inference we did in Step 5 of the informal example

presented in Section 8.3.7: from ‘ (¬S ^ ¬H) ’ we inferred ‘ ¬H ’.

In fact, we have the corresponding entailments:

P ^ Q ✏ P and P ^ Q ✏ Q

which you can easily verify using truth tables.

This gives us two ^ -Elimination rules:

^ E Rules

m P ^ Q

k P ^E m

m P ^ Q

j Q ^E m

• Notice the line numbers m, k, j , the scope line, the sentences, and the justifications.

• The name of the rule is ‘ ^ E’ and it requires one line to justify its use.

• This rule is sometimes called ‘simplification’.

8.5.2 Introducing ‘ ^ ’

To introduce ‘ ^ ’ between two sentences, we need to have these sentences individually.

Again, this corresponds to an entailment:

P, Q ✏ P ^ Q

For instance, if we have ‘Sara doesn’t enjoy hiking’ and also ‘Mariusz doesn’t enjoy hiking’

we can conclude ‘Sara doesn’t enjoy hiking and Mariusz doesn’t enjoy hiking’,

or, in short: ‘Neither Sara nor Mariusz enjoys hiking’.

As a rule in the Fitch system, we get:

^ I Rule

m P

n Q

k P ^ Q ^I m, n

• Again, notice line numbers m, n, k , the scope line, the sentences, and the justification.

• The name of the rule is ‘ ^ I’ and it requires two lines to justify its use.

• This rule is sometimes simply called ‘conjunction’.