translation

PHIL 220. Introduction to Logic

Gabriel Uzquiano

translation

We know how to use truth tables to assess the validity of arguments written in the language of propositional logic.

but arguments come to us in English

To assess the validity of a natural language argument, we first translate its premises and conclusion into the language of propositional logic.

Only then can we use a truth table on it.

Three steps, applied to each sentence in turn:

  1. paraphrase the sentence to reveal how it is built from simpler sentences by connectives: ‘not’, ‘and’, ‘or’, ‘if … then …’
  2. replace each connective with its formal counterpart
  3. replace the simplest sentences that remain with propositional variables

example 1: the British are coming

the argument

The British are coming by sea. For if they are coming by land, then Paul Revere will light two lamps. And if the British are not coming by land, then they are coming by sea. However, Paul Revere will not light two lamps.

in premise-conclusion form

  1. If the British are coming by land, Paul Revere will light two lamps.
  2. If the British are not coming by land, then they are coming by sea.
  3. Paul Revere will not light two lamps.
  1. The British are coming by sea.

step 1. paraphrase

We use parentheses to reveal how each sentence is built from simpler ones.

  1. (If the British are coming by land, then Paul Revere will light two lamps).
  2. (If the British are not coming by land, then they are coming by sea.)
  3. Paul Revere will not light two lamps.
  1. The British are coming by sea.

step 2. propositional connectives

We replace each connective with its formal counterpart:

  1. (The British are coming by land \(\to\) Paul Revere will light two lamps).
  2. (\(\neg\) The British are coming by land \(\to\) they are coming by sea.)
  3. \(\neg\) Paul Revere will light two lamps.
  1. The British are coming by sea.

step 3. propositional variables

We now specify a translation key:

\(p\): The British are coming by land.

\(q\): Paul Revere will light two lamps.

\(r\): The British are coming by sea.

  1. \((p \to q)\)
  2. \((\neg p \to r)\)
  3. \(\neg q\)
  1. \(r\)

the question of validity

We are now in a position to use a truth table to check for validity:

The argument is valid.

summary

One last pass over the natural language argument. This time, do all three steps yourself in a single interactive worksheet. Select your own translation key and translate the argument:

Open the full exercise in the translation app

example 2: interest rates

the argument

Interest rates will rise or they will stay the same. If interest rates rise, then inflation will not rise. But if they stay the same, then inflation will rise. Inflation will not rise. Therefore, interest rates will rise.

in premise-conclusion form

  1. Interest rates will rise or they will stay the same.
  2. If interest rates rise, then inflation will not rise.
  3. If interest rates stay the same, then inflation will rise.
  4. Inflation will not rise.
  1. Interest rates will rise.

translation key

We specify a translation key:

\(p\): Interest rates will rise.

\(q\): Interest rates will stay the same.

\(r\): Inflation will rise.

premise 1

\(p\): Interest rates will rise.

\(q\): Interest rates will stay the same.

\(r\): Inflation will rise.

premise 2

\(p\): Interest rates will rise.

\(q\): Interest rates will stay the same.

\(r\): Inflation will rise.

premise 3

\(p\): Interest rates will rise.

\(q\): Interest rates will stay the same.

\(r\): Inflation will rise.

premise 4

\(p\): Interest rates will rise.

\(q\): Interest rates will stay the same.

\(r\): Inflation will rise.

conclusion

\(p\): Interest rates will rise.

\(q\): Interest rates will stay the same.

\(r\): Inflation will rise.

the argument translated

\(p\): Interest rates will rise.

\(q\): Interest rates will stay the same.

\(r\): Inflation will rise.

  1. \(p \vee q\)
  2. \(p \to \neg r\)
  3. \(q \to r\)
  4. \(\neg r\)
  1. \(p\)

the question of validity

issues with translation

Some issues require special discussion.

structural ambiguity

English is fraught with ambiguity. Sometimes the ambiguity is lexical, e.g., ‘I will wait for you near the bank’, but sometimes it is a matter of syntactic structure:

The British are coming and Paul Revere will light one lamp or he will light two lamps.

There are at least two potential interpretations:

  1. ((The British are coming and Paul Revere will light one lamp) or Paul Revere will light two lamps)

  2. (The British are coming and (Paul Revere will light one lamp or Paul Revere will light two lamps))

\(p\): The British are coming.

\(q\): Paul Revere lights one lamp.

\(r\): Paul Revere lights two lamps.

  1. \(((p \wedge q) \vee r)\)

  2. \((p \wedge (q \vee r))\)

they are not equivalent …

neither … nor …

Paul Revere will light neither one lamp nor two lamps.

There are at least two interpretations with identical truth conditions:

  1. (Paul Revere will not light one lamp and Paul Revere will not light two lamps)

  2. It is not the case that (Paul Revere will light one or two lamps)

\(p\): Paul Revere will light one lamp.

\(q\): Paul Revere will light two lamps.

  1. \((\neg p \wedge \neg q)\)

  2. \(\neg (p \vee q)\)

they are equivalent …

… only if … versus … if …

… only if …

Paul Revere will light two lamps only if the British come by sea.

If Paul Revere lights two lamps, then the British must have come by sea.

\(p\): Paul Revere lights two lamps.

\(q\): The British come by sea.

\[ p \to q \] Coming by sea is necessary for lighting two lamps. Compare:

I will win the lottery only if I buy a ticket.

… if …

Paul Revere will light two lamps if the British come by sea.

If the British come by sea, then Paul Revere lights two lamps.

\(p\): Paul Revere lights two lamps.

\(q\): The British come by sea.

\[ q \to p \] Coming by sea is sufficient for lighting two lamps. Compare:

I will win the lottery if I buy a ticket.

… if, and only if, …

Paul Revere will light two lamps if, and only if, the British are coming by sea.

A conjunction of the two conditionals from the previous slide:

(Paul Revere will light two lamps if the British are coming by sea) and (Paul Revere will light two lamps only if the British are coming by sea)

\(p\): Paul Revere lights two lamps.

\(q\): The British come by sea.

\[ (q \to p) \wedge (p \to q) \]

We sometimes use \(\leftrightarrow\) to abbreviate formulas of this form: \[ (\varphi \leftrightarrow \psi) := (\varphi \to \psi) \wedge (\psi \to \varphi) \]

unless

Paul Revere will light one lamp unless the British are coming by sea.

Two equivalent paraphrases:

If the British are not coming by sea, then Paul Revere lights one lamp.

\(p\): Paul Revere lights one lamp.

\(q\): The British come by sea.

\[ \neg q \to p \]

Paul Revere lights one lamp or the British are coming by sea.

\(p\): Paul Revere lights one lamp.

\(q\): The British come by sea.

\[ p \vee q \]

they are equivalent …