Formalization in Propositional Logic

Question 1

What are connectives?

Answer 1

Expressions that can be used to combine or modify English sentences to form a sentence

Question 2

What is the issue with because

Answer 2

If A and B are both true in A because B this could yield a true or false result

This is why because differs from and

Connectives like because are not truth functional

Question 3

What is truth functionality?

Answer 3

A connective is truth-functional iff the truth-value of the compound sentence can not be changed by replacing a direct sub sentence with another sentence having the same truth-value

Question 4

What is the issue with ‘if…then’

Answer 4

Is usually translated as the arrow but some of it’s occurrences are not truth functional

It cannot deal with counter factuals, those that describe what would have happened under circumstances that are not actual
‘If Giovanni hadn’t gone to England, he would not have caught a cold in Cambridge.’

It’s questionable to use it for indicative conditionals
‘If Jones gets to the airport an hour late, his plane will wait for him’

This is false even if Jones gets there on time

Question 5

The definition of truth functionality also applies to unary connectives

Answer 5

A unary connective is truth functional iff the truth-value of the sentence with the connective cannot he changed by replacing the direct sub sentence with a sentence with the same truth value

It is necessary that is a unwary connective that is not truth functional. If A is false then it is necessary that A is false but if A is true, it is necessary that A could be true or false

Question 6

Formulations of conjunction

Answer 6

But
And
Although

Question 7

Formulations of disjunction?

Answer 7

Or

Unless

Question 8

Formulations of negation

Answer 8

It is not the case that
Not
None
Never

Question 9

Formulations of arrow

Answer 9

If…then
Provided that…,…
Only if

Question 10

Formulations of double arrow

Answer 10

If and only if
Precisely if
Exactly if

Question 11

Ambiguity

Answer 11

A sentence like ‘Brown is in Barcelona and Jones owns a Ford or Smith owns a Form.’ Is ambiguous

It is a scope ambiguity

Whilst sentences of English are often ambiguous in their structure, sentences of L1 are never structurally ambiguous

Question 12

What is the scope of a connective in L1

Answer 12

The scope of an occurrence of a connective in A sentence # of L1 is the occurrence of the smallest sub sentence of # that contains this occurrence of the connective

Question 13

An English sentence is a tautology iff

Answer 13

It’s formalisation in propositional logic is logically true (that is, iff it is a tautology)

Instead of saying that a sentence is a tautology one can also describe it as propositionally valid

Question 14

An English sentence is a propositional contradiction iff

Answer 14

It’s formalisation in propositional logic is a contradiction

Question 15

A set of English sentences is propositionally consistent if

Answer 15

the set of all their formalisations in propositional logic is semantically consistent

Question 16

In the formal language L1 of propositional logic, the logically true sentences are exactly the tautologies. In contrast, in English there are…

Answer 16

logically true sentences that are not tautologies

The sentence ‘All logicians are logicians’ is logically true but is not a tautology, because the formalisation in propositional logic is a single sentence letter. A single sentence letter is never logically true, that is, it is never a tautology.

Similarly there may be a English sentence that is a contradiction without being a propositional contradiction ‘there is an oak that is not an oak.’

Question 17

An argument in English is propositionally valid iff it’s

Answer 17

formalisation in L1 is valid

Question 18

What is ex falso quodlibet

Answer 18

With contradictory premisses anything follows. The argument is propositionally valid