**Gerard Beirne** is an Irish writer who has lived in Canada for over ten years. He has recently been appointed Writer-in-Residence at the University of New Brunswick (UNB) for the forthcoming academic year.

His collection of poems *Digging My Own Grave*** **was published by Dedalus Press, Ireland. An earlier version won Second Place in the Patrick Kavanagh Award. His novel *The Eskimo in the Net* (Marion Boyars Publishers, London) was short listed for The Kerry Group Irish Fiction Award 2004. He is a past recipient of the Sunday Tribune/Hennessy *New Irish Writer of the Year *award. A CD of his poetry *if it’s words you’re after…* was released in 2005. His story *Sightings of Bono *was adapted into a short film (Parallel Productions) featuring Bono.

*Extract from* Introduction to Mathematical Logic

** **

**7. Biconditional Statements**

for every conditional statement

there corresponds a converse

rehearse:

*for every converse*

*there corresponds *

*a conditional statement*

(but worse

the converse is not

necessarily true

an unresponsive correspondence

from me to you)

(less despondent) the biconditional statement

(the conjunction of a conditional

and its converse)

for this

read *equivalence*

(the argument not always reversible)

hint:

henceforth enhance your expression

by reducing your statements

to those of simpler equivalence

(call this mathematical manipulation

a stipulation of natural prosody)

Call this

authorial intrusion:

if a conditional has a conjunction

in either the hypothesis or conclusion

it is possible to

form

* partial converses*

(Faraday’s method for discovering

electromagnetic induction -

knowing

if a current flows in a wire

and the wire forms

a closed circuit

then a magnetic field is created

translated

if a magnetic field is created

around a wire

and the wire forms

a closed circuit

then a current flows)

generating your own electricity

be aware of both of those

**8. Necessary and Sufficient Conditions**

“Necessary and sufficient

conditions”

a frequently occurring phrase

in mathematics

(the power of repetition)

conditional and biconditional

sentences

enabling us to symbolise

their meaning

for example:

“if two angles are right-angled

then they are equal”

the two angles being right-angled

is *sufficient* for them to be equal

but not *necessary* for equal angles

since they may be equal

but not right angled

( a right tangle -

but necessary and sufficient

for our understanding

our comprehension

of the tension

implicit in our words

the underlying intent

struggling to be heard)

thus *p *Þ* q*

can be read

“*p* is sufficient for *q*”

but in order to

state

“If *p* is not true

then *q* is not true”

( ~*p *Þ* *~*q*)

* p* must be

a necessary condition of *q*

(for that read

indispensable

one sentence dependent on the other

what follows made comprehensible

only by what came before

words sufficiently

conveying the truth

afterwards

necessitating lies

in other words

a poem

not fully realised)

Onto

the biconditional *p* Û *q*

meaning

* p *Þ* q* and *q *Þ* p*

all things as they are meant

to be

for this read

“*p* is sufficient and necessary for *q*”

or “*q* is necessary and sufficient for *p*”

or

“*p* if and only if *q*”

*p* and *q* logically equivalent

our minds spent

now

what to do?

remember

*p* and *q*

being variables

lines, sentences

of your construction

insert your own intentions

the sufficient and necessary essentials

with little or no instruction

words, phrases

interchangeable

logical and tangible

contrapositives searching out

our own ambivalence