Deduction, Validity, Soundness

the next topic for discussion is
deduction validity and soundness each of these previous topics that we’ve
considered so far builds toward a better understanding and explication of these
ideas as I said in the last segment deduction validity and soundness are
absolutely central concepts in terms of understanding logic now we contrast
deduction with something called induction okay induction is the study of
arguments that deal with probability or likelihood how likely is it that a
conclusion follows so many inductive inferences deal with statistics
statistical information surveys generalizations and so on the focus of
our course is not on induction but on deduction which deals with certainty and
that concept of certainty is actually captured in this notion of validity okay
so in brief deduction deals with something we call logical certainty does
the conclusion follow necessarily from the evidence or the premises that are
offered in its support okay induction deals with questions of
probability how likely is it that a conclusion follows based on the evidence
that we have all right with that said let’s turn to this topic of validity we
used these two examples earlier to illustrate argument structure and
substitution instances I’d like to stick with these examples for the sake of
simplicity so we can clarify this notion of validity as I said these two
arguments have something in common and what’s in common here is the arguments
underlying structure okay that structure looks like this all frg all G or H and
all frh now even though we would accept that argument about whales and mammals
and reject that argument about squares and triangles they have something else
in common besides the structure or something about the structure that’s
very important and that is this property if these claims are true then the conclusion here comes a big emphasis
must the conclusion must be true also I’ll try not to knock down the board and
my excitement here okay when we interpret F G and H that is when we give
them a substitution instance and we talk about whales and mammals and
warm-blooded things we ask this question well if the premises are true is it
necessary that the conclusion follows if all whales are mammals and all mammals
are warm-blooded can we imagine a world in which it’s false that all whales are
warm-blooded clearly not we know that that conclusion has to be the case okay
that is at the essence of what we call validity all right an argument a
deductive argument is valid if and only if when the premises are true the
conclusion must be true also okay validity only applies to deduction
remember the induction is a matter of probability or likelihood alright
validity is about the structure of deductive arguments alright so this is a
deductive notion and that’s important to keep in mind so when an argument is
valid and we’re always referring to the structure when we talk about validity
it’s an issue of whether or not under interpretation these premises guarantee
the truth of this conclusion when an argument is valid it’s valid period it’s
valid for all times just like two plus two equals four doesn’t change whether
we’re counting apples or we’re counting oranges or we’re counting people okay an
argument is valid once and for all so those previous arguments we were
considering all whales are mammals all mammals are warm-blooded therefore all
whales are warm-blooded this argument is valid just as the other argument we
looked at below all squares or triangles all triangles or rectangles so all
squares or rectangles this is valid – okay
now the the usual question that comes up here is this well if an argument is
valid why do I have this judgment that this argument is a good one and that’s a
poor one and the answer to that question is we’re not just concerned with
validity when we look at deductive arguments we’re also concerned with
something called soundness validity has to do with argument structure in the
abstract if you remember back to the first segment we did on arguments non
arguments and logic I I suggested with a diagram that logic is about studying
argument structure in the abstract well validity deals with these kinds of
abstract structures all right soundness is something that takes into
consideration the particular interpretation or substitution instance
that we’re dealing with okay in the real world it is false that all squares are
triangles right so even though we’re dealing with something of the form all
frg under that particular substitution instance we have created a false
statement and that’s the reason we rejected that second argument and
accepted the first argument because those premises happen to be true all
right so to bring this together a little bit what we what we know is that
soundness can be understood in the following way an argument and again
we’re talking only about deductive arguments here an argument is sound if
and only if it is valid which means it has this structure that’s an integrity
in the structure and and I’ll use the plus sign for ant here and the premises
are true in fact in the real world okay now this is a frequent source of
confusion for people because when we look at this issue of validity we’ve
we’ve raised the question are the premiere if the premises are true does
the conclusion necessary we follow okay but notice we’re focused
on this hypothetical if if the premises are true in in any possible world that I
can imagine if the premises are true would the conclusion have to be true
also we are not concerned with whether or not the premises are true in fact
when we’re asking this question about validity you see the difference if the
moon were made of green cheese then I could eat it I’m not saying that the
moon is made of green cheese right I’m creating this kind of hypothetical
scenario well the same is true when we deal with validity we ask in the
abstract well if these were true if all squares are triangles and if all
triangles are rectangles would it follow that all squares are rectangles and the
answer is yes all right so for validity we’re dealing
with this hypothetical case of the premises being true when we’re dealing
with soundness we’re going on to ask the question well are the premises true in
the real world all right so if we can go back to our original two arguments what
we would say is this both of these arguments have the same underlying
logical structure and both of these arguments because they have this
structure are valid because we know the structure is is a valid argument
structure if the premises are true then the conclusion would have to be true
also but we we distinguish this as a good argument from this as a poor
argument because this one is sound all right let me write that here this
argument is sound which means it’s valid and when we plug in the terms whales and
mammals and warm-blooded things for F G and H we get true statements in the real
world because lo and behold whales are mammals and mammals are warm-blooded
things right so the difference between that argument and this one is that this
is what we call unsound alright it has a an argument structure that is valid but
under this substitution instance this interpretation these claims come out
false in the real world okay and that’s why we have this
intuition that well this is a good argument and this is a poor argument
even though they have the same structure okay so just to recap given this notion
of argument structure we need to be clear that validity deals with this
hypothetical relationship between premises and conclusion in other words
if the premises are true then the conclusion must be true also if it meets
that requirement it’s valid if it doesn’t then it’s invalid it’s not an
invalid argument it’s an invalid argument okay if in addition to having a
valid argument structure when we interpret that structure we have a
substitution instance and the premises come out true in fact in the real world
then we have what’s called a sound argument and we contrast that with an
unsound argument in terms of accepting a deductive argument as a good one we only
would accept those arguments that have soundness as a property okay valid and
true premises in fact

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