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