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4.13

University of York

Skip to 0 minutes and 0 seconds Now we’re going to look at evaluating two arguments with forms involving conditionals.

Skip to 0 minutes and 10 seconds Consider this conditional: if I have symptoms then I am self-isolating. That’s a compound sentence involving two simple sentences.

Skip to 0 minutes and 21 seconds And Q: I am self-isolating. So our conditional here is ‘(P arrow Q)’. Now I want to consider one argument which goes like this. ‘(P arrow Q), P therefore Q’. And a second argument we’ll consider ‘(P arrow Q), Q, therefore P’.

Skip to 0 minutes and 47 seconds So this one (the first) says if I have symptoms than I am self- 00:50.500 –> 00:54.200 isolating I have symptoms therefore I’m self isolating.

Skip to 0 minutes and 54 seconds This one (the second) says: if I have symptoms then I’m self-isolating. I’m self-isolating, therefore I have symptoms. Let’s do truth tables for those two arguments. So, argument number

Skip to 1 minute and 7 seconds one first: We’ll list our simple sentences ‘P’ and ‘Q’. They can be both true. ‘P’ can be true and ‘Q’ false. [P]false [and Q]true [P]false [and Q] false. And then put our argument two premises ‘(P arrow Q)’, ‘P’ and our conclusion, ‘Q’. Now, what we’re going to do is first of all, the truth table for ‘(P arrow Q)’ and that, you’ll remember, is true, false, true, true. ‘P arrow Q’ is false where ‘P’ is true, and ‘Q’ is false, otherwise it’s true.

Skip to 1 minute and 34 seconds And for P we just copy over: true, true, false, false. Now we are looking to see whether this argument is valid. If it’s valid then whenever the premises are true, the conclusion is also true. Here, the only row where all the premises are true is the first, and on that line the conclusion is also true, so it’s valid. We don’t need to look further at the other rows in order to test for validity because we can see none of them have all the premises true. So, argument one is valid. Now let’s do argument two.

Skip to 2 minutes and 10 seconds Again our basic sentences are ‘P’ and ‘Q’ and we have the possibilities true, true, true, false, false, true, false, false, and the argument is ‘(P arrow Q), Q therefore P’. The truth table for ‘(P arrow Q)’ is true, false, true, true.

Skip to 2 minutes and 31 seconds Then for Q we carry over: true, false, true, false. And now we can see two lines on which all of the premises are true. So let’s see what the conclusion is on those two lines. On the first row, the conclusion ‘P’ is true. So the premises are true and the conclusion’s true. On the third row though, ‘P’ is false. So, there’s a way for the premises to be true and the conclusion false. So what we’ve learnt there Is that in one kind of case in which the premises are true the conclusion is also true. But in another kind of case in which the premises are true, the conclusion is false. And that’s a counter-example to the validity of this argument.

Skip to 3 minutes and 13 seconds So this argument is invalid. Now these two

Skip to 3 minutes and 18 seconds forms of argument have names: The first argument,

Skip to 3 minutes and 21 seconds the valid form, has a Latin name: modus ponens. It’s a very common form of argument. The second one is a common fallacy, It’s quite tempting to think, say, ‘That person is self isolating so they must have symptoms’, making an argument with the second form. But that’s a fallacy called affirming the consequent, because there could be another reason they’re self-isolating. Perhaps they’re at risk and can’t mix.

Another example of testing for validity

This video shows another example of testing for validity using a truth-table. This time looking at an argument involving a conditional. Let’s pick up on a couple of key points from the video …

First, we’ve seen a truth-table demonstration of the validity of the form:

• (α $$\rightarrow$$ β), α; therefore, β

This form is called modus ponens. In the video we assume (believably enough, from what we’ve seen so far) a match between the truth-conditions associated with arrow and with ‘If … then … ’ in English. If that assumption is correct, we now have a clear understanding of the validity of the form ‘If α then β; α; therefore β’ in English.

Secondly, we’ve seen a truth-table demonstration of the invalidity of the form:

(α $$\rightarrow$$ β), β; therefore, α

This form is called affirming the consequent. If we’re right about the match between arrow and ‘If … then … ’ in English, we now have a clear understanding of the invalidity of the form ‘If α then β; β; therefore α’ in English.

Logical equivalence and entailment

Before we move on to the next step, note that we can express logical equivalence in terms of mutual entailment: where two claims are such that each entails the other.

Two claims will be logically equivalent if and only if they are mutually entailing.

If they’re logically equivalent, then where one is true so is the other, so there won’t be any situation in which one, standing as premise, is true, and the other, standing as conclusion, is false; so, they’ll both entail each other.

If two claims are mutually entailing, then where one is true, so is the other (and where one is false, so must the other be, otherwise we’d have a counterexample to one of the entailments). So, they must be true in exactly the same circumstances, so they’re logically equivalent.

We’ll look at some cases of testing for mutual entailment/logical equivalence in the next two short steps. These demonstrate a couple of important logical laws.