Logical Validity

Logic is defined simply as the art of reasoning well. Logic consists of arguments, and an argument is simply a set of two or more sentences, one of which is designated as the conclusion. An argument can be Valid, which is often defined as follows: “an argument is valid if and only if the conclusion follows logically and inescapably from the truth of the premises.” This is the definition I’ve heard from people like Dr. William Lane Craig, and in reflecting on it, I think it may be a very good definition of ‘validity’. However, this semester for the purposes of my Logic class, where we are looking at sentential logic, we have defined validity in such a way that there are many arguments which qualify as valid even when and where we would never consider countenancing them seriously. Validity, for the purposes of sentential Logic, is defined precisely as:

An argument is deductively valid if and only if it is not possible for all the premises to be true and the conclusion false

This definition is so broad that there are these special cases of valid arguments which are very curious. For instance, consider the following argument:

  1. The sky is brown
  2. Therefore, it is either raining in Boston or it is not raining in Boston

Here, since “it is not possible for all the premises to be true and the conclusion false” (namely because it is not possible for the conclusion to be false) the argument presented is valid.

Consider the following:

  1. Jane is taller than Anna
  2. Anna is taller than Cynthia
  3. Cynthia is taller than Jane
  4. Therefore, the Thesaurus is heavy

In this argument, “it is not possible for the premises to be true and the conclusion false” (namely because it is not possible for the premises to be true), and therefore the argument is valid.

Consider the following argument:

  1. Bod will not go to the office if Jane does
  2. Jane will not go to the office if Bob does
  3. Therefore, it is raining in Boston and it is not raining in Boston.

Since in this argument the conclusion is logically false, it it is possible for the premises to be true and the conclusion false. The argument is invalid, and it seems to be because the conclusion is not only false, but logically false, thus always false. Therefore it would seem as though, for any argument, if the conclusion is logically false we are dealing with an invalid argument… But we would be wrong.

Consider the following:

  1. It is raining in Boston
  2. It is not raining in Boston
  3. Jane will show up to the office
  4. Therefore it is both raining in Boston and it is not raining in Boston

In this argument we have a conclusion which is logically false, and yet it is not possible for the conclusion to be false and the premises true. Therefore, since the premises are inconsistent, the argument is always automatically valid. There are two ways for premises to be inconsistent: either two or more premises contradict each other, or else at least one of the premises is itself logically false (imagine, for instance, if we had the conclusion of the previous argument act as a premise).

This definition of validity is a curious one which I will have to ameliorate myself to over the course of the semester, but I think that I will likely want, by the end of the class, to retain the definition of validity proposed by Dr. Craig.

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About tylerjourneaux

I am an aspiring Catholic theologian and philosopher, and I have a keen interest in apologetics. I am creating this blog both in order to practice and improve my writing and memory retention as I publish my thoughts, and in order to give evidence of my ability to understand and communicate thoughts on topics pertinent to Theology, Philosophy, philosophical theology, Catholic (Christian) Apologetics, philosophy of religion and textual criticism.
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One Response to Logical Validity

  1. Yousuf says:

    Hey Tyler

    The article is quite interesting. I find the special cases of logical validity rather intriguing. The broad sense of ‘logical validity’ do present us with counter-intuitive results. Some of those results are also insightful. For instance, that, say, “There is a pink unicorn,” follows logically from a contradiction or inconsistent set of premises. Mathematicians and logicians better stay away from contradictions!

    I find the broad sense of ‘logical validity’ appealing, yet I am uneasy with some special cases. A logically valid inference is a truth-preserving inference, in that the form of the inference does not allow us to reach false conclusions from true premises. Yet some results of logically valid inferences appear perplexing.

    I am more troubled by the purported sense of ‘logical validity’ by Craig. Perhaps, he was speaking casually; I suspect he would have a more rigorous explication of ‘logical validity’ elsewhere. You stated that Craig said, “An argument is [logically] valid if and only if the conclusion follows logically”. This is the first part of the definition that Craig gave for logical validity.The notion of ‘logical validity’ is explicated by the phrase ‘follows logically’. ‘Follows logically’ itself needs explication. Otherwise there is a vicious circle since ‘logical validity’ is, upon inspection, defined by ‘logical’. This cannot be done if our task is to give a clear sense of ‘logical validity’.

    The second part of the explication of ‘logical validity’ that Craig gave is as follows: “An argument is [logically] valid if and only if the conclusion [is] inescapable from the truth of the premises”. He has explicated ‘logical validity’ by using the notion of ‘inescapability’. And in doing so, there is no apparent circularity. Could you perhaps clarify the sense of the word ‘escapable’ over here? I would like to know how strongly this word is used; in other words, what does ‘escapable’ mean?

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