Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.
Sentential logic (SL) uses sentences as the building blocks for its work in proving logical arguments, which can have far-reaching implications for truth. Each of the sentences in SL can only be true of false, but not both at the same time and not nothing either (28). Throughout SL, we want to link sentences together to create arguments which can be used to find new conclusions about given premises. To make such links, we use sentential connectives such as “and,” “or,” “although,” “unless,” “before,” and “if and only if” (Bergmann 29).
When we link two or more simple, or singular, sentences with a sentential connective, we have created a compound sentence (29). Such compounds are powerful because they rely on the simple sentences within them to give them their total truth value that can exceed the parts that make it. When a sentential connective is performing in such a way that the compound’s truth value is based off the truth values of the simple sentences that make up that compound, we call it truth functional (29).
In SL, we use capital letters to abbreviate sentences in an effort to look at the compound structure and derive from it some conclusions which we can ultilize. The letters allow us to present that information in an easy-to-manage manner. Rewriting the same sentence over and over is tiresome, so just abbreviate it with a letter that makes sense. When we have a sentence that is abbreviated, we refer to the letter that abbreviates it as an atomic sentence. Any combination of letters is referred to as a molecular sentence (30).
So how do we go about converting sentences so we can work with them in SL? Generally, we need to paraphrase the sentence into its appropriate form. If it has a sentential connective, we need to split it up into simple sentences, and then use an appropriate letter relating to the information in the sentence to make it atomic. Take note that we cannot always convey all the information we desire in the paraphrase. That is okay, just make the closest approximation you can (32-3). For example, “All my dogs are brown” could be simplified with a D for dogs or a B for brown. Usually the other sentences we are paraphrasing will push us to choose certain letters over others. If the other sentence had been, “None of my dogs are puppies,” then we would definitely use B for the first sentence and P for the second, the P representing puppies.
It is nice to know what a paraphrase will look like once in molecular form. Anything that can be replaced with, “Neither A nor B,” which is the same as, “Both it is not the case that A and it is not the case that B,” will take on the form ~A ^ ~B in SL, with the tilde meaning a negation of the sentence and the carat meaning, “And.” It is interesting to note that “neither A nor B” can also be paraphrased as “it is not the case that either A or B,” which in SL is ~ (A v B), with the wedge meaning, “Or.” So, ~A ^ ~B is truth equivalent to ~ (A v B) (39). This is one of De Morgan’s Laws. We will frequently encounter more equivalent statements in SL.
Another example of what a paraphrase will look like in SL is “not both A and B.” This could be paraphrased as “it is not the case that both A and B,” which in SL is ~ (A ^ B). “Not both A and B” could also be paraphrased as “either it is not the case that A or it is not the case that B,” which in SL is ~A v ~B. Hence, ~ (A ^ B) is equivalent to ~A v ~B (39-40). This is the other De Morgan’s Law.
In general, when you paraphrase: (51)
- Anything that you find to be already a simple sentence can be put directly into SL
- Use those connectives when necessary to join atomic sentences
- Ensure that clarity is preserved! Parenthesis can be our friend
- If dealing with an argument, be sure to have premises separated from conclusion
- If a sentence repeats a main idea, do not include it twice
When we work with an argument, we have multiple sentences that may have different levels of reliability. If sentence A guarantees the truth of sentence B but B does not ensure A, then we say that B is the weaker sentence. If instead B had led to truth of A but A did not guarantee truth of B then B would be the stronger sentence. What we aim for are equivalent sentences, where both lead to the truth of the other (62).
Finally, it is important to make the distinction between an object language and a meta-language. The object language is what we are talking about, but the meta-language is used to do the actual talking. This distinction is necessary because we will be aiming to make generalizations of sentences and need to have a broader plane to work with. Those broader statements in the meta-language are often bolded as a meta-variable, while sentences in the object language are put in quotations. (67-9).
Bergmann, Merrie, James Moor, and Jack Nelson. The Logic Book. New York: McGraw-Hill Higher Education, 2003. Print. 28-30, 32, 33, 39, 40, 51, 62, 67-9.
© 2013 Leonard Kelley