By Adam J. Pearson
Fig 1.1: A truth tree from sentential logic. Truth trees are a way of demonstrating that sentences of a certain form can be logically derived from other sentences.
A common misconception is that there are such things as ‘valid sentences’ and ‘invalid beliefs’ in logic. This misconception is a product of the way we use these words ‘valid’ and ‘invalid’ in daily conversation as synonyms for ‘true’ and ‘false.’ In logic, however, these words have more precise meanings.
‘Validity‘ is not a property of either beliefs or of sentences in formal logic. It is a property of certain kinds of relationships between sentences; if a conclusion can be logically derived from a given set of premises, that argument is called valid. If it cannot, that argument is called invalid.
Validity is a property of the relationship between premises and conclusions or of arguments: collections of premises and conclusions. A premise is a sentence used to derive a conclusion in an argument; a conclusion is a sentence that is derived from a premise or set of premises in an argument. Logicians do not speak of ‘valid and invalid’ sentences and beliefs, but only of ‘true’ and ‘false’ sentences.
Moreover, logicians distinguish between ‘validity‘ and ‘soundness.’ An argument can be valid, but still unsound. This is because, to be valid, all an argument needs to have is a set of sentences that logically force the conclusion; that is, a set of premises and a conclusion that can be logically derived from them. A valid argument does not have to have premises and conclusions that are actually true. This is where soundness comes in; a sound argument is an argument that is (1) logically valid and (2) composed of premises that are all true. Thus, a valid argument need not be sound, but a sound argument must be valid.
Science aims not only for valid arguments, but for sound ones. Because of the logical relationships between the parts of sentences, if your premises are all true and they are connected to a conclusion that can be validly derived from then, then that conclusion must be true. If some of your premises are true, but others are false, though, there is no guarantee that the conclusion you draw from them will be true. This is a point at which where error can creep in.
Formal logicians tend not to be very interested in beliefs as such; instead, they tend to be interested in the logical forms that underlie sentences and relationships between sentences. Indeed, logicians like to strip all of the factual content out of sentences and just present them in their logical forms. For example, instead of writing “Tom is an ant and Ricky is a bee,” they’d write, in the system of sentential logic, “A & B.”
The logical form of a sentence in sentential logic reduces it to its basic logical structures: atomic sentences (the sentences that combine with other sentences, e.g. ‘Tom is an ant’ and ‘Ricky is a bee’) and logical relation words (e.g. ‘and’ or &). Other systems of logic also include specific terms within larger sentences in their descriptions of the logical forms of sentences. For instance, predicate logic includes logical operators such as the ‘existential operator’ (which means ‘there is a something x with a given property Y’) or the ‘universal operator‘ (which means ‘ALL things x have a property Y).
I mention this little tidbit only to help explain the meaning of ‘logical form.’ As you can see, in formal logic, specific sentences are replaced with variable letters (e.g. A or B, x or y); empirical content and particular examples are cut out of the sentences in order to leave them in their most basic, general, and logical form, emptied of factual content.
Beliefs have content of some kind; if I believe “cats are aliens and monkeys are made of metal,” then I am claiming that it is true that cats are aliens and monkeys are metallic. But here’s the important point: unlike zoologists, for example, logicians wouldn’t particularly care about what I’m claiming about cats and monkeys; all they’d be interested in looking at is the logical form of my sentence: A&B (in the system of sentential logic).
The content is unimportant to the logical analysis of the sentence; it is the basic logical relations between the parts of the sentence that is of interest to logic. Thus, when we’re looking at the logical relations between sentences, it’s helpful to totally empty them of their content and just look at their logical form. And this is precisely what logicians do with tremendous analytic detail.
To explain the irrelevance of beliefs to logic a little more, let’s return to the A&B example. “A & B” is a type of sentence called a conjunction: a composite sentence that joins two other sentences with the word ‘and’ symbolized by the symbol & (ampersand).
A&B is a composite sentence because it is a sentence made up of other sentences. Every composite sentence a certain truth-table associated with it. A truth-table is a table that defines all of the possible cases in which a sentence of a given form is true and all of the possible cases in which it is false.
For example, with A & B, regardless of what ‘A’ and ‘B’ actually refer to–that is, regardless of what factual content we plug in to the logical form–the sentence A & B is true if and only if the sentence ‘A’ is true and the sentence ‘B’ is true.
The truth-table for this sentence would look like this:
A B A&B
T T T
T F F
F T F
F F F
This truth-table outlines all of the possible combinations of truth and falsity values for the component sentences A and B and the result of these combinations for the overall truth of sentence ‘A&B.’ I
f A and B are both individually true, then A&B is true. If either A or B is false, then A&B is false. If both A and B are false, then A&B is false.
This truth table holds regardless of what content we plug in to the logical form for A and B (e.g. “Donald Trump hates Muslims and all dogs are snakes,” “Some people hate the Simpsons and people who hate the Simpsons need to lighten up,” or “Tom likes cycling and Richard likes spinning in circles so many times that he creates small tornadoes”).
In all of these cases, the overall sentence is true if and only if both of its component sentences–the two statements on either side of the “and”–are true. If one or both of the component sentences is false, then the conjunction is false too. This is what logic tells us and nothing more. It does not evaluate the content of beliefs; that’s the job of other sciences.
Therefore, as you can see, logicians do not usually study factual content (e.g. particular sentences that we can input into A and B in the conjunction A&B). Instead, they study the general, formal structures of particular types of sentences and the combinations of those sentences into arguments and chain of reasoning. Thus, particular beliefs are outside of the scope of logical investigation.