A syllogism is a basic formula for argument, in which a conclusion is derived from two (or more) premises. Having a bad day, Stoppard's character Guildernstern says:

• Probability is a factor that operates within natural forces.
• Probability is not operating as a factor.
• We are now within un-, sub-, or supernatural forces.

Syllogisms are often based on inclusion of one case within a group, exclusion of one case from a group, or overlap between a case and two or more groups. Inclusion suggests that all the group characteristics are shared by group members. Exclusion suggests that no common characteristics are shared. Overlap suggests that some characteristics of one group may match some characteristics of another, while other characteristics are not shared. Fish rely on oxygen; people rely on oxygen; yet people are not fish: they overlap without inclusion happening. Vertebrates have backbones; fish have backbones; humans have backbones; humans and fish are vertebrates by inclusion, since both shares the characteristic of the larger class. Insects, having no bones, are not vertebrates by exclusion, since they do not share the defining characteristic.

Formal logic is a tool for assessing argument. Logical argument is obviously a complex process, which depends (like a mathematical equation) on both the structure (formula) and the content (constants and variables) to obtain a right answer. Logic, in its simplest form (the syllogism), is just addition: because this, and that, and the other are true, this is the only, or most likely, conclusion. At least two pieces of information (or premises) are necessary to add up to a third. An argument can be deemed sound if its structure is valid and its premises are true or at least acceptable.

An argument cannot be valid unless:

• its structure follows this convention of addition
• the reasoning conforms to rules of logic
• the conclusion follows necessarily from the premises
  

Its premises are:

• true if they can be demonstrated in absolute or mathematical terms.
• acceptable (and may be more open to question and dispute) if they are widely accepted
• as true and can withstand a test of reasonable doubt

Deductive reasoning is the purest and most mathematical form of logic, because in the process of adding up premises, the only possible solution is reached. Those premises are already available, and the argument typically moves from the general to the specific; the specific case is an instance of the general cases and it must be true if they are. In a classic example: All men are mortal, and Socrates is a man, so Socrates is mortal. The premises are both presumably true, the structure is valid, and the conclusion is necessary, so the argument is sound.

Another example:

• blood on the glove matched the murder victims' DNA;
• a trail led from the crime scene to a man's bedroom;
• the gloves, allowing for shrinkage, fit;
• no one else had entered the bedroom;
• the man is therefore guilty of the murder

Inductive reasoning depends on probability more than certainty, perhaps because the available premises are not as definite. In one of Conan Doyle's Sherlock Holmes stories, Holmes argues: a crime was committed; the dog, curiously, did not bark on the night of the crime; the dog was unharmed and would normally bark at strangers; therefore the criminal was not a stranger. No one could dispute the first two claims; the third is more uncertain ('normally' is not 'always') so the conclusion depends to a degree on probability. (Perhaps the dog was asleep.)

As in this case, the premises may be based on experience rather than proven fact, and are more likely to be acceptable than true. Evidence may be circumstantial, for instance. The premises may be quite disconnected or related in less obvious ways; no one else found it curious that the dog had not barked in the night, so only an intuitive leap connected the event and the non-event. In total, though, they add up to a reasonable conclusion.

Another, more complex example with some preliminary conclusions leading up to a final one:

• the man had a history of wife abuse;
• the victim was the man's ex-wife;
• the man had abused the victim in the past;
• the man had recently bought a knife;
• the knife was missing after the murder;
• the man lost, gave away or disposed of the knife
• the man had a history of jealousy;
• the victim was with another man at the time of the murder;
• the man was likely to have been jealous seeing them together;
• the man left town for no obvious reason within minutes of the murder;
• the airport is not far from the murder scene and the bedroom with the glove;
• the man ran from police when he returned;
• the man was evading police;
• the man is therefore guilty of the murders.