As part of Buridan's nominalism, he took the primary bearers of truth to be mental propositions "construed as singular acts of individual human minds" [xxxiv], expressed by written, spoken and otherwise communicated token propositions.
First, as a matter of convention, they are associated with a cognitive act (concept) of a human mind. Second, objects fall under concepts (or concepts represent objects) and in so doing, a meaningful token utterance will signify such objects. It is a important to note that token utterances receive their meaning partly as a matter of convention, since the labels we assign to mental acts is entirely a matter of convention for Buridan.
Complex concepts are made up of simple concepts by means of complexive concepts, that is, concepts which do not represent anything by themselves, but which modify the representative function of representative concepts. Non-representative concepts are called syncategormatic while representative ones are called categormatic. Some examples of syncategormatic expressions include the copula 'to be', 'and', 'not', and other sentential connectives. It is important that there are such complexive concepts so that one may maintain, for example, that a sentence and its negation do not signify for the same thing, as they unintuitively do on some medieval theories of signification. What Buridan says is that 'God is God' and 'God is not God' both signify for the same thing outside the mind (ad extra) because their categormatic concepts represent likewise, but they signify differently inside the mind (apud mentem) because of the difference in syncategormatic concepts.
Infinitizing negation is the result of applying the concept of negation to another concept, as in 'A unicorn is a non-horse', whereas sentential negation is the result of applying negation to a proposition, as in 'It is not the case that a unicorn is a horse'. The difference between them is drawn out in the examples just given: the first is false since it requires the existence of unicorns (because it's affirmative), while the second is true since it does not (because it's negative). This has application to the square of opposition. At the top corners we have the universal affirmative (Every S is P) and negative (No S is P) forms, and at the bottom corners the existential affirmative (Some S is P) and negative (Some S is not P) forms. Since 'Every unicorn is a horse' is affirmative, it is false if there are no unicorns. So it is false. Its contradictory is formed using sentential negation, which is 'It is not the case that every unicorn is a horse'. Since the affirmative is false, the contradictory is true. But it is in turn equivalent to 'Some unicorn is not a horse' which strikes us as intuitively false. Buridan chalks this up to "sloppy usage" (Klima's words). This marks an important difference to modern logic according to which universal affirmatives have no "existential import"--they do not require the existence of things which fall under the subject term. Hence, according to modern logic, 'Every unicorn is a horse' is, perhaps unintuitively, true. One has to pick their poison: concede existential import and 'Every unicorn is a horse' is true or deny it and 'Some unicorn is a non-horse' is true. One might today wish to explain away the unintuitiveness of either of these choices in terms of pragmatics rather than semantics, e.g. by holding that 'Every unicorn is a horse' is true but not usually assertible since it presupposes (a pragmatic phenomenon) the existence of unicorns.
An affirmative propositions such as 'Every man is an animal' seem not only true but necessary. Yet, its truth depends on the existence of man, which is contingent. Thus, the universal proposition cannot be necessary, 'Every man is animal' is possibly false, viz. relative to a possibility in which there is no man. This is where the notion of ampliation comes in. In the explicitly modal proposition 'Every man is necessarily an animal', the modal 'necessarily' ampliates for, not only actual men, but possible men as well. Similarly, in the tensed statement 'Every man was, is or will be an animal', the tense modifiers 'was' and 'will be' ampliate for past and future men. As such, the modal proposition is true, since what is says is that everything that is or can be a man is necessarily an animal. Indeed, the non-modal affirmative proposition may be read in the modal way, as expressing a law-like claim. Note that the usual duality between necessity and possibility must be rejected on this view, that duality being the equivalence of 'Necessarily p' with 'It is not the case that, possibly, not-p', for we have that 'Necessarily, man is an animal' is true (due to ampliation) while 'Every man is an animal' is possibly false. Note also that ampliation should not commit a nominalist like Buridan to the existence of possible men, as it does on some theories of ampliation.
Suppose that every man is an animal is necessary. If there are no tokenings of 'A man is an animal' (say Earth has gone out of existence), then 'Every that a man is an animal is true' is false, even if there are men!
An appellative term is one that signifies its significata in relation to some things, as with 'father', since someone is a father in virtue of standing in relation to their children, whereas 'man' is not appellative in this way. The distinction in modern terminology would be roughly drawn by the intrinsic/extrinsic divide, although this is debatable. E.g. being a man is an intrinsic property since one has it in virtue of the way they are and not anything else, whereas being a father is extrinsic since one has it in virtue of standing in relation to distinct things. Actually, appellation is more complicated than this, since it depends on the occurrence on a term in a statement. E.g. in 'A man's donkey is an animal', the term 'man' appellates for men as donkey owners, not men generally. So both relative and absolute terms can appellate.
One puzzle solved concerns particular existential vs non-particular existential quantification, as one might call it. When I say 'Someone has to show up for the game or it's canceled', I do not mean of any particular person that they must show up for the game or it's canceled. And yet, the existential sentence must be true, according to modern logic, just in case there is at least one person such that they must show up to the game. Similarly, when I promise you a horse, there is no particular horse such that of it, it is promised to you by me. Any horse will do. Klima says that in 'I promise (that I give you) a horse', the term 'horse' does not supposit for a "universal horse" or any particular horse, but for "any singular horse along with the appellation of the universal concept of horses" [lii] by which he must mean that 'horse' does not supposit at all, but only appellates (as Buridan's quoted passage suggests).